Operator Method for Solution of PDEs Based on Their Symmetries

Operator Theory: Advances andApplicationsVol. 157

Editor:I. Gohberg

H. G. Kaper (Argonne)S. T. Kuroda (Tokyo)P. Lancaster (Calgary)L. E. Lerer (Haifa)B. Mityagin (Columbus)V. V. Peller (Manhattan, Kansas)L. Rodman (Williamsburg)J. Rovnyak (Charlottesville)D. E. Sarason (Berkeley)I. M. Spitkovsky (Williamsburg)S. Treil (Providence)H. Upmeier (Marburg)S. M. Verduyn Lunel (Leiden)D. Voiculescu (Berkeley)H. Widom (Santa Cruz)D. Xia (Nashville)D. Yafaev (Rennes)

Honorary and AdvisoryEditorial Board:C. Foias (Bloomington)P. R. Halmos (Santa Clara)T. Kailath (Stanford)P. D. Lax (New York)M. S. Livsic (Beer Sheva)

Editorial Office:School of MathematicalSciencesTel Aviv UniversityRamat Aviv, Israel

Editorial Board:D. Alpay (Beer-Sheva)J. Arazy (Haifa)A. Atzmon (Tel Aviv)J. A. Ball (Blacksburg)A. Ben-Artzi (Tel Aviv)H. Bercovici (Bloomington)A. Böttcher (Chemnitz)K. Clancey (Athens, USA)L. A. Coburn (Buffalo)K. R. Davidson (Waterloo, Ontario)R. G. Douglas (College Station)A. Dijksma (Groningen)H. Dym (Rehovot)P. A. Fuhrmann (Beer Sheva)B. Gramsch (Mainz)G. Heinig (Chemnitz)J. A. Helton (La Jolla)M. A. Kaashoek (Amsterdam)

Birkhäuser VerlagBasel . Boston . Berlin

Operator Theory, Systems Theory

and Scattering Theory:

Multidimensional Generalizations

Daniel AlpayVictor VinnikovEditors

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Library of Congress, Washington D.C., USA

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in the Internet at <http://dnb.ddb.de>.

ISBN 3-7643-7212-5 Birkhäuser Verlag, Basel – Boston – Berlin

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Editors:

Daniel Alpay

Victor Vinnikov

Department of Mathematics

Ben-Gurion University of the Negev

P.O. Box 653

Beer Sheva 84105

Israel

e-mail: dany@math.bgu.ac.il

vinnikov@math.bgu.ac.il

2000 Mathematics Subject Classification 47A13, 47A40, 93B28

Contents

Editorial Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

J. Ball and V. VinnikovFunctional Models for Representations of the Cuntz Algebra . . . . . . . . . 1

T. Banks, T. Constantinescu and J.L. JohnsonRelations on Non-commutative Variables andAssociated Orthogonal Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

M. BessmertnyıFunctions of Several Variables in the Theory of FiniteLinear Structures. Part I: Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

S. Eidelman and Y. KrasnovOperator Methods for Solutions of PDE’sBased on Their Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

D.S. Kalyuzhnyı-VerbovetzkiıOn the Bessmertnyı Class of Homogeneous PositiveHolomorphic Functions on a Product of Matrix Halfplanes . . . . . . . . . . . 139

V. Katsnelson and D. VolokRational Solutions of the Schlesinger System andIsoprincipal Deformations of Rational Matrix Functions II . . . . . . . . . . . 165

M.E. Luna–Elizarraras and M. ShapiroPreservation of the Norms of Linear Operators Actingon some Quaternionic Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

P. Muhly and B. SolelHardy Algebras Associated with W ∗-correspondences(Point Evaluation and Schur Class Functions) . . . . . . . . . . . . . . . . . . . . . . . . 221

M. PutinarNotes on Generalized Lemniscates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

M. Reurings and L. RodmanOne-sided Tangential Interpolation for Hilbert–SchmidtOperator Functions with Symmetries on the Bidisk . . . . . . . . . . . . . . . . . . 267

F.H. SzafraniecFavard’s Theorem Modulo an Ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Editorial Introduction

Daniel Alpay and Victor Vinnikov

La seduction de certains problemes vient de leur

defaut de rigueur, comme des opinions discor-

dantes qu’ils suscitent: autant de difficultes dont

s’entiche l’amateur d’Insoluble.

(Cioran, La tentation d’exister, [29, p. 230])

This volume contains a selection of papers on various aspects of operator theoryin the multi-dimensional case. This last term includes a wide range of situationsand we review the one variable case first.

An important player in the single variable theory is a contractive analytic func-tion on the open unit disk. Such functions, often called Schur functions, have arich theory of their own, especially in connection with the classical interpolationproblems. They also have different facets arising from their appearance in differentareas, in particular as:

• characteristic operator functions, in operator model theory. Pioneering worksinclude the works of Livsic and his collaborators [54], [55], [25], of Sz. Nagyand Foias [61] and of de Branges and Rovnyak [23], [22].• scattering functions, in scattering theory. We mention in particular the Lax–

Phillips approach (see [53]), the approach of de Branges and Rovnyak (see[22]) and the inverse scattering problem of network theory [38]; for a solutionof the latter using reproducing kernel Hilbert space methods, see [8], [9].• transfer functions, in system theory. It follows from the Bochner–Chandra-

sekharan theorem that a system is linear, time-invariant, and dissipative ifand only if it has a transfer function which is a Schur function. For moregeneral systems (even multi-dimensional ones) one can make use of Schwartz’kernel theorem (see [76], [52]) to get the characterisation of invariance undertranslation; see [83, p. 89, p. 130].

There are many quite different approaches to the study of Schur functions, theirvarious incarnations and related problems, yet it is basically true that there is onlyone underlying theory.

viii D. Alpay and V. Vinnikov

One natural extension of the single variable theory is the time varying case, whereone (roughly speaking) replaces the complex numbers by diagonal operators andthe complex variable by a shift operator; see [7], [39].

The time varying case is still essentially a one variable theory, and the variousapproaches of the standard one variable theory generalize together with their in-terrelations. On the other hand, in the multi-dimensional case there is no longera single underlying theory, but rather different theories, some of them looselyconnected and some not connected at all. In fact, depending on which facet ofthe one-dimensional case we want to generalize we are led to completely differentobjects and borderlines between the various theories are sometimes vague. Thedirections represented in this volume include:

• Interpolation and realization theory for analytic functions on the polydisk.This originates with the works of Agler [2], [1]. From the view point of sys-tem theory, one is dealing here with the conservative version of the systemsknown as the Roesser model or the Fornasini–Marchesini model in the multi-dimensional system theory literature; see [71], [46].• Function theory on the free semigroup and on the unit ball of CN . From

the view point of system theory, one considers here the realization problemfor formal power series in non-commuting variables that appeared first inthe theory of automata, see Schutzenberger [74], [75] and Fliess [44], [45](for a good survey see [17]), and more recently in robust control of linearsystems subjected to structured possibly time-varying uncertainty (see Beck,Doyle and Glover [15] and Lu, Zhou and Doyle [59]). In operator theory, twomain parallel directions may be distinguished; the first direction is along thelines of the works of Drury [43], Frazho [47], [48], Bunce [26], and especiallythe vast work of Popescu [65], [63], [64], [66], where various one-dimensionalmodels are extended to the case of several non-commuting operators. Anotherdirection is related to the representations of the Cuntz algebra and is alongthe line of the works of Davidson and Pitts (see [36] and [37]) and Bratelliand Jorgensen [24]. When one abelianizes the setting, one obtains resultson the theory of multipliers in the so-called Arveson space of the ball (see[12]), which are closely related with the theory of complete Nevanlinna–Pickkernels; see the works of Quiggin [70], McCullough and Trent [60] and Aglerand McCarthy [3]. We note also connections with the theory of wavelets andwith system theory on trees; see [16], [10].• Hyponormal operators, subnormal operators, and related topics. Though nom-

inally dealing with a single operator, the theory of hyponormal operators andof certain classes of subnormal operators has many features in common withmultivariable operator theory. We have in mind, in particular, the works ofPutinar [68], Xia [81], and Yakubovich [82]. For an excellent general survey ofthe theory of hyponormal operators, see [80]. Closely related is the principalfunction theory of Carey and Pincus, which is a far reaching developmentof the theory of Kreın’s spectral shift function; see [62], [27], [28]. Another

Editorial Introduction ix

closely related topic is the study of multi-dimensional moment problems; ofthe vast literature we mention (in addition to [68]) the works of Curto andFialkow [33], [34] and of Putinar and Vasilescu [69].• Hyperanalytic functions and applications. Left (resp. right) hyperanalytic

functions are quaternionic-valued functions in the kernel of the left (resp.right) Cauchy–Fueter operator (these are extensions to R4 of the operator∂∂x + i ∂

∂y ). The theory is non-commutative and a supplementary difficulty

is that the product of two (say, left) hyperanalytic functions need not beleft hyperanalytic. Setting the real part of the quaternionic variable to bezero, one obtains a real analytic quaternionic-valued function. Conversely,the Cauchy–Kovalevskaya theorem allows to associate (at least locally) toany such function a hyperanalytic function. Identifying the quaternions withC2 one obtains an extension of the theory of functions of one complex variableto maps from (open subsets of) C2 into C2. Rather than two variables thereare now three non-commutative non-independent hyperanalytic variables andthe counterparts of the polynomials zn1

1 zn22 are now non-commutative poly-

nomials (called the Fueter polynomials) in these hyperanalytic variables. Theoriginal papers of Fueter (see, e.g., [50], [49]) are still worth a careful reading.• Holomorphic deformations of linear differential equations. One approach to

study of non-linear differential equations, originating in the papers of Schle-singer [73] and Garnier [51], is to represent the non-linear equation as thecompatibility condition for some over-determined linear differential systemand consider the corresponding families (so-called deformations) of ordinarylinear equations. From the view point of this theory, the situation when thelinear equations admit rational solutions is exceptional: the non-resonanceconditions, the importance of which can be illustrated by Bolibruch’s coun-terexample to Hilbert’s 21st problem (see [11]), are not met. However, anal-ysis of this situation in terms of the system realization theory may lead toexplicit solutions and shed some light on various resonance phenomena.

The papers in the present volume can be divided along these categories as follows:

Polydisk function theory:

The volume contains a fourth part of the translation of the unpublished thesis [18]of Bessmertnyı, which foreshadowed many subsequent developments and containsa wealth of ideas still to be explored. The other parts are available in [20], [19]and [21]. The paper of Reurings and Rodman, One-sided tangential interpolationfor Hilbert–Schmidt operator functions with symmetries on the bidisk, deals withinterpolation in the bidisk in the setting of H2 rather than of H∞.

Non-commutative function theory and operator theory:

The first paper in this category in the volume is the paper of Ball and Vinnikov,Functional models for representations of the Cuntz algebra. There, the authorsdevelop functional models and a certain theory of Fourier representation for a rep-resentation of the Cuntz algebra (i.e., a row unitary operator). Next we have the

x D. Alpay and V. Vinnikov

paper of Banks, Constantinescu and Johnson, Relations on non-commutative vari-ables and associated orthogonal polynomials, where the authors survey varioussettings where analogs of classical ideas concerning orthogonal polynomials andassociated positive kernels occur. The paper serves as a useful invitation and ori-entation for the reader to explore any particular topic more deeply. In the paperof Kalyuzhnyı-Verbovetzkiı, On the Bessmertnyı class of homogeneous positiveholomorphic functions on a product of matrix halfplanes, a recent investigationof the author on the Bessmertnyı class of operator-valued functions on the openright poly-halfplane which admit a so-called long resolvent representation (i.e., aSchur complement formula applied to a linear homogeneous pencil of operatorswith positive semidefinite operator coefficients), is generalized to a more general“non-commutative” domain, a product of matrix halfplanes. The study of the Bess-mertnyı class (as well as its generalization) is motivated by the electrical networkstheory: as shown by M.F. Bessmertnyı [18], for the case of matrix-valued func-tions for which finite-dimensional long resolvent representations exist, this classis exactly the class of characteristic functions of passive electrical 2n-poles whereimpedances of the elements of a circuit are considered as independent variables.Finally, in the paper Hardy algebras associated with W ∗-correspondences (pointevaluation and Schur class functions), Muhly and Solel deal with an extension ofthe non-commutative theory from the point of view of non-self-adjoint operatoralgebras.

Hyponormal and subnormal operators and related topics:

The paper of Putinar, Notes on generalized lemniscates, is a survey of the theoryof domains bounded by a level set of the matrix resolvent localized at a cyclicvector. The subject has its roots in the theory of hyponormal operators on the onehand and in the theory of quadrature domains on the other. While both topics arementioned in the paper, the main goal is to present the theory of these domains(that the author calls “generalized lemniscates”) as an independent subject matter,with a wealth of interesting properties and applications. The paper of Szafraniec,Orthogonality of polynomials on algebraic sets, surveys recent extensive work ofthe author and his coworkers on polynomials in several variables orthogonal on analgebraic set (or more generally with respect to a positive semidefinite functional)and three term recurrence relations. As it happens often the general approachsheds new light also on the classical one-dimensional situation.

Hyperanalytic functions:

In the paper Operator methods for solutions of differential equations based ontheir symmetries, Eidelman and Krasnov deal with construction of explicit solu-tions for some classes of partial differential equations of importance in physics, suchas evolution equations, homogeneous linear equations with constant coefficients,and analytic systems of partial differential equations. The method used involvesan explicit construction of the symmetry operators for the given partial differen-tial operator and the study of the corresponding algebraic relations; the solutions

Editorial Introduction xi

of the partial differential equation are then obtained via the action of the sym-metry operators on the “simplest” solution. This allows to obtain representationsof Clifford-analytic functions in terms of power series in operator indeterminates.Luna–Elizarraras and Shapiro in Preservation of the norms of linear operators act-ing on some quaternionic function spaces consider quaternionic analogs of someclassical real spaces and in particular compare the norms of operators in the orig-inal space and in the quaternionic extension.

Holomorphic deformations of linear differential equations:

This direction is represented in the present volume by the paper of Katsnelson andVolok, Rational solutions of the Schlesinger system and rational matrix functionsII, which presents an explicit construction of the multi-parametric holomorphicfamilies of rational matrix functions, corresponding to rational solutions of theSchlesinger non-linear system of partial differential equations.

There are many other directions that are not represented in this volume. Withoutthe pretense of even trying to be comprehensive we mention in particular:

• Model theory for commuting operator tuples subject to various higher-ordercontractivity assumptions; see [35], [67].• A multitude of results in spectral multivariable operator theory (many of

them related to the theory of analytic functions of several complex variables)stemming to a large extent from the discovery by Taylor of the notions of thejoint spectrum [78] and of the analytic functional calculus [77] for commutingoperators (see [32] for a survey of some of these).• The work of Douglas and of his collaborators based on the theory of Hilbert

modules; see [42], [40], [41].• The work of Agler, Young and their collaborators on operator theory and

realization theory related to function theory on the symmetrized bidisk, withapplications to the two-by-two spectral Nevanlinna–Pick problem; see [5], [4],[6].• Spectral analysis and the notion of the characteristic function for commuting

operators, related to overdetermined multi-dimensional systems. The mainnotion is that of an operator vessel, due to Livsic; see [56], [57], [58]. Thisturns out to be closely related to function theory on a Riemann surface; see[79],[13].• The work of Cotlar and Sadosky on multievolution scattering systems, with

applications to interpolation problems and harmonic analysis in several vari-ables; see [30], [31], [72].

Acknowledgments

This volume has its roots in a workshop entitled Operator theory, system theoryand scattering theory: multi-dimensional generalizations, 2003, which was held atthe Department of Mathematics of Ben-Gurion University of the Negev during theperiod June 30–July 3, 2003. It is a pleasure to thank all the participants for an

xii D. Alpay and V. Vinnikov

exciting scientific atmosphere and the Center of Advanced Studies in Mathemat-ics of Ben-Gurion University of the Negev for its generosity and for making theworkshop possible.

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Daniel Alpay and Victor VinnikovDepartment of MathematicsBen-Gurion University of the NegevBeer-Sheva, Israele-mail: dany@math.bgu.ac.ile-mail: vinnikov@math.bgu.ac.il

Functional Models for Representationsof the Cuntz Algebra

Joseph A. Ball and Victor Vinnikov

Abstract. We present a functional model, the elements of which are formalpower series in a pair of d-tuples of non-commuting variables, for a row-unitaryd-tuple of operators on a Hilbert space. The model is determined by a weight-ing matrix (called a “Haplitz” matrix) which has both non-commutative Han-kel and Toeplitz structure. Such positive-definite Haplitz matrices then serveto classify representations of the Cuntz algebra Od with specified cyclic sub-space up to unitary equivalence. As an illustration, we compute the weightingmatrix for the free atomic representations studied by Davidson and Pitts andthe related permutative representations studied by Bratteli and Jorgensen.

Mathematics Subject Classification (2000). Primary: 47A48; Secondary: 93C35.

1. Introduction

Let U be a unitary operator on a Hilbert space K and let E be a subspace of K.Define a map Φ from K to a space of formal Fourier series f(z) =

∑∞n=−∞ fnzn

Φ: k →∞∑

n=−∞(PEU∗nk)zn

where PE is the orthogonal projection onto the subspace E ⊂ K. Note that Φ(k) = 0if and only if k is orthogonal to the smallest reducing subspace for U containingthe subspace E ; in particular, Φ is injective if and only if E is ∗-cyclic for U ,i.e., the smallest subspace reducing for U and containing E is the whole space K.Denote the range of Φ by L; note that we do not assume that Φ maps K into normsquare-summable series L2(T, E) = f(z) =

∑∞n=−∞ fnzn :

∑∞n=−∞ ‖fn‖2 <∞.

The first author is supported by NSF grant DMS-9987636; both authors are support by a grantfrom the US-Israel Binational Science Foundation.

2 J.A. Ball and V. Vinnikov

Nevertheless, we may assign a norm to elements of L so as to make Φ a coisometry:

‖Φk‖2L = ‖P(kerΦ)⊥k‖2K.

Moreover, we see that if we set Φk =∑∞

n=−∞ fnzn for a k ∈ K (so fn = PEU∗nk),then

ΦUk =∞∑

n=−∞(PEU∗n−1k)zn

=∞∑

n=−∞fn−1z

= z ·∞∑

n=−∞fn−1z

= z ·∞∑

n=−∞fnzn = MzΦk,

i.e., the operator U is now represented by the operator Mz of multiplication bythe variable z on the space L.

We can make this representation more explicit as follows. The standard ad-joint Φ[∗] of Φ with respect to the L2-inner product on the target domain is definedat least on polynomials:

⟨Φk,

j=−N

⟨k, Φ[∗]

⎛⎝

j=−N

⎞⎠⟩

Kwhere we have set

Φ[∗]

⎛⎝

j=−N

⎞⎠ =

j=−N

Ujpj .

Furthermore, the range Φ[∗]P of Φ[∗] acting on polynomials (where we use Pto denote the subspace of L2(T, E) consisting of trigonometric polynomials withcoefficients in E) is dense in (ker Φ)⊥, and for Φ[∗]p an element of this dense set(with p ∈ P), we have

〈ΦΦ[∗]p, ΦΦ[∗]p〉L = 〈Φ[∗]p, Φ[∗]p〉K= 〈ΦΦ[∗]p, p〉L2 .

This suggests that we set W = ΦΦ[∗] (well defined as an operator from the spaceof E-valued polynomials P to the space L(Z, E) of formal Fourier series with co-efficients in E) and define a Hilbert space LW as the closure of WP in the innerproduct

〈Wp, Wq〉LW = 〈Wp, q〉L2 .

The Toeplitz structure of W (i.e., the fact that Wi,j = PEUj−i|E depends only onthe difference i− j of the indices) implies that the operator Mz of multiplication

Functional Models 3

by z is isometric (and in fact unitary) on LW . Conversely, starting with a positive

semidefinite Toeplitz matrix [Wi,j ] with Wi,j = Wi−j , we may form a space LW

and associated unitary operator UW equal to the multiplication operator Mz actingon LW as a functional model for a unitary operator. While the space LW ingeneral consists only of formal Fourier series and there may be no bounded pointevaluations for the elements of the space, evaluation of any one of the Fouriercoefficients is a bounded operator on the space, and gives the space at least thestructure of a formal reproducing kernel Hilbert space, an L2-version of the usualreproducing kernel Hilbert spaces of analytic functions arising in many contexts;we develop this idea of formal reproducing kernel Hilbert spaces more fully in theseparate report [4].

Note that a unitary operator can be identified with a unitary representationof the circle group T or of the C∗-algebra C(T). Given any group G or C∗-algebraA, there are two natural problems: (1) classification up to unitary equivalence ofunitary representations of G or of A, and (2) classification up to unitary equivalenceof unitary representations which include the specification of a ∗-cyclic subspace.While the solution of the first problem is the loftier goal, the second problem isarguably also of interest. Indeed, there are problems in operator theory where a∗-cyclic subspace appears naturally as part of the structure; even when this is notthe case, a solution of the second problem often can be used as a stepping stoneto a solution of the first problem. In the case of G = T or A = C(T), the theory ofLW spaces solves the second problem completely: given two unitary operators U onK and U ′ on K′ with common cyclic subspace E contained in both K and K′, thenthere is a unitary operator U : K → K′ satisfying UU = U ′U and U |E = IE if andonly if the associated Toeplitz matrices Wi,j = PEUj−i|E and W ′

i,j = PEU ′j−i|Eare identical, and then both U and U ′ are unitarily equivalent to UW on LW withcanonical cyclic subspace W · E ⊂ LW . A little more work must be done to analyzethe dependence on the choice of cyclic subspace E and thereby solve the firstclassification problem. Indeed, if we next solve the trigonometric moment problemfor W and find a measure µ on T (with values equal to operators on E) for whichWn =

zn dµ(z), then we arrive at a representation for the original operator U as

the multiplication operator Mz on the space L2(µ). Alternatively, one can use thetheory of the Hellinger integral (see [5]) to make sense of the space of boundaryvalues of elements of LW as a certain space of vector measures (called “charts” in[5]), or one can view the space LW as the image of the reproducing kernel Hilbertspace L(ϕ) appearing prominently in work of de Branges and Rovnyak in theirapproach to the spectral theory for unitary operators (see, e.g., [6]), where

ϕ(z) =

λ + z

λ− zdµ(z) for z in the unit disk D,

under the transformation (f(z), g(z)) → f(z) + z−1g(z−1). In any case, the first(harder) classification problem (classification of unitary representations up to uni-tary equivalence without specification of a ∗-cyclic subspace) is solved via use ofthe equivalence relation of mutual absolute continuity on spectral measures. For

this classical case, we see that the solution of the second problem serves as astepping stone to the solution of the first problem, and that the transition fromthe second to the first involves some non-trivial mathematics (e.g., solution of thetrigonometric moment problem and measure theory).

The present paper concerns representations of the Cuntz algebra Od (see,e.g., [8] for the definition and background), or what amounts to the same thing, ad-tuple of operators U = (U1, . . . ,Ud) on a Hilbert space K which is row-unitary,i.e.,

⎡⎢⎣U∗

1...U∗

⎤⎥⎦[U1 . . . Ud

⎡⎢⎣I

⎤⎥⎦ ,

[U1 . . . Ud

]⎡⎢⎣U∗

1...U∗

⎤⎥⎦ = I.

Equivalently, U = (U1, . . . ,Ud) is a d-tuple of isometries on K with orthogonalranges and with span of the ranges equal to the whole space K. It is known thatOd is NGCR, and hence the first classification problem for the case of Od is in-tractable in a precise sense, although particular special cases have been workedout (see [7, 9]). The main contribution of the present paper is that there is a sat-isfactory solution of the second classification problem (classification up to unitaryequivalence of unitary representations with specification of ∗-cyclic subspace) forthe case of Od via a natural multivariable analogue of the spaces LW sketchedabove for the single-variable case.

In detail, the functional calculus for a row-unitary d-tuple U = (U1, . . . ,Ud),involves the free semigroup Fd on a set of d generators g1, . . . , gd; elements ofthe semigroup are words w of the form w = gin . . . gi1 with i1, . . . , in ∈ 1, . . . , d.If w = gin . . . gi1 , set Uw = Uin · · · Ui1 . The functional model for such a row-unitaryd-tuple will consist of formal power series of the form

f(z, ζ) =∑

v,w∈Fd

fv,wzvζw (1.1)

where z = (z1, . . . , zd) and ζ = (ζ1, . . . , ζd) is a pair of d non-commuting variables.The formalism is such that zizj = zjzi and ζiζj = ζjζi for i = j but ziζj = ζjzi forall i, j = 1, . . . , d. In the expression (1.1), for w = gin · · · gi1 we set zw = zin · · · zi1

and similarly for ζ. The space LW of non-commuting formal power series whichserves as the functional model for the row-unitary U = (U1, . . . ,Ud) with cyclicsubspace E will be determined by a weighting matrix

Wv,w;α,β = PEUwU∗vUα⊤U∗β⊤ |Ewith row-index (v, w) and column index (α, β) in the Cartesian product Fd ×Fd. On the space LW is defined a d-tuple of generalized shift operators UW =(UW,1, . . . ,UW,d) (see formula (2.12) below) which is row-unitary and which havethe subspace W · E as a ∗-cyclic subspace. Matrices W (with rows and columnsindexed by Fd×Fd) arising in this way from a row-unitary U can be characterizedby a non-commutative analogue of the Toeplitz property which involves both anon-commutative Hankel-like and non-commutative Toeplitz-like property along

Functional Models 5

with a non-degeneracy condition; we call such matrices “Cuntz weights”. SuchCuntz weights serve as a complete unitary invariant for the second classificationproblem for the Cuntz algebraOd: given two row-unitary d-tuples U = (U1, . . . ,Ud)on K and U ′ = (U ′

1, . . . ,U ′d) on K′ with common ∗-cyclic subspace E contained in

both K and K′, then there is a unitary operator U : K → K′ such that UUj = U ′jU

and UU∗j = U ′∗

j U for j = 1, . . . , d and U |E = IE if and only if the associated Cuntz

weights Wv,w;α,β = PEUwU∗vUα⊤U∗β⊤ |E and W ′v,w;α,β = PEU ′wU ′∗vU ′α⊤U ′∗β⊤ |E

are identical, and then both U and U ′ are unitarily equivalent to the model row-unitary d-tuple UW = (UW,1, . . . ,UW,d) acting on the model space LW with canon-ical ∗-cyclic subspace W · E ⊂ LW .

The parallel with the commutative case can be made more striking by view-ing LW as a non-commutative formal reproducing kernel Hilbert space, a naturalgeneralization of classical reproducing kernel Hilbert spaces to the setting wherethe elements of the space are formal power series in a collection of non-commutingindeterminates; we treat this aspect in the separate report [4].

A second contribution of this paper is the application of this functional modelfor row-unitary d-tuples to the free atomic representations and permutative rep-resentations of Od appearing in [9] and [7] respectively. These representations areof two types: the orbit-eventually-periodic type, indexed by a triple (x, y, λ) wherex and y are words in Fd and λ is a complex number of modulus 1, and the orbit-non-periodic case, indexed by an infinite word x = gk1gk2 · · · gkn · · · . Davidson andPitts [9] have identified which pairs of parameters (x, y, λ) or x give rise to unitarilyequivalent representations of Od, which parameters correspond to irreducible rep-resentations, and how a given representation can be decomposed as a direct sum ordirect integral of irreducible representations. The contribution here is to recoverthese results (apart from the identification of irreducible representations) as anapplication of the model theory of LW spaces and the calculus of Cuntz weights.The approach shares the advantages and disadvantages of the de Branges-Rovnyakmodel theory for single operators (see [6]). Once Cuntz weights W are calculated,identifying unitary equivalences is relatively straightforward and obtaining decom-positions is automatic up to the possible presence of overlapping spaces. There issome hard work involved to verify that the overlapping space is actually trivial inspecific cases of interest. While these results are obtained in an elementary wayin [9], our results here show that a model theory calculus, a non-commutativemultivariable extension of the single-variable de Branges-Rovnyak model theory,actually does work, and in fact is straightforward modulo overlapping spaces.

The paper is organized as follows. After the present Introduction, Section2 lays out the functional models for row-isometries and row-unitary operator-tuples in particular. We show there that the appropriate analogue for a bi-infiniteToeplitz matrix is what we call a “Haplitz operator”. Just as Toeplitz operators

W = [Wi−j ]i,j=...,−1,0,1,... have symbols W (z) =∑∞

n=−∞ Wnzn, it is shown that

associated with any Haplitz operator W is its symbol W (z, ζ), a formal power seriesin two sets of non-commuting variables (z1, . . . , zd) and ζ1, . . . , ζd). These symbols

serve as the set of free parameters for the class of Haplitz operators; many questionsconcerning a Haplitz operator W can be reduced to easier questions concerning

its symbol W (z, ζ). In particular, positivity of the Haplitz operator W is shown to

be equivalent to a factorization property for its symbol W (z, ζ) and for the Cuntzdefect D

W(z, ζ) of its symbol (see Theorem 2.8). Cuntz weights are characterized

as those positive semidefinite Haplitz operators with zero Cuntz defect.

Section 3 introduces the analogue of L∞ and H∞, namely, the space of in-

tertwining maps LW,W∗

T between two row-unitary model spaces LW and LW∗, and

the subclass of such maps (“analytic intertwining operators”) which preserve thesubspaces analogous to Hardy subspaces. The contractive, analytic intertwiningoperators then form an interesting non-commutative analogue of the “Schur class”which has been receiving much attention of late from a number of points of view(see, e.g., [2]). These results can be used to determine when two functional modelsare unitarily equivalent, or when a given functional model decomposes as a directsum or direct integral of internal pieces (modulo overlapping spaces). Section 4gives the application of the model theory and calculus of Cuntz weights to freeatomic and permutative representations of Od discussed by Davidson and Pitts [9]and Bratteli and Jorgensen [7] mentioned above.

In a separate report [3] we use the machinery developed in this paper (es-pecially the material in Section 3) to study non-commutative analogues of Lax-Phillips scattering and unitary colligations, how they relate to each other, andhow they relate to the model theory for row-contractions developed in the workof Popescu ([12, 13, 14, 15]).

2. Models for row-isometries and row-unitaries

Let F be the free semigroup on d generators g1, . . . , gd with identity. A genericelement of Fd (apart from the unit element) has the form of a word w = gin · · · gi1 ,i.e., a string of symbols αn · · ·α1 of finite length n with each symbol αk belongingto the alphabet g1, . . . , gd. We shall write |w| for the length n of the wordw = αn · · ·α1. If w = αn · · ·α1 and v = βm · · ·β1 are words, then the product vwof v and w is the new word formed by the juxtaposition of v and w:

vw = βm · · ·β1αn · · ·α1.

We define the transpose w⊤ of the word w = gin · · · gi1 by w⊤ = gi1 · · · gin . Wedenote the unit element of Fd by ∅ (corresponding to the empty word). In partic-ular, if gk is a word of unit length, we write gkw for gkαn · · ·α1 if w = αn · · ·α1.Although Fd is a semigroup, we will on occasion work with expressions involvinginverses of words in Fd; the meaning is as follows: if w and v are words in Fd, theexpression wv−1 means w′ if there is a w′ ∈ Fd for which w = w′v; otherwise wesay that wv−1 is undefined. An analogous interpretation applies for expressionsof the form w−1v. This convention requires some care as associativity can fail: ingeneral it is not the case that (wv−1) · w′ = w · (v−1w′).

Functional Models 7

For E an auxiliary Hilbert space, we denote by ℓ(Fd, E) the set of all E-valuedfunctions v → f(v) on Fd. We will write ℓ2(Fd, E) for the Hilbert space consistingof all elements f in ℓ(Fd, E) for which

‖f‖2ℓ2(Fd,E) :=∑

v∈Fd

‖f(v)‖2E <∞.

Note that the space ℓ2(Fd, E) amounts to a coordinate-dependent view of theFock space studied in [1, 9, 10, 12, 13]. It will be convenient to introduce the

non-commutative Z-transform f → f(z) on ℓ(Fd, E) given by

f(z) =∑

v∈Fd

f(w)zw

where z = (z1, . . . , zd) is to be thought of as a d-tuple of non-commuting variables,and we write

zw = zin · · · zi1 if w = gin · · · gi1 .

We denote the set of all such formal power series f(z) also as L(Fd, E) (or L2(Fd, E)for the Hilbert space case). The right creation operators SR

1 , . . . , SRd on ℓ2(Fd, E)

are given by

SRj : f → f ′ where f ′(w) = f(wg−1

with adjoint given by

SR∗j : f → f ′ where f ′(w) = f(wgj).

(Here f(wg−1j ) is interpreted to be equal to 0 if wg−1

j is undefined.) In the non-

commutative frequency domain, these right creation operators (still denoted bySR

1 , . . . , SRd for convenience) become right multiplication operators:

SRj : f(z) → f(z) · zj, SR∗

j : f(z) → f(z) · z−1j .

In the latter expression zw · z−1j is taken to be 0 in case the word w is not of the

form w′gj for some w′ ∈ Fd. The calculus for these formal multiplication operatorsis often easier to handle; hence in the sequel we will work primarily in the non-commutative frequency-domain setting L(Fd, E) rather than in the time-domainsetting ℓ(Fd, E).

Let K be a Hilbert space and U = (U1, . . . ,Ud) a d-tuple of operators on K.We say that U is a row-isometry if the block-operator row-matrix

[U1 · · · Ud

]: ⊕d

k=1 K → Kis an isometry. Equivalently, each of U1, . . . ,Ud is an isometry on K and the imagespaces imU1, . . . , imUd are pairwise orthogonal. There are two extreme cases ofrow-isometries U : (1) the case where U is row-unitary, i.e.,

[U1 . . . Ud

]is uni-

tary, or equivalently, imU1, . . . , imUd span the whole space K, and (2) the casewhere U is a row-shift, i.e.,

spanimUv : |v| = n = 0;

here we use the non-commutative multivariable operator notation

Uv = Uin . . .Ui1 if v = gin · · · gi1 .

A general row-isometry is simply the direct sum of these extreme cases by theWold decomposition for row-isometries due to Popescu (see [14]). It is well knownthat the operators SR

1 , . . . , SRd provide a model for any row-shift, as summarized

in the following.

Proposition 2.1. The d-tuple of operators (SR1 , . . . , SR

d ) on the space L2(Fd, E) isa row-shift. Moreover, if U = (U1, . . . ,Ud) is any row-shift on a space K, then Uis unitarily equivalent to (SR

1 , . . . , SRd ) on L2(Fd, E), with E = K⊖

k=1UkK].

To obtain a similar concrete model for row-unitaries, we proceed as follows.Denote by ℓ(Fd ×Fd, E) the space of all E-valued functions on Fd ×Fd:

f : (v, w) → f(v, w).

We denote by ℓ2(Fd×Fd, E) the space of all elements f ∈ ℓ(Fd×Fd, E) for which

‖f‖2ℓ2(Fd×Fd,E) :=∑

v,w∈Fd

‖f(v, w)‖2 <∞.

The Z-transform f → f for elements of this type is given by

f(z, ζ) =∑

f(v, w)zvζw.

Here z = (z1, . . . , zd) is a d-tuple of non-commuting variables as before, and ζ =(ζ1, . . . , ζd) is another d-tuple of non-commuting variables, but we specify thateach ζi commutes with each zj for i, j = 1, . . . , d. For the case d = 1, note thatℓ2(F1, E) is the standard ℓ2-space over the non-negative integers ℓ2(Z+, E), while

ℓ2(F1 ×F1, E) = ℓ2(Z+ × Z+, E)appears to be a more complicated version of ℓ2(Z, E). Nevertheless, we shall seethat the weighted modifications of ℓ2(Fd×Fd, E) which we shall introduce below docollapse to ℓ2(Z, E) for the case d = 1. Similarly, one should think of L2(Fd, E) as anon-commutative version of the Hardy space H2(D, E) over the unit disk D, and ofthe modifications of L2(Fd×Fd, E) to be introduced below as a non-commutativeanalogue of the Lebesgue space L2(T, E) of measurable norm-square-integrableE-valued functions on the unit circle T.

In the following we shall focus on the frequency domain setting L2(Fd ×Fd, E) rather than the time-domain setting ℓ2(Fd,×Fd, E), where it is convenientto use non-commutative multiplication of formal power series; for this reason we

shall write simply f(z, ζ) for elements of the space rather than f(z, ζ). Unlike theunilateral setting L2(Fd, E) discussed above, there are two types of shift operatorson L2(Fd ×Fd, E) of interest, namely:

SRj : f(z, ζ) → f(z, ζ) · zj , (2.1)

URj : f(z, ζ) → f(0, ζ) · ζ−1

j + f(z, ζ) · zj (2.2)

Functional Models 9

where f(0, ζ) is the formal power series in ζ = (ζ1, . . . , ζd) obtained by formallysetting z = 0 in the formal power series for f(z, ζ):

f(0, ζ) =∑

w∈Fd

f∅,wζw if f(z, ζ) =∑

v,w∈Fd

fv,wzvζw.

One can think of SRj as a non-commutative version of a unilateral shift (even

in this bilateral setting), while URj is some kind of bilateral shift. We denote by

SR[∗]j and U

R[∗]j the adjoints of SR

j and URj in the L2(Fd × Fd, E)-inner product

(to avoid confusion with the adjoint with respect to a weighted inner product toappear below). An easy computation shows that

SR[∗]j : f(z, ζ) → f(z, ζ) · z−1

j , (2.3)

UR[∗]j : f(z, ζ) → f(0, ζ) · ζj + f(z, ζ) · z−1

j . (2.4)

Note that

UR[∗]i SR

j : f(z, ζ) → UR[∗]i (f(z, ζ) · zj) = δi,jf(z, ζ)

and hence we have the useful identity

UR[∗]i SR

j = δi,jI. (2.5)

On the other hand

R[∗]j : f(z, ζ) →SR

j (f(0, ζ)ζj + f(z, ζ)z−1j )

= f(0, ζ)ζjzj + [f(z, ζ)z−1j ]zj

and hence⎛⎝I −

R[∗]j

⎞⎠ : f(z, ζ) →f(z, ζ)−

f(0, ζ)ζjzj −d∑

[f(z, ζ)z−1j ]zj

= f(0, ζ)−d∑

f(0, ζ)ζjzj

and hence⎛⎝I −

R[∗]j

⎞⎠ : f(z, ζ) → f(0, ζ) ·

⎛⎝1−

⎞⎠ . (2.6)

Now suppose that U = (U1, . . . ,Ud) is a row-unitary d-tuple of operators on aHilbert spaceK, E is a subspace ofK, and we define a map Φ: K → L(Fd×Fd, E) by

Φk =∑

v,w∈Fd

(PEUwU∗vk)zvζw. (2.7)

Φ Ujk =∑

v,w∈Fd

(PEUwU∗vUjk)zvζw

w∈Fd

(PEUwgj k)ζw +∑

v,w : v =∅(PEUwU∗vg−1

j k)zvζw

= (URj Φk)(z, ζ) (2.8)

Φ U∗j k =

v,w∈Fd

(PEUvU∗wU∗j k)zvζw

v,w∈Fd

(PEUvU∗wgj k)zvζw

= (SR[∗]j Φk)(z, ζ). (2.9)

If we let W = ΦΦ[∗] (where Φ[∗] is the adjoint of Φ with respect to the Hilbertspace inner product on K and the formal L2-inner product on L(Fd×Fd, E)), then

Φ[∗] : zαζβe → Uα⊤U∗β⊤

and W := ΦΦ[∗] = [Wv,w;α,β]v,w,α,β∈Fdwhere

Wv,w;α,β = PEUwU∗vUα⊤Uβ⊤ |E . (2.10)

If imΦ is given the lifted norm ‖ · ‖ℓ,‖Φk‖ℓ = ‖P(kerΦ)⊥k‖K ,

then one easily checks that W · P(Fd ×Fd, E) ⊂ imΦ and

‖Wp‖2ℓ = ‖Φ[∗]p‖2K= 〈Φ[∗]p, Φ[∗]p〉K= 〈Wp, p〉L2 .

Thus, if we define a space LW as the closure of W · P(Fd ×Fd, E) in the norm

‖Wp‖2LW= 〈Wp, p〉L2 , (2.11)

then LW = imΦ isometrically. From the explicit form (2.10) of Wv,w;α,β it is easyto verify the intertwining relations

URj W = WSR

j , SR[∗]j W = WU

R[∗]j on P(Fd ×Fd, E).

If we define UW = (UW,1, . . . ,UW,d) on LW by

UW,j : Wp → URj Wp = WSjp, (2.12)

then, from the intertwining relations

ΦUj = URj Φ, ΦU∗

j = SR[∗]j Φ for j = 1, . . . , d

Functional Models 11

and the fact that Φ is coisometric, we deduce that UW is row-unitary on LW withadjoint U∗

J = (U∗W,1, . . . ,U∗

W,d) on LW given by

U∗W,j : Wp → S

R[∗]j Wp = WU

R[∗]j p. (2.13)

If Φ is injective (or, equivalently, if E is ∗-cyclic for the row-unitary d-tupleU = (U1, . . . ,Ud)), then UW = (UW,1, . . . ,UW,d) on LW is a functional modelrow-unitary d-tuple for the abstractly given row-unitary d-tuple U = (U1, . . .Ud).

Our next goal is to understand more intrinsically which weights

W = [Wv,w;α,β]

can be realized in this way as (2.10) and thereby lead to functional models forrow-unitary d-tuples U . From identity (2.5) we see that (SR

1 , . . . , SRd ) becomes a

row-isometry if we can change the inner product on L2(Fd × Fd, E) so that the

adjoint SR∗j of Sj in the new inner product is U

R[∗]j . Moreover, if we in addition

arrange for the new inner product to have enough degeneracy to guarantee that

all elements of the form f(0, ζ)(1 −∑dj=1 zjζj) have zero norm, then the d-tuple

(SR1 , . . . , SR

d ) in this new inner product becomes row-unitary. These observationssuggest what additional properties we seek for a weight W so that it may be ofthe form (2.10) for a row-unitary U .

Let W be a function from four copies of Fd into bounded linear operators onE with value at (v, w, α, β) denoted by Wv,w;α,β. We think of W as a matrix withrows and columns indexed by Fd ×Fd; thus Wv,w;α,β is the matrix entry for row(v, w) and column (α, β). Denote by P(Fd,×Fd, E) the space of all polynomials inthe non-commuting variables z1, . . . , zd, ζ1, . . . , ζd:

P(Fd,×Fd, E) =p(z, ζ) =∑

v,w∈Fd

pv,wzvζw : pv,w ∈ E and

pv,w = 0 for all but finitely many v, w.

Then W can be used to define an operator from P(Fd×Fd, E) into L(Fd×Fd, E)by extending the formula

W : e zαζβ →∑

v,w∈Fd

Wv,w;α,βe zvζw .

for monomials to all of P(Fd,×Fd, E) by linearity. Note that computation of theL2-inner product 〈Wp, q〉L2 involves only finite sums if p and q are polynomials,and therefore is well defined. We say that W is positive semidefinite if

〈Wp, p〉L2 ≥ 0 for all p ∈ P(Fd ×Fd, E).

Under the assumption that W is positive semidefinite, define an inner product onW · P(Fd ×Fd, E) by

〈Wp, Wq〉LW = 〈Wp, q〉L2 . (2.14)

Modding out by elements of zero norm if necessary, W · P(Fd × Fd, E) is a pre-Hilbert space in this inner product. We define a space

LW = the completion of W · P(Fd × Fd, E) in the inner product (2.14). (2.15)

Note that the (v, w)-coefficient of Wp ∈ WP(Fd ×Fd, E) is given by

〈[Wp]v,w, e〉E = 〈Wp, W (zvζwe)〉LW (2.16)

and hence the map Φv,w : f → fv,w extends continuously to the completion LW

of W · P(Fd × Fd, E) and LW can be identified as a space of formal power seriesin the non-commuting variables z1, . . . , zd and ζ1, . . . , ζd, i.e., as a subspace ofL(Fd×Fd, E). (This is the main advantage of defining the space as the completionof W · P(Fd×Fd, E) rather than simply as the completion of P(Fd×Fd, E) in theinner product given by the right-hand side of (2.14).) Note that then, for f ∈ LW

and α, β ∈ FD we have

〈Φα,βf, e〉E = 〈fα,β , e〉E = 〈f, W [zαζβe]〉LW

from which we see that Φ∗α,β : E → LW is given by

Φ∗α,β : e →W [zαζβe].

By using this fact we see that

〈Φv,wΦ∗α,βe, e′〉E = 〈Φ∗

α,βe, Φ∗v,we′〉LW

= 〈W [zαζβe], W [zvζwe′]〉LW

= 〈W [zαζβe], zvζwe′〉LW

= 〈W [zαζβe], zvζwe′〉L2

= 〈Wv,w;α,βe, e′〉Eand we recover the operator matrix entries Wv,w;α,β of the operator W from thefamily of operators Φα,β (α, β ∈ Fd) via the factorization

Wv,w;α,β = Φv,wΦ∗α,β .

Conversely, one can start with any such factorization of W (through a generalHilbert space K rather than K = LW as in the construction above). The followingtheorem summarizes the situation.

Theorem 2.2. Assume that W = [Wv,w;α,β ]v,w,α,β∈Fdis a positive semidefinite

(Fd×Fd)×(Fd×Fd) matrix of operators on the Hilbert space E with a factorizationof the form

Wv,w;α,β = Φv,wΦ∗α,β

for operators Φv,w : K → E for some intermediate Hilbert space K for all v, w, α,β ∈ Fd. Define an operator Φ: K → L(Fd ×Fd, E) by

Φ: k →∑

v,w∈Fd

(Φv,wk)zvζw .

Let LW be the Hilbert space defined as the completion of WP(Fd × Fd, E) in thelifted inner product

〈Wp, Wp〉LW = 〈Wp, p〉L2 .

Then Φ is a coisometry from K onto LW with adjoint given densely by

Φ∗ : Wp →∑

v,w∈Fd

Φ∗α,βpα,β

for p(z, ζ) =∑

α,β pα,βzαζβ a polynomial in P(Fd × Fd, E). In particular, LW isgiven more explicitly as LW = imΦ.

Proof. Note than W (as a densely defined operator on L2(Fd×Fd, E) with domaincontaining at least P(Fd ×Fd, E)) factors as W = ΦΦ[∗], where Φ[∗] is the formalL2-adjoint of Φ defined at least on polynomials by

Φ[∗] : p(z, w) →∑

Φ∗α,βpα,β for p(z, w) =

pα,βzαζβ .

In particular, Φ[∗]P(Fd × Fd, E) is contained in the domain of Φ when Φ is con-sidered as an operator from K into LW with domΦ = k ∈ K : Φk ∈ LW . SinceΦ is defined in terms of matrix entries Φv,w and evaluation of Fourier coefficientsp → pv,w is continuous in LW , it follows that Φ as an operator from K into LW

with domain as above is closed. For an element k ∈ K of the form Φ[∗]p for apolynomial p ∈ P(Fd ×Fd, E), we have

〈Wp, Wp〉LW = 〈Wp, p〉L2

= 〈ΦΦ[∗]k, k〉L2

= 〈Φ[∗]k, Φ[∗]k〉K.

Hence Φ maps Φ[∗]P(Fd × Fd, E) isometrically onto the dense submanifold W ·P(Fd ×Fd, E) of LW . From the string of identities

〈k, Φ[∗]p〉K = 〈Φk, p〉L2

= 〈Φk, Wp〉LW

and the density of W · P(Fd × Fd, E) in LW , we see that kerΦ = (Φ[∗]P(Fd ×Fd, E))⊥. Hence Φ is isometric from a dense subset of the orthogonal complement ofits kernel (Φ[∗]P(Fd×Fd, E)) onto a dense subset of LW (namely, WP(Fd×Fd, E)).Since Φ is closed, it follows that necessarily Φ: K → LW is a coisometry. Finally,notice that

〈Φk, Wp〉LW = 〈Φk, p〉L2

= 〈k, Φ[∗]p〉Kfrom which we see that

Φ∗ : Wp → Φ[∗]p for p ∈ P(Fd ×Fd, E)and the formula for Φ∗ follows. This completes the proof of Theorem 2.2.

We seek to identify additional properties to be satisfied by W so that theoperators UW,j defined initially only on W · P(Fd ×Fd, E) by

UW,j : Wp → WSRj p

and then extended to all of LW by continuity become a row-isometry, or even arow-unitary operator-tuple. From (2.5), the row-isometry property follows if it can

be shown that U∗W,j : Wp →WU

R[∗]j p, or equivalently,

WSRj = UR

j W on P(Fd ×Fd, E). (2.17)

Similarly, from (2.6) we see that the row-unitary property will follow if we showin addition that

[p(0, ζ)

)]= 0 for all p ∈ P(Fd ×Fd, E). (2.18)

The next theorem characterizes those operator matrices W for which (2.17) and(2.18) hold.

Theorem 2.3. Let W be a (Fd ×Fd)× (Fd ×Fd) matrix with matrix entries equalto operators on the Hilbert space E. Then:

1. W satisfies (2.17) if and only if

W∅,w;αgj ,β = W∅,wgj ;α,β, (2.19)

Wv,w;αgj ,β = Wvg−1j ,w;α,β for v = ∅ (2.20)

for all v, w, α, β ∈ Fd, j = 1, . . . , d, where we interpret Wvg−1j ,w;α,β to be 0

in case vg−1j is not defined.

2. Assume that W is selfadjoint. Then W satisfies (2.17) and (2.18) if and onlyif W satisfies (2.19), (2.20) and in addition

W∅,w;∅,β =

W∅,wgj ;∅,βgj(2.21)

for all w, β ∈ Fd.

Proof. By linearity, it suffices to analyze the conditions (2.17) and (2.18) on mono-mials f(z, ζ) = zαζβe for some α, β ∈ Fd and e ∈ E . We compute

WSRj (zαζβe) = W (zαgj ζβe)

v,w∈Fd

Wv,w;αgj ,βzvζw (2.22)

while, on the other hand,

URj W (zαζβe) = UR

⎛⎝ ∑

v,w∈Fd

Wv,w;α,βezvζw

⎞⎠

w∈Fd

W∅,w;α,βζwg−1j +

v,w∈Fd

Wv,w;α,βzvgj ζw . (2.23)

Equality of (2.22) with (2.23) for all α, β and e in turn is equivalent to (2.19) and(2.20).

Assume now that W satisfies (2.19) and is selfadjoint. Then we have

Wv,w;∅,βgj= [W∅,βgj ;v,w]∗

= [W∅,β;vgj ,w]∗

= Wvgj ,w;∅,β

and hence

Wvgj ,w;∅,β = Wv,w;∅,βgj. (2.24)

To verify the condition for (2.18), if f(z, ζ) = zαζβe, then f(0, ζ) = 0 if

α = ∅ and W(f(0, ζ)

(1−∑d

j=1 zjζj

))= 0 trivially. Thus it suffices to consider

the case α = ∅ and f(z, ζ) = ζβe. Then f(0, ζ) = ζβe and

f(0, ζ)

⎛⎝1−

⎞⎠ = ζβe−

zgj ζβgj e.

⎡⎣f(0, ζ)

⎛⎝1−

⎞⎠⎤⎦ = W

⎡⎣ζβe−

zjζβgj e

⎤⎦

v,w∈Fd

⎛⎝Wv,w;∅,β −

Wv,w;gj ,βgj

⎞⎠ e zvζw

This last quantity set equal to zero for all β ∈ Fd and e ∈ E is equivalent to

Wv,w;∅,β =

Wv,w;gj ,βgj . (2.25)

If v = ∅, we may use (2.19) to see that

W∅,w;gj ,βgj= W∅,wgj ;∅,βgj

and (2.25) collapses to (2.21). If v = ∅, write v = v′gk for some k. From (2.20) wehave

Wv,w;gj ,βgj = δj,kWv′,w;∅,βgk

and henced∑

Wv,w;gj ,βgj = Wv′,w;∅,βgk

= Wv′gk,w;∅,β (by (2.24))

= Wv,w;∅,β

and hence (2.25) for this case is already a consequence of (2.20) and (2.24). Thiscompletes the proof of Theorem 2.3.

Note that (2.19) is a Hankel-like property for the operator matrix W for thisnon-commutative setting, while (2.20) is a Toeplitz-like property. We shall there-fore refer to operators W : P(Fd × Fd, E) → L(Fd × Fd, E) for which both (2.19)and (2.20) are valid as Haplitz operators. We shall call positive semidefinite Haplitzoperators having the additional property (2.21) Cuntz weights. As an immediatecorollary of the previous result we note the following.

Corollary 2.4. Let W be a positive semidefinite operator

W : P(Fd ×Fd, E) → L(Fd ×Fd, E)and consider the d-tuple UW = (UW,1, . . . ,UW,d) of operators on LW defineddensely by

UW,j : Wp →WSRj p for p ∈ P(Fd ×Fd, E). (2.26)

1. W is Haplitz if and only if UW is a row-isometry.2. W is a Cuntz weight if and only if UW is row-unitary.

In either case, UW,j and U∗W,j are then given on arbitrary elements f ∈ LW by

UW,j : f(z, ζ) → f(0, ζ) · ζ−1j + f(z, ζ) · zj , (2.27)

U∗W,j : f(z, ζ) → f(0, ζ) · ζj + f(z, ζ) · z−1

j . (2.28)

Proof. The results are immediate consequences of (2.5) and (2.6). To prove (2.27)and (2.28), note the two expressions for UW,j and for U∗

W,j on polynomials in case

W = W ∗ is a Haplitz operator (as in (2.12) and (2.13)), and then note that thefirst formula necessarily extends by continuity to all of LW since the map f → fv,w

is continuous on any space LW .

Remark 2.5. If W = [Wv,w;α,β]v,w,α,β∈Fdwhere Wv,w;α,β = PEUwU∗vUα⊤Uβ⊤ |E

for a row-unitary d-tuple U as in (2.10), then it is easily checked that W is a Cuntzweight. To see this, note

W∅,w;αgj ,β = PEUw · UjUα⊤U∗β⊤∣∣∣E

= PEUwgjUα⊤U∗β⊤∣∣∣E

= W∅,wgj ;α,β

∣∣E

simply from associativity of operator composition, and (2.19) follows. Similarly,for v = ∅, we have

Wv,w;αgj ,β = PEUwU∗v · UjUα⊤U∗β⊤∣∣∣E

= PEUwU∗vg−1j Uα⊤U∗β⊤

∣∣∣E

= Wvg−1j ,w;α,β

from the row-isometry property of U :

U∗i Uj = δi,jI.

Hence (2.20) follows and W is Haplitz. To check (2.21), we use the row-coisometry

property of U (∑d

j=1 UjU∗j = I) to see that

W∅,wgj ;∅,βgj=

PEUwUjU∗j U∗β⊤

∣∣∣E

= PEUwU∗β⊤∣∣∣E

= W∅,w;∅,β.

From the formula (2.10) we see that W has the additional normalization propertyW∅,∅;∅,∅ = IE .

We shall be particularly interested in [∗]-Haplitz operators, i.e., (Fd ×Fd)×(Fd×Fd) operator matrices W for which both W and W [∗] are Haplitz. (Of coursea particular class of [∗]-Haplitz operators are the selfadjoint Haplitz operators –Haplitz operators W with W = W [∗].) For these operators the structure can bearranged in the following succinct way.

Proposition 2.6. Suppose that W = [Wv,w;α,β] is a [∗]-Haplitz operator matrix, and

define a Fd ×Fd operator matrix W = [Wv,w] by

Wv,w = Wv,w;∅,∅.

Then W is completely determined from W according to the formula

Wv,w;α,β =

W(vα−1)β⊤,w if |v| ≥ |α|,Wβ⊤,w(αv−1)⊤ if |v| ≤ |α|. (2.29)

Conversely, if W is any Fd × Fd matrix, then formula (2.29) defines a [∗]-Haplitz operator matrix W .

Proof. Suppose first that W = [Wv,w;α,β] is [∗]-Haplitz. Then we compute, for|v| ≥ |α|,

Wv,w;α,β = Wvα−1,w;∅,β by (2.20)

[∗]∅,β;vα−1,w

[∗]∅,∅;(vα−1)β⊤,w

)∗by (2.19) for W [∗]

= W(vα−1)β⊤,w;∅,∅ = W(vα−1)β⊤,w

while, for |v| ≤ |α| we have

Wv,w;α,β =(W

[∗]α,β;v,w

[∗]αv−1,β;∅,w

)∗by (2.20) for W [∗]

= W∅,w;αv−1,β

= W∅,w(αv−1)⊤;∅,β by (2.19)

[∗]∅,β;∅,w(αv−1)⊤

[∗]∅,∅;β⊤,w(αv−1)⊤

)∗by (2.19)

= Wβ⊤,w(αv−1)⊤;∅,∅ = Wβ⊤,w(αv−1)⊤

and the first assertion follows.Conversely, given W = [Wv,α], define W = [Wv,w;α,β] by (2.29). Then verify

W∅,w;αgj ,β = Wβ⊤,wα⊤ versus W∅,wgj ;α,β = Wβ⊤,wα⊤ and (2.19) follows for W .Similarly, compute, for v = ∅,

Wv,w;αgj ,β =

W(vg−1

j α−1)β⊤,w if |v| > |α|,Wβ⊤,w(αgjv−1)⊤ if |v| ≤ |α|

versus, again for v = ∅,

Wvg−1j ,w;α,β =

0 if v = v′gj,

Wv′,w;α,β if v = v′gj

Wv′,w;α,β =

Wv′α−1β⊤,w if |v′| ≥ |α|,Wβ⊤,w(αv′−1)⊤ if |v′| ≤ |α|

and (2.20) follows for W .

From (2.29) we see that W [∗] is given by

W[∗]v,w;α,β = (Wα,β;v,w)

⎧⎨⎩

(W(αv−1)w⊤,β

)∗if |α| ≥ |v|,(

Ww⊤,β(vα−1)⊤

)∗if |α| ≤ |v|.

(2.30)

Using (2.30) we see that

W[∗]∅,w;αgj ,β =

(Wαgjw⊤,β

W[∗]∅,wgj ;α,β =

(Wαgjw⊤,β

and (2.19) follows for W [∗]. Similarly, for v = ∅,

W[∗]v,w;αgj ,β =

⎧⎨⎩

(Wαgjv−1)w⊤,β

)∗if |v| ≤ |α|,(

Ww⊤,β(vg−1j α−1)⊤

)∗if |v| > |α|

versus

W[∗]vg−1

j ,w;α,β=

0 if v = v′gj,

W[∗]v′,w;α,β if v = v′gj

W[∗]v′,w;α,β =

⎧⎨⎩

(W(αv′−1)w⊤,β

)∗if |v′| < |α|,(

Ww⊤,β(v′α−1)⊤

)∗if |v′| ≥ |α|

and (2.20) follows for W [∗] as well. We conclude that W as defined by (2.29) is[∗]-Haplitz as asserted.

The formula (2.29) motivates the introduction of the symbol W (z, ζ) for the[∗]-Haplitz operator W defined by

W (z, ζ)e = (We)(z, ζ) =∑

v,w∈Fd

Wv,w;∅,∅ezvζw .

For any [∗]-Haplitz W and given any α, β ∈ Fd, we have∑

Wv,w;α,βezvζw = W (ezαζβ)

= W ((SR)α⊤

(UR[∗])β⊤

= (UR)α⊤

W ((UR[∗])β⊤

= (UR)α⊤

(SR[∗])β⊤

Hence the matrix entries Wv,w;α,β for W are determined from a knowledge ofmatrix entries of the special form Wv,w;∅,∅ (i.e., the Fourier coefficients of the

symbol W (z, ζ)) via

v,w∈Fd

Wv,w;α,βezvζw = (UR)α⊤

(SR[∗])β⊤

⎛⎝ ∑

v,w∈Fd

Wv,w;∅,∅ezvζw

⎞⎠ . (2.31)

This gives a method to reconstruct a [∗]-Haplitz operator directly from its symbol(equivalent to the reconstruction formula (2.29) from the matrix entries Wv,w;∅,∅).This can be made explicit as follows.

Proposition 2.7. Let W = [Wv,w;α,β ] be a [∗]-Haplitz operator as above with sym-

bol W (z, ζ) =∑

v,w∈FdWv,w;∅,∅zvζw. Then, for a general polynomial f(z, ζ) =∑

α,β fα,βzαζβ ∈ P(Fd ×Fd, E), we have

W [f ](z, ζ) = W (z′, ζ)kper(z, ζ)f(z, z′−1)|z′=0 + W (z′, ζ)f(z, z′−1)|z′=z, (2.32)

where we have set kper(z, ζ) equal to the “perverse Szego kernel”

kper(z, ζ) =∑

v′ =∅(z−1)v′⊤

(ζ−1)v′

(2.33)

and where z′ = (z′1, . . . , z′d) is another set of non-commuting indeterminants, each

of which commutes with z1, . . . , zd, ζ1, . . . , ζd, and where it is understood that theevaluation at z′ = 0 is to be taken before the multiplication with kper(z, ζ).

In particular, for the case where p(z) =∑

α pαzα is an analytic polynomialin P(Fd × ∅, E), then

W [p](z) = W (0, ζ)kper(z, ζ)p(z) + W (z, ζ)p(z). (2.34)

If in addition W (0, ζ) = I, then (2.34) simplifies further to

W [p](z) = W (z, ζ)p(z). (2.35)

Proof. We compute

W [zαζβe] = W (SR)α⊤

(UR[∗])β⊤

= (UR)α⊤

(SR[∗])β⊤(W (z, ζ)e

= (UR)α⊤(W (z, ζ)(z−1)β

Note that [W (z, ζ)(z−1)β ](0, ζ) =∑

w Wβ⊤,w;∅,∅ζw. Therefore

URj : W (z, ζ)(z−1)βe →

Wβ⊤,w;∅,∅ζwζ−1

j e +∑

Wvβ⊤,w;∅,∅zvζwzje,

URk UR

j : W (z, ζ)(z−1)β →URk

j e +∑

Wvβ⊤,w;∅,∅zvζwzje

j ζ−1k e +

Wvβ⊤,w;∅,∅zvζwzjzke

and then by induction we have

W : zαζβe →∑

α′,α′′ : α=α′α′′,α′ =∅

Wβ⊤,w;∅,∅ζw(ζ−1)α′

zα′′

e +∑

Wvβ⊤,w;∅,∅zvζwzαe

w∈Fd

Wβ⊤,w;∅,∅ζw

)· kper(z, ζ) · zαe +

(W (z, ζ)(z−1)β

= W (z′, ζ)kper(z, ζ)zα(z′−1)βe|z′=0 + W (z′, ζ)zα(z′−1)βe|z′=z

= W (z′, ζ)kper(z, ζ)f(z, z′−1)|z′=0 + W (z′, ζ)f(z, z′−1)|z′=z

for the case f(z, ζ) = zαζβe, and formula (2.32) now follows by linearity. Theformulas (2.34) and (2.35) are specializations of (2.32) to the case where f = p ∈P(F × ∅, E) and where also W (0, ζ) = I respectively.

Positive semidefiniteness of a selfadjoint Haplitz operator W as well as thevalidity of (2.21) required to be a Cuntz weight can be characterized directly

in terms of the symbol W (z, ζ) as follows. For a general formal power series

A(z) =∑

w∈FdAwzw with operator coefficients, we set A(z)∗ =

∑w∈Fd

A∗wzw⊤

=∑w∈Fd

A∗w⊤zw. For a formal power series K(z, ζ) in the two sets of non-commuting

variables z = (z1, . . . , zd) and ζ = (ζ1, . . . , ζd), we already have the notion of Kis a positive kernel; by this we mean that the NFRKHS H(K) is well defined,or equivalently, that K has a factorization of the form K(z, ζ) = Y (z)Y (ζ)∗ forsome formal non-commuting power series Y (z) =

∑v∈Fd

Yvzv. See [4]. The next

result suggests that we introduce the following related notion: we shall say thatthe formal power series K(z, ζ) is a positive symbol if K ′(z, ζ) = K(ζ, z) is a pos-itive kernel, i.e., if K has a factorization of the form K(z, ζ) = Y (ζ)Y (z)∗. This

terminology will be used in the sequel. In addition, for any symbol W (z, ζ), wedefine the Cuntz defect D

W(z, ζ) by

(z, ζ) = W (z, ζ)−d∑

z−1k W (z, ζ)ζ−1

Theorem 2.8. A Haplitz operator W : Fd × Fd → L(E) is positive semidefinite if

and only if both its symbol W (z, ζ) and the Cuntz defect of its symbol DW

(z, ζ)are positive symbols, i.e., there exist formal power series Y (z) =

∑w∈Fd

Ywzw and

Γ(z) =∑

w∈FdΓwzw so that

W (z, ζ) = Y (ζ)Y (z)∗, (2.36)

W (z, ζ)−d∑

z−1j W (z, ζ)ζ−1

j = Γ(ζ)Γ(z)∗. (2.37)

The Haplitz operator W is a Cuntz weight if and only if its symbol W (z, ζ) is apositive symbol and its Cuntz defect D

W(z, ζ) is zero:

W (z, ζ)−d∑

z−1j W (z, ζ)ζ−1

j = 0. (2.38)

Proof. Suppose that W is a positive semidefinite Haplitz operator. From the theoryof reproducing kernels, this means that W has a factorization

Wv,w;α,β = Xv,wX∗α,β (2.39)

for some operators Xv,w : L → E . As W is selfadjoint Haplitz, we have

Wv,w = Wv,w;∅,∅

=(W∅,∅;v,w

=(W∅,v⊤;∅,w

= W∅,w;∅,v⊤

= X∅,wX∗∅,v⊤

orWv,w = YwY ∗

v⊤ (2.40)

where we have set Yw = X∅,w. The identity (2.40) in turn is equivalent to (2.36)with Y (z) =

∑w∈Fd

Derivation of the necessity of the factorization (2.37) lies deeper. From Corol-lary 2.4 we know that the operators (UW,1, . . . ,UW,d) given by (2.26) form a row-isometric d-tuple on LW . Furthermore, factorization (2.39) implies that the map

Φ: ℓ →∑

v,w,∈Fd

(Xv,wℓ)zvζw

is a coisometry. Without loss of generality, we may assume that Φ is actuallyunitary. Hence there is a row-isometric d-tuple (V1, . . . ,Vd) of operators on Ldefined by

ΦVj = UW,jΦ for j = 1, . . . , d.

SetL = closed span Y ∗

β E : β ∈ Fd.We claim: V∗

j : Y ∗β e → Y ∗

βgje for e ∈ E and j = 1, . . . , d. Indeed, note that

ΦY ∗β e =

(Xv,wY ∗β e)zvζw

(Xv,wX∗∅,βe)zv, ζw

(Wv,w;∅,βe)zvζw

= [W (ζβe)](z, ζ)

from which we see that

U∗W,jΦY ∗

β e = U∗W,jW (ζβe)

= W (UR[∗]j (ζβe))

= W (ζβgj e)

(Wv,w;∅,βgje)zvζw

= ΦY ∗βgj

and the claim follows.Thus L is invariant for V∗

j for 1 ≤ j ≤ d. As (V1, . . . ,Vd) is a row-isometry,it follows that (V∗

1 , . . . ,V∗d )|L is a column contraction, or

∥∥∥∥∥∥∥

⎡⎢⎣

Y ∗βjg1

...Y ∗

⎤⎥⎦ ej

∥∥∥∥∥∥∥

≤ ‖n∑

Y ∗βj

ej‖2

for all β1, . . . , βn ∈ Fd and e1, . . . , en ∈ E . Thus

[Wv⊤j ,vi

]i,j=1,...,n = [YviY∗vj

]i,j=1,...,n

⎡⎢⎣

...Yvn

⎤⎥⎦[U1 . . . Ud

]⎡⎢⎣U∗

1...U∗

⎤⎥⎦[Y ∗

v1. . . Y ∗

⎡⎢⎣

Ygℓv1

...Ygℓvn

⎤⎥⎦[Yv1gℓ

. . . Y ∗vngℓ

[Wgℓv⊤j ,vigℓ

]i,j=1,...,n (2.41)

and hence we have the inequality[Wv⊤

j ,vi−

Wgℓv⊤j ,vigℓ

i,j=1,...,d

≥ 0. (2.42)

This is exactly the matrix form of the inequality (2.37), and hence (2.37) follows.Suppose now that W is a Cuntz weight. Then (UW,1, . . . ,UW,d) is row-unitary.

It follows that (V1, . . . ,Vd) is row-unitary, and hence that (V∗1 , . . . ,V∗

d )|L is a col-umn isometry. It then follows that equality holds in (2.42), from which we get theequality (2.38) as asserted.

Conversely, suppose that W is a selfadjoint Haplitz for which the two factor-izations (2.36) and (2.37) hold. Then we have

Wv,w = YwY ∗v⊤ where Yv : L → E .

We may assume that L = closed span Y ∗v E : v ∈ Fd. From (2.37), by reversing

the order of the steps in (2.41) we see that the operators (T ∗1 , . . . , T ∗

d ) defined by

T ∗j Y ∗

v e = Y ∗vgj

extend to define a column contraction on L. By induction, T ∗uY ∗v = Y ∗

vu⊤e or

YvT u⊤

= Yvu⊤ for all u ∈ Fd and e ∈ E , and

[T1 . . . Td

⎡⎢⎣L...

⎤⎥⎦ → L

is a contraction. Let (V1, . . . ,Vd) on L ⊃ L be the row-isometric dilation of(T1, . . . , Td) (see [14]), so

[V1 . . . Vd

⎡⎢⎣L...L

⎤⎥⎦ → L is isometric and PLVu = T uPL.

Xv,w = YwPLV∗v : L → Efor v, w ∈ Fd. Then, for |v| ≥ |α| we have

Xv,wX∗α,β = YwPLV∗wVα⊤

Y ∗β

= YwPLV∗vα−1

Y ∗β

= YwT ∗vα−1

Y ∗β

= YwY ∗β(vα−1)⊤

= W(vα−1)β⊤,w = Wv,w;α,β

where we used (2.29) for the last step. Similarly, if |v| ≤ |α| we have

Xv,wX∗α,β = YwPLV∗wVα⊤

Y ∗β

= YwPLV(αv−1)⊤Y ∗β

= YwT (αv−1)⊤Y ∗β

= Yw(αv−1)⊤Y ∗β

= Wβ⊤,w(αv−1)⊤

= Wv,w;α,β

and the factorization Wv,wα,β = Xv,wX∗α,β shows that W is positive semidefinite

as wanted.If the symbol W of the Haplitz operator W satisfies (2.36) and (2.38), then

we see that the d-tuple (T ∗1 ,...,T ∗

d ) in the above construction is a row-isometry, inwhich case the row-isometric extension (V1,...,Vd) of the row-coisometry (T1,...,Td)

is in fact row-unitary. The construction above applies here to give Wv,w;α,β =YwPLV∗vVαY ∗

β , but now with V equal to a row-unitary d-tuple. Hence

W∅,wgj ;∅,βgj=

Ywgj PLY ∗βgj

YwPLVjV∗j Y ∗

= YwY ∗β = W∅,w;∅,β

and (2.21) follows, i.e., W is a Cuntz weight in case (2.37) is strengthened to (2.38).The theorem now follows.

Remark 2.9. If W is a [∗]-Haplitz operator, then the adjoint W [∗] of W has symbol

W [∗](z, ζ) =∑

W ∗v,w;∅,∅z

W ∗v,w∅,∅ = (W∅,∅;v,w)∗

= (W∅,v⊤;∅,w)∗

= W[∗]∅,w;∅,v⊤

= W[∗]∅,∅;w⊤,v⊤

= (Ww⊤,v⊤;∅,∅)∗

W [∗](z, ζ) = W (ζ, z)∗. (2.43)

(where we use the convention that (zv)∗ = zv⊤

and (ζw)∗ = ζw⊤

). Thus theselfadjointness of a [∗]-Haplitz operator W can be expressed directly in terms ofthe symbol: W = W [∗] (as a [∗]-Haplitz operator) if and only if

W (z, ζ) = W (ζ, z)∗. (2.44)

Note that (2.44) is an immediate consequence of the factorization (2.36) requiredfor positive semidefiniteness of the Haplitz operator W .

In case W is a Cuntz weight with the additional property

W∅,∅;∅,∅ = IE , (2.45)

then the coefficient space E can be identified isometrically as a subspace of LW viathe isometric map VW : E → LW given by VW : e→ We. Let us say that a Cuntzweight W with the additional property (2.45) is a normalized Cuntz weight. Thenwe have the following converse of Corollary 2.4 for the case of normalized Cuntzweights. A more precise converse to Corollary 2.4, using the notion of “row-unitaryscattering system”, is given in [4].

Theorem 2.10. Let U = (U1, . . . ,Ud) be a row-unitary d-tuple of operators on theHilbert space K and let E be a subspace of K. Let W be the (Fd ×Fd)× (Fd×Fd)matrix of operators on E defined by

Wv,w;α,β = PEUwU∗vUα⊤U∗β⊤∣∣∣E

. (2.46)

Then W is a normalized Cuntz weight and the map Φ defined by

Φ: k →∑

v,w∈Fd

(PEUwU∗vk)zvζw (2.47)

is a coisometry from K onto LW which satisfies the intertwining property

ΦUj = UW,jΦ for j = 1, . . . , d.

In particular, if the span of UwU∗ve : v, w ∈ Fd and e ∈ E is dense in K, then Φis unitary and the row-unitary d-tuple U is unitarily equivalent to the normalizedCuntz-weight model row-unitary d-tuple UW via Φ. Moreover, if W ′ is any othernormalized Cuntz weight and Φ′ any other coisometry from K onto LW ′ such thatΦ′Uj = UW ′,jΦ

′ for j = 1, . . . , d and Φ′e = W ′(z∅ζ∅e) for all e ∈ E, then W ′ = Wand Φ′ = Φ.

Proof. Apart from the uniqueness statement, Theorem 2.10 has already been de-rived in the motivational remarks at the beginning of this section and in Remark2.5. We therefore consider only the uniqueness assertion.

Suppose that W ′ is another normalized Cuntz weight on E for which thereis a coisometry Φ′ : K → LW ′ with Φ′ : e → W ′e for e ∈ E for all e ∈ E and withΦUj = UW ′,jΦ

′ for j = 1, . . . , d.

We claim that W ′ = W , or W ′v,w;α,β = Wv,w;α,β for all v, w, α, β ∈ Fd.

Since W ′ is also normalized, Φ′ in fact is isometric on E , and hence also onspanv,w∈Fd

UwU∗vE . Hence

〈Wv,w;α,βe, e′〉E = 〈UwU∗vUα⊤U∗β⊤

e, e′〉K= 〈Uα⊤U∗β⊤

e,Uv⊤U∗w⊤

e′〉K= 〈Φ′(Uα⊤U∗β⊤

e), Φ′(Uv⊤U∗w⊤

e′)〉LW ′

= 〈Uα⊤

W ′ U∗β⊤

W ′ Φ′e,Uv⊤

W ′U∗w⊤

W ′ Φ′e′〉LW ′

= 〈Uα⊤

W ′ U∗β⊤

W ′ W ′e,Uv⊤

W ′U∗w⊤

W ′ W ′e′〉LW ′

= 〈W ′(zαζβe), W ′(zvζwe′)〉LW ′

= 〈W ′(zαζβe), zvζwe′〉L2

= 〈W ′v,w;α,βe, e′〉E .

We conclude that W = W ′ as claimed. This completes the proof of Theorem2.10.

For purposes of the following corollary, let us say that two normalized Cuntzweights W and W ′ on Hilbert spaces E and E ′ respectively are unitarily equivalentif there is a unitary operator γ : E → E ′ such that Wv,w;α,β = γ∗W ′

v,w;α,βγ for allv, w, α, β ∈ Fd.

Corollary 2.11. The assignment

(U1, . . . ,Ud; E) → [Wv,w;α,β ] = [PEUwU∗vUα⊤Uβ⊤ |E ]

of an equivalence class of normalized Cuntz weights to a row-unitary d-tuple(U1, . . . ,Ud) of operators on a Hilbert space K (or, equivalently, an isometric rep-resentation of the Cuntz algebra Od) together with a choice of ∗-cyclic subspace Eprovides a complete unitary invariant for the category of row-unitary d-tuples to-gether with cyclic subspace. Specifically, if (U1, . . . ,Ud) is row-unitary on the Hilbertspace K with cyclic subspace E ⊂ K and (U ′

1, . . . ,U ′d) is another row-unitary on

the Hilbert space K′ with cyclic subspace E ′, then there is a unitary transformationΓ: K → K′ with ΓUj = U ′

jΓ for j = 1, . . . , d and such that ΓE = E ′ if and only ifWv,w;α,β = γ∗Wv,w;α,βγ for all v, w, α, β ∈ Fd, where

Wv,w;α,β = PEUwU∗vUα⊤Uβ⊤ |E ,

W ′v,w;α,β = P ′

EU ′wU ′∗vU ′α⊤U ′β⊤ |E′ , and

the unitary γ : E → E ′ is given by γ = Γ|E .

Remark 2.12. Theorem 2.10 can be formulated directly in terms of the modelrow-unitary UW = (UW,1, . . . ,UW,d) on the model space LW as follows. For W aCuntz weight on E, the matrix entries Wv,w;α,β (for v, w, α, β ∈ Fd) are given by

Wv,w;α,β = i∗WUwWU∗v

W Uα⊤

W U∗β⊤

W iW (2.48)

where iW : E → LW is the injection operator i : e → We. To see (2.48) directly,note that

〈Wv,w;α,βe, e′〉E = 〈W (zαζβe), zvζwe′〉L2

= 〈W (zαζβe), W (zvζwe′)〉LW

= 〈WSRα⊤

(UR[∗])β⊤

e, WSRv⊤

(UR[∗])w⊤

e′〉LW

= 〈Uα⊤

W U∗β⊤

W We,Uv⊤

W U∗w⊤

W We′〉LW

= 〈i∗WUwWU∗v

W Uα⊤

W U∗β⊤

W iW e, e〉Eand (2.48) follows.

Remark 2.13. From the point of view of classification and model theory for rep-resentations of the Cuntz algebra, the weakness in Corollary 2.11 is the demandthat a choice of cyclic subspace E be specified. To remove this constraint whatis required is an understanding of when two Cuntz weights W and W ′ are such

that the corresponding Cuntz-algebra representations UW and UW ′ are unitarilyequivalent, i.e., when is there a unitary intertwining map S : LW → LW ′ such that

SUW,j = UW ′,jS for j = 1, . . . , d.

Preliminary results on this problem appear in Section 3 where the “analytic”intertwining maps S (S : HW → HW ′ ) are analyzed.

Remark 2.14. An instructive exercise is to sort out the model for the case d = 1.Then the alphabet consists of a single letter g and the semigroup Fd can beidentified with the semigroup Z+ of non-negative integers (with the empty word ∅set equal to 0 and a word w = g . . . g set equal to its length |w| ∈ N ⊂ Z+). Hencea Haplitz weight W is a matrix with rows and columns indexed by Z+ ×Z+. TheHaplitz property (2.19) means

W0,n;k+1,ℓ = W0,n+1;k,ℓ (2.49)

while (2.20) meansWm+1,n;k+1,ℓ = Wm,n;k,ℓ. (2.50)

Condition (2.21) reduces to

W0,n;0,k = W0,n+1;0,k+1. (2.51)

and (2.29) becomes

Wm,n;k,ℓ =

Wm−k+ℓ,n if m ≥ k,

Wℓ,n+k−m if m ≥ k

Wi,j = Wi,j;0,0 = (W0,0;i,j)∗ = (W0,i;0,j)

∗ = W0,j;0,i

W0,j−i;0,0 = W0,j−i if j ≥ i,

W0,0;0,i−j if j ≤ i

where, for j ≤ i,

W0,0;0,i−j = (W0,i−j;0,0)∗ = (W0,i−j)

or alternatively

W0,0;0,i−j = (W0,i−j;0,0)∗ = (W0,0;i−j,0)

∗ = Wi−j,0;0,0 = Wi−j,0.

Hence if we set

W0,k if k ≥ 0,

W−k,0 = (W0,−k)∗ if k ≤ 0,

then Wi,j = Ti−j and Wm,n;k,ℓ collapses to

Wm,n;k,ℓ = Tn+k−m−ℓ.

The space LW is spanned by elements of the form W (zkζℓe) for k, ℓ ∈ Z+ ande ∈ E . However, from (2.51) we see that W (zkζℓe) = W (zk+1ζℓ+1e) and hencewe may identify W (zkζℓe) simply with W (zk−ℓe) (where now k− ℓ in general lies

in Z). The reduced weight W has row and columns indexed by Z and is given by

Wm,k = Tk−m. In this way, we see that a Haplitz weight for the case d = 1 reducesto a Laurent matrix [Tk−m]m,k∈Z. If we then solve the trigonometric momentproblem to produce a operator-valued measure µ on the unit circle such that

zjµ(z),

we then obtain a version of the spectral theorem for the unitary operator U .

3. Analytic intertwining operators between model spaces

Let W and W∗ be two Cuntz weights (or more generally positive semidefiniteHaplitz operators) with block-matrix entries equal to operators on Hilbert spacesE and E∗ respectively. Let us say that a bounded operator S : LW → LW∗

is anintertwining operator if

SUW,j = UW∗,jS, SU∗W,j = U∗

W∗,jS

for j = 1, . . . , d. While LW is the analogue of the Lebesgue space L2, the subspaceHW := closure in LW of WP(Fd × ∅, E) is the analogue of the Hardy spaceH2. Let us say that an intertwining operator S : LW → LW∗

with the additionalproperty S : HW → HW∗

is an analytic intertwining operator. Note that for d = 1with LW and LW∗

simply taken to be L2(Z, E) and L2(Z, E∗), the contractiveanalytic intertwining maps are multiplication operators with multiplier T fromthe Schur class S(E , E∗) of contractive, analytic, operator-valued functions on theunit disk. In analogy with this classical case, we denote by Snc(W, W∗) (the non-commutative Schur class associated with Cuntz weights W and W∗) the class ofall contractive, analytic, intertwining operators from LW to LW∗

. A particularlynice situation is when there is a formal power series T (z) so that

S(We) = W∗[T (z)e] for each e ∈ E . (3.1)

When we can associate a formal power series T (z) with the operator S∈Snc(W,W∗)in this way, we think of T (z) as the symbol for S and write S(z) = T (z). Thepurpose of this section is to work out some of the calculus for intertwining mapsS ∈ Snc(W, W∗) and their associated symbols T (z), and understand the conversedirection: given a power series T (z), when can the formula (3.1) be used to definean intertwining operator S ∈ Snc(W, W∗)? Two particular situations where thisoccurs are: (i) W∗ is a general positive Haplitz operator and T (z) is an analyticpolynomial (so we are guaranteed that T (z)e is in the domain of W∗), and (ii) W∗is a Haplitz extension of the identity and T (z)e ∈ L2(Fd, E∗) for each e ∈ E (soagain T (z)e is guaranteed to be in the domain of W∗). Here we say that the Haplitzoperator W∗ is a Haplitz extension of the identity if W∗;v,∅;α,∅ = δv,αIE∗

where δv,α

is the Kronecker delta-function equal to 1 for v = α and equal to 0 otherwise. Thesetting (ii) plays a prominent role in the analysis of Cuntz scattering systems andmodel and dilation theory for row-contractions in [3]. Our main focus here is onthe setting (i).

In general, given a formal power series T (z) =∑

w∈FdTwzw in the non-

commuting variables z = (z1, . . . , zd) with coefficients Tw equal to bounded oper-ators between two Hilbert spaces E and E∗ (Tw ∈ L(E , E∗) for w ∈ Fd), we defineMT : P(Fd, E) → L(Fd, E∗) by

(MT p)(z) = T (z) · p(z) :=∑

w∈Fd

⎛⎝ ∑

v,v′ : vv′=w

Tvpv′

⎞⎠ zw

if p(z) =∑

w∈Fdpwzw is a polynomial with coefficients in E . There are various

modes of extension of MT to spaces of formal power series in a double set of non-commuting variables (z, ζ) (with z = (z1, . . . , zd) and ζ = (ζ1, . . . , ζd) – here againwe are assuming that z’s do not commute among themselves, ζ’s do not commuteamong themselves, but ziζj = ζjzi for i, j = 1, . . . , d). One such operator is theLaurent operator LT defined as follows. We take the domain of LT to be the spaceP(Fd×Fd, E) of polynomials f(z, ζ) =

∑v,w∈Fd

fv,wzvζw ∈ L2(Fd×Fd, E) wherefv,w ∈ E vanishes for all but finitely many v, w ∈ Fd. Define LT on monomials by

LT (zvζwe) = SRv⊤

UR[∗]w⊤

T (z)e

and extend by linearity to P(Fd×Fd, E). Then LT |P(Fd×∅,E) = MT |P(Fd×∅,E).The defining properties of LT : P(Fd ×Fd, E) → L(Fd ×Fd, E) are:

LT |P(Fd×∅,E) = MT |P(Fd×∅,E), (3.2)

LT UR[∗]j = U

R[∗]j LT for j = 1, . . . , d on P(Fd ×Fd, E) (3.3)

LT SRj = SR

j LT for j = 1, . . . , d on P(Fd ×Fd, E). (3.4)

Viewing L(Fd × Fd, E) as the dual of P(Fd × Fd, E) in the L2-inner product, we

see that L[∗]T is well defined as an operator of the form L

[∗]T : P(Fd × Fd, E∗) →

L(Fd ×Fd, E).The following proposition gives some useful formulas for the action of LT and L

[∗]T .

Proposition 3.1. Let T (z) =∑

v∈FdTvz

v be a formal power series.

1. Then the action of the Laurent operator LT on a polynomialf(z, ζ) =

∑α,β fα,βzαζβ in P(Fd ×Fd, E) is given by

LT [f ](z, ζ) = T (ζ−1)f(z, ζ)− [T (ζ−1)f(z, ζ)]|ζ=0

+ T (z′)f(z, z′−1)|z′=z

where the z′-variables are taken to the left of the z-variables in the last termbefore the evaluation z′ = z.

2. The action of L[∗]T on a general polynomial g(z, ζ) =

∑v,w gv,wzvζw ∈ P(Fd×

Fd, E∗) is given by

L[∗]T [g](z, ζ) = T (ζ)∗ [g(z, ζ) + kper(z, ζ)g(z, 0)] (3.6)

where we use the convention T (ζ)∗ =∑

v∈FdT ∗

v ζv⊤

and where kper(z, ζ) is

the “perverse Szego kernel” as in (2.33).

Remark 3.2. We see that the formulas (3.5) and (3.6) actually define LT and L[∗]T

from all of L(Fd×Fd, E) into L(Fd×Fd, E∗) and from L(Fd×Fd, E∗) into L(Fd×Fd, E) respectively. This is analogous to the fact that the operator MT : f(z) →T (z)f(z) is well defined as an operator on formal power series MT : L(Fd, E) →L(Fd, E∗), as only finite sums are involved in the calculation of a given coefficientof T (z) · f(z) in terms of the (infinitely many) coefficients of f(z).

Proof. By definition, for e ∈ E and β ∈ Fd, we have

LT (ζβe) = UR[∗]β⊤

= UR[∗]β⊤

v∈Fd

UR[∗]k (T (z)e) = T∅ζke + T (z)z−1

UR[∗]j U

R[∗]k LT e = U

R[∗]j (T∅ζke + T (z)z−1

= T∅ζkζje + Tkζje + T (z)z−1k z−1

and then by induction

LT (ζβe) = (UR[∗])β⊤

LT (e) =∑

β′,β′′ : β=β′β′′,β′′ =∅Tβ′⊤ζβ′′

e + T (z)(z−1)β

β′∈Fd

Tβ′⊤(ζ−1)β′⊤

ζβe− Tβ⊤e + T (z)(z−1)βe

= T (ζ−1) · ζβe− [T (ζ−1) · ζβe]|ζ=0 + T (z′)(z′−1)β |z′=z

and formula (3.5) follows for the case where f(z, ζ) = ζβ .

For a general monomial f(z, ζ) = zαζβ , we have

LT (zαζβe) = LT (SR)α⊤

(UR[∗])β⊤

= (SR)α⊤((UR[∗])β⊤

= (SR)α⊤

⎛⎝ ∑

β′∈Fd

Tβ′⊤(ζ−1)β′

ζβe− Tβ⊤e + T (z)(z−1)β

⎞⎠ e

β′∈Fd

Tβ′⊤(ζ−1)β′

zαζβe− Tβ⊤zαe +(T (z)(z−1)β

= T (ζ−1) · zαζβe− [T (ζ−1)zαζβe]|ζ=0 + T (z′)zα(z′−1)β |ζ=z

and formula (3.5) follows for the case of a general monomial. The general case of(3.5) now follows by linearity.

To verify (3.6), we first compute the action of L[∗]T on a monomial; for

u, v, α, β ∈ Fd, e∗ ∈ E∗ and e ∈ E , compute

〈L[∗]T zvζwe∗, z

αζβe〉L2 = 〈zvζwe∗, LT zαζβe〉L2

= 〈zvζwe∗, (M∗T∗ζβ)zαe− Tβ⊤zαe + (T (z)(z−1)β)zαe〉L2

= δα,v〈MT∗ζwe∗, ζβe〉L2 − δα,vδw,∅〈T ∗

β⊤e∗, e〉E+ δw,∅〈zve∗, (T (z)(z−1)β)zαe〉L2 . (3.7)

From this we read off that, for w = ∅ we have

L[∗]T (zvζwe∗) = (MT∗ζw)zve∗ = T (ζ)∗zvζwe∗

By linearity we see that the formula (3.6) holds for the case when f ∈ P(Fd ×(Fd \ ∅), E), i.e., for the case where f(z, 0) = 0.

For the case where w = ∅, the second term on the right-hand side of (3.7)(with the sign reversed) is equal to

δα,v〈T ∗β⊤e∗, e〉E = δα,v〈T ∗

β⊤zve∗, zαe〉L2 = δα,v〈T (ζ)∗zve∗, z

αζβe〉L2

and thus exactly cancels with the first term. We are therefore left with

〈L[∗]T zve∗, z

αζβe〉L2 = 〈zve∗, T(vα−1)β⊤zve〉L2

= 〈e∗, T(vα−1)β⊤e〉E∗

= 〈(T(vα−1)β⊤)∗e∗, e〉Efrom which we deduce that

L[∗]T (zve∗) =

(T(vα−1)β⊤)∗zαζβe∗

v′,α : v=v′α

(Tv′β⊤)∗ζβzαe∗

v′,α : v=v′α

T ∗βv′⊤ζβzαe∗

v′,α : v=v′α

T (ζ)∗(ζ−1)v′

zαe∗

= T (ζ)∗[∑

(ζ−1)v′

(z−1)v′⊤

= T (ζ)∗ [zv + kper(z, ζ)zv]

which is formula (3.6) for the case where f(z, ζ) = zv. By linearity (3.6) holds iff(z, ζ) = f(z) =

∑v fv,∅zvζ∅. Since any f ∈ P(Fd×Fd, E) can be decomposed as

f(z, ζ) = [f(z, ζ)− f(z, 0)] + f(z, 0)

where f(z, ζ) − f(z, 0) ∈ P(Fd × (Fd \ ∅), E) and f(z, 0) ∈ P(F × ∅, E), wecan use linearity once again to see that (3.6) holds for all f ∈ P(Fd ×Fd, E).

There is a second type of extension of a multiplication operator MT definedas follows. Suppose that W∗ is a positive semidefinite Haplitz operator on E∗, W isa positive semidefinite Haplitz operator on E , and T (z) is a formal power series inthe z-variables only such that T (z)e is in the domain of the Haplitz operator W∗for each e ∈ E ; as was mentioned above, two cases where this occurs are (i) whenW∗ is general and T (z) is a polynomial, and (ii) when W∗ is a Haplitz extensionof the identity and MT is a bounded multiplier from L2(Fd, E) to L2(Fd, E∗). We

then define LW,W∗

T : WP(Fd ×Fd, E) →W∗P(Fd ×Fd, E∗) by

LW,W∗

T : Wp →W∗LT p. (3.8)

If it happens that ‖W∗LT p‖LW∗≤ M‖Wp‖LW for some M <∞, or equivalently,

M2W − L[∗]T W∗LT ≥ 0,

then LW,W∗

T is well defined and has a continuous extension (also denoted by LW,W∗

LW,W∗

T : LW → LW∗.

Moreover one easily checks that

LW,W∗

T UW,j = UW∗,jLW,W∗

T and LW,W∗

T U∗W,j = U∗

W∗,jLW,W∗

We shall be particularly interested in the case where LW,W∗

T defines a contraction,isometry or unitary operator between LW and LW∗

; the following propositionsummarizes the situation. The proof is an elementary continuation of the ideassketched above.

Proposition 3.3. Let W and W∗ be positive semidefinite Haplitz operators with ma-trix entries Wv,w;α,β and W∗;v,w;α,β equal to bounded operators on Hilbert spacesE and E∗ respectively, let T (z) =

∑v∈Fd

Tvzv be a non-commutative analytic poly-

nomial with coefficients Tv equal to bounded operators from E to E∗, and define

LW,W∗

T : WP(Fd ×Fd, E) → P(Fd ×Fd, E∗)by (3.8). Then:

1. LW,W∗

T extends by continuity to define a contraction operator from LW →LW∗

if and only if the Haplitz operator W−L[∗]T W∗LT is positive semidefinite.

2. LW,W∗

T extends by continuity to define an isometry from LW into LW∗if and

only if W − L[∗]T W∗LT = 0.

3. LW,W∗

T extends by continuity to define a unitary operator from LW onto LW∗

if and only if W −L[∗]T W∗LT = 0 and the subspace span W∗(z, ζ)T (z)e : e ∈

E is ∗-cyclic for UW∗.

In each of these cases, the extended operator LW,W∗

T : LW → LW∗has the additional

properties

LW,W∗

T UW,k = UW∗,kLW,W∗

T and LW,W∗

T U∗W,k = U∗

W∗,kLW,W∗

T for k = 1, . . . d.(3.9)

The next sequence of results will help us compute the symbol V (z, ζ) for a

Haplitz operator V of the form V = L[∗]T W∗LT .

Proposition 3.4. Assume that W∗ = [W∗;v,w;α,β ] is a selfadjoint Haplitz operator(with matrix entries W∗;v,w;α,β equal to bounded operators on the Hilbert space E∗)and that T (z) =

∑v∈Fd

Tvzv is a polynomial (so Tv = 0 for all but finitely many

v ∈ Fd) with coefficients equal to bounded operators from E into E∗. Then W∗LT ,

L[∗]T W∗ and L

[∗]T W∗LT are well defined [∗]-Haplitz operators with respective symbols

W∗LT (z, ζ) =W∗(z, ζ)T (z) + W∗(0, ζ)kper(z, ζ)T (z) (3.10)

[∗]T W∗(z, ζ) =T (ζ)∗W∗(z, ζ) + T (ζ)∗kper(z, ζ)W∗(z, 0) (3.11)

[∗]T W∗LT (z, ζ) =T (ζ)∗W∗(z, ζ)T (z) + T (ζ)∗[W∗(0, ζ)kper(z, ζ)T (z)] (3.12)

+ T (ζ)∗kper(z, ζ)[W∗(z, 0)T (z) + (W∗(0, z−1)− W∗;∅,∅)T (z)](3.13)

The latter formula can also be written in the more selfadjoint form

[∗]T W∗LT (z, ζ) = T (ζ)∗W∗(z, ζ)T (z)

+ T (ζ)∗[W∗(0, ζ)kper(z, ζ)T (z)] + [T (ζ)∗kper(z, ζ)W∗(z, 0)]T (z)

+ T (ζ)∗kper(z, ζ)[(W∗(0, z−1)− W∗;∅,∅)T (z)]

+ [T (ζ)∗(W∗(ζ−1, 0)− W∗;∅,∅)]kper(z, ζ)T (z)

+ T (ζ)∗W∗;∅,∅kper(z, ζ)T (z). (3.14)

In case W∗ is a [∗]-Haplitz extension of the identity, formulas (3.10), (3.11)and (3.13) simplify to

W∗LT (z, ζ) = W∗(z, ζ)T (z), (3.15)

[∗]T W∗(z, ζ) = T (ζ)∗W∗(z, ζ), (3.16)

[∗]T W∗LT (z, ζ) = T (ζ)∗W∗(z, ζ)T (z) + T (ζ)∗kper(z, ζ)T (z). (3.17)

Proof. The formula (3.10) for W∗LT (z, ζ) is an immediate consequence of for-

mula (2.34) in Proposition 2.7. Since L[∗]T W∗ = (W∗LT )[∗], the formula (3.11) for

[∗]T W∗(z, ζ) then follows by applying the general formula (2.43) to W∗LT .

To verify the formula (3.13) for

L[∗]T W∗LT (z, ζ), we compute

[∗]T W∗LT (z, ζ) = L

[∗]T

(W∗LT (z, ζ)

)(z, ζ)

= T (ζ)∗W∗(z, ζ)T (z) + T (ζ)∗[W∗(0, ζ)kper(z, ζ)T (z)]

+ T (ζ)∗kper(z, ζ)[W∗(z, 0)T (z) + [W∗(0, ζ)kper(z, ζ)T (z)]|ζ=0

= T (ζ)∗W∗(z, ζ)T (z) + T (ζ)∗[W∗(0, ζ)kper(z, ζ)T (z)]

+ T (ζ)∗kper(z, ζ)[W∗(z, 0)T (z) + (W∗(0, z−1)− W∗;∅,∅)T (z)

and (3.13) follows.

To see that (3.15), (3.16) and (3.17) follows from (3.10), (3.11) and (3.13),

observe that, for W∗(z, 0) = W∗(0, ζ) = I,

W∗(0, ζ)kper(z, ζ)T (z) = 0

T (ζ)∗kper(z, ζ)W∗(z, 0) = 0

W∗(0, z−1)− W∗;∅,∅ = 0.

To see that (3.14) is equivalent to (3.13), proceed as follows. First we notethe general identity, for g(ζ) =

∑w gwζw a formal power series in the variables

ζ = (ζ1, . . . , ζd) and f(z) =∑

v fvzv a formal power series in the variables z =

(z1, . . . , zd),

g(ζ)kper(z, ζ)f(z) =∑

⎡⎣∑

v′ =∅gwv′⊤fv′v

⎤⎦ zvζw. (3.18)

Next, let us write the right-hand side of (3.13) as k0(z, ζ) + k1(z, ζ) + k2(z, ζ) +k3(z, ζ) where

k0(z, ζ) = T (ζ)∗W∗(z, ζ)T (z)

k1(z, ζ) = T (ζ)∗[W∗(0, ζ)kper(z, ζ)T (z)]

k2(z, ζ) = T (ζ)∗kper(z, ζ)[W∗(z, 0)T (z)]

k3(z, ζ) = T (ζ)∗kper(z, ζ)[(W∗(0, z−1)− W∗;∅,∅)T (z)].

We use the general identity (3.18) to compute

k2(z, ζ) = T (ζ)∗kper(z, ζ)[W∗(z, 0)T (z)

= T (ζ)∗kper(z, ζ)

α′α′′=v

W∗;α′,∅Tα′′

⎡⎣ ∑

α′,α′′,v′∈S2

T(wv′⊤)⊤W∗;α′,∅Tα′′

⎤⎦ zvζw (3.19)

where we have set

S2 = (α′, α′′, v′) : α′α′′ = v′v, v′ = ∅.

Similarly,

k1(ζ, z)∗ =[T (ζ)∗kper(z, ζ)W∗(z, 0)

]T (z)

⎛⎝∑

⎡⎣∑

v′ =∅T ∗

(wv′⊤)⊤W∗;v′v

⎤⎦ zvζw

⎞⎠ ·(∑

Tαzα

⎡⎣ ∑

(β′,β′′,v′)∈S1

T ∗(wv′⊤)⊤W∗;v′β′Tβ′′

⎤⎦ zvζw

S1 = (β′, β′′, v′) : β′β′′ = v, v′ = ∅.Note that the map

ι : (β′, β′′, v′) → (α′, α′′, v′) := (v′β′, β′′, v′)

maps S1 injectively into S2, with the remainder set S2 \ ιS1 be given by

S2 \ ιS1 = (α′, α′′, v′) : α′α′′′ = v′ for some α′′′ = ∅, α′′ = α′′′v.

When forming the difference k2(z, ζ) − k1(ζ, z)∗, the terms in k2(z, ζ) over in-dices in ι(S1) exactly cancel with the terms of k1(ζ, z)∗ and we are left with the“associativity defect”

k2(z, ζ)− k1(ζ, z)∗ =∑

⎡⎣ ∑

(α′,α′′,v′)∈S2\ιS1

T(wv′⊤)⊤Wα′,∅Tα′′

⎤⎦ zvζw. (3.20)

On the other hand, we compute, again using (3.18),

k3(ζ, z)∗ =[T (ζ)∗(W (ζ−1, 0)− W∅,∅)

]kper(z, ζ)T (z)

⎛⎝∑

⎡⎣∑

v′′ =∅T(wv′′⊤)⊤Wv′′,∅

⎤⎦ ζw

⎞⎠ kper(z, ζ)T (z)

⎡⎣∑

v′ =∅

v′′ =∅T ∗

(wv′⊤v′′⊤)⊤Wv′′,∅Tv′v

⎤⎦ zvζw. (3.21)

A close comparison of (3.20) and (3.21) shows that

k2(z, ζ)− k1(ζ, z)∗ − k3(ζ, z)∗ =∑

⎡⎣ ∑

α′′′ =∅T ∗

α′′′⊤w⊤W∗;∅,∅Tα′′′v

⎤⎦ zvζw

= T (ζ)∗W∗;∅,∅kper(z, ζ)T (z) =: k4(z, ζ). (3.22)

By using this last identity (3.22) we now get

k0(z, ζ) + k1(z, ζ) + k2(z, ζ) + k3(z, ζ) = k0(z, ζ) + [k1(z, ζ) + k1(ζ, z)∗]

+ [k2(z, ζ)− k1(ζ, z)∗] + k3(z, ζ)

= k0(z, ζ) + [k1(z, ζ) + k1(ζ, z)∗]

[k3(z, ζ)∗ + k4(z, ζ)] + k3(z, ζ)

= k0(z, ζ) + [k1(z, ζ) + k1(ζ, z)∗]

+ [k3(z, ζ), +k3(ζ, z)∗] + k4(z, ζ)

where the last line is exactly equal to the right-hand side of (3.14). Thus (3.13)and (3.14) are equivalent as asserted.

Finally, suppose that W∗ is a Haplitz extension of the identity, i.e., W∗;v,∅;α,∅=δv,αIE∗

. From (2.19) we see that

W∗;v,w = W∗;v,w;∅,∅ = W∗;v,∅;w⊤,∅ = δv,wIE∗.

In particular W∗;∅,w = δ∅,wIE∗and W∗;v,∅ = δv,∅IE∗

from which we get W∗(z, 0) =

IE∗z∅ and W∗(0, ζ) = IE∗

ζ∅. Hence we are in the situation where formula (2.35)

applies for the action of W on an analytic polynomial. With this simplification,formulas (3.15), (3.16) and (3.17) follow as specializations of the more generalformulas (3.10), (3.11) and (3.13).

Remark 3.5. From the identity (2.43) we know that the kernel

k(z, ζ) :=

L[∗]T W∗LT (z, ζ)

must satisfy k(z, ζ) = k(ζ, z)∗ (since L[∗]T W∗LT = (L

[∗]T W∗LT )[∗]). While this prop-

erty is not apparent from the first formula (3.13) for k(z, ζ), it is apparent fromthe second formula (3.13).

4. Application: Free atomic/permutative representationsof the Cuntz algebra Od

We illustrate the machinery developed in the preceding sections by applying it tofree atomic representations of the Cuntz algebra Od studied in [9], a class more gen-eral than but closely related to the permutative representations of Od studied in [7].

We first remark that a representation π of the C∗-algebra known as the CuntzalgebraOd amounts to a specification of a Hilbert space Kπ together with a d-tupleUπ = Uπ,1, . . . ,Uπ,d of operators on Kπ which form a row-unitary operator:

U∗π,iUπ,j = δi,jIK,

Uπ,kU∗π,k = IK

(see, e.g., [8]). To simplify the terminology, we generally shall say “row-unitary”rather than “representation of the Cuntz algebra Od”.

We introduce the class of free atomic representations of Od studied by David-son and Pitts [9] in the framework used by Bratteli and Jorgensen [7] for the studyof the special case of permutative representations. We let K be a Hilbert space withorthonormal basis ei : i ∈ I indexed by some index set I. Let σ = (σ1, . . . , σd)be a function system of order d on I; by this we mean that each σk : I → I is aninjective function from I to I

σk(i) = σk(i′) for some i, i′ ∈ I =⇒ i = i′ for each k = 1, . . . , d, (4.1)

the σk’s have pairwise disjoint images

imσk ∩ imσℓ = ∅ for k = ℓ in 1, . . . , d, (4.2)

and the union of the images of the σk’s is all of I:∪d

k=1 imσk = I. (4.3)

In addition assume that we are given a collection of numbers

λ = λk,i ∈ T : k = 1, . . . , d; i ∈ Ieach of modulus 1 indexed by 1, . . . , d×I. For convenience we shall refer to thecombined set (σ, λ) as an iterated function system. We then define a weighted shiftoperator Uk (for k = 1, . . . , d) on K by extending by linearity the action on thebasis vectors ei : i ∈ I given by

Uk : ei → λk,ieσk(i). (4.4)

Then, by using that each λk,i has modulus 1 and the properties (4.1)–(4.3) of aniterated function system, it is easy to see that U = (U1, . . . ,Ud) is row-unitary.When we wish to make the dependence on σ and λ = λk,i : k = 1, . . . , d; i ∈ Iexplicit, we write Uσ,λ = (Uσ,λ

1 , . . . ,Uσ,λd ) in place of U = (U1, . . . ,Ud).

We shall need a certain calculus associated with an iterated function system(σ, λ). For a given i ∈ I there is a unique k = k(i) ∈ 1, . . . , d such that i = σk(i′)for some (necessarily unique) i′ ∈ I. Then we write i′ = σ−1

k (i). For v ∈ Fd a wordof the form v = gkn · · · gk1 (with k1, . . . , kd ∈ 1, . . . , d), we define σv as thecomposition of maps σv = σkn · · · σk1 . If k′ is an element of 1, . . . , d notequal to this particular k, then we say that σ−1

k (i) is undefined (or empty). Moregenerally, given i ∈ I and a natural number n, there is a unique word v = gkn . . . gk1

in Fd of length n so that σv(i′) = i for some (necessarily unique) i′ ∈ I. When

this is the case we then write i′ = (σv)−1(i) = (σ−1)v⊤

(i). If v′ is another word

in Fd of length n not equal to this particular v, then we say that (σ−1)v′⊤

(i) isundefined or empty.

This calculus extends to the set of modulus-1 multipliers λ as follows. If gk isa single letter (word of length 1) in Fd and i ∈ I, we define λgk

i = λk,i. Inductively,if v = gkn · · · gk1 ∈ Fd is a word of length n and i ∈ I, we define

λvi = λkn,σkn−1

···σk1(i) · · ·λk2,σk1

(i)λk1,i.

Similarly, given v = gkn · · · gk1 ∈ Fd and i ∈ I, if there is some (necessarily unique)i′ ∈ I so that σv(i′) = i, then we define

λv⊤

i = λk1,(σ−1)v⊤ (i) · · ·λkn−1,σ−1kn

(i)λkn,i;

otherwise, we set λv⊤

i equal to 1. Properties of this calculus then are

λvv′

i = λvσv′ (i)

λv′

i , λvi′ = (λ

i ) if σv(i′) = i. (4.5)

This calculus gives us a simple notation for calculating products of Uσ,λ1 , . . . ,Uσ,λ

on basis vectors ei; indeed, if we simplify the notation by writing simply U forUσ,λ, we have

Uvei = λvi eσv(i),

U∗v⊤

ei = λv⊤

i e(σ−1)v⊤ (i). (4.6)

Here it is understood that e(σ−1)v⊤ (i) = 0 if (σ−1)v⊤

(i) is undefined.

The calculus is further streamlined if we introduce some more formalism.As was already used above, we know that, for a given i ∈ I, there is a uniquek = k(i) ∈ 1, . . . , d such that i = σk(i′) for some (necessarily unique) i′ ∈I. Iteration of this observation enables to define inductively a function ρ : I →(1,...,d×I)N by

ρ(i) = (k1, i1), (k2, i2), . . . , (kn, in), . . . (4.7)

by the conditions

σk1(i1) = i, σk2(i2) = i1, . . . , σkn+1(in+1) = in, . . . .

On occasion we need to separate the first and second components of ρ as fol-lows. Define ρ1 : I → g1, . . . , gdN (viewed as the set of infinite words in thealphabet g1, . . . , gd having a beginning on the left but no end on the right) andρ2 : I → IN by

ρ1(i) = gk1gk2 · · · gkn · · · , (4.8)

ρ2(i) = i1, i2, . . . , in, . . . (4.9)

if ρ(i) is as in (4.7). After a cosmetic change of notation, the function ρ1 is exactlythe object associated with the iterated function system σ termed the coding func-tion by Bratteli-Jorgensen in [7]. The function ρ (4.7) has already appeared in animplicit way in the (σ, λ)-calculus described in the previous paragraph. Indeed, if

v is a word of length n in Fd, then (σ−1)v⊤

(i) is defined exactly when ρ1(i) has the

form ρ1(i) = vv′ for some v′ ∈ g1, . . . , gdN. Also, one can compute λv⊤

i directlyin terms of the original set of multipliers λ = λk,i : k ∈ 1, . . . , d, i ∈ I via

λv⊤

i = λk1,i1 · · ·λkn,in

where (k1, i1), (k2, i2), . . . , (kn, in) are the first n terms of the sequence ρ(i).

As a consequence of (4.6) it is easy to see that: given a subset I ′ ⊂ I,the smallest subspace H of K containing each of the basis vectors ei for i ∈ I′and reducing for Uσ,λ is the subspace H := closed span ei : i ∈ I′′ where I ′′ =

∪v,w∈Fdσw(σ−1)v⊤

(I ′). An application of this general principle tells us that a sub-set I ′′ ⊂ I has the property that the associated subspace H := closed span ei : i ∈I ′′ ⊂ K is reducing for Uσ,λ if and only if I ′′ has the property

I ′′ =

σk(I ′′). (4.10)

A consequence of this latter statement is: for a given i ∈ I, the basis vector ei

is ∗-cyclic for Uσ,λ if and only if the action of σ on I is ergodic, i.e., the onlysubset I ′′ ⊂ I having the property (4.10) is I ′′ = ∅ and I ′′ = I. Note that thislatter statement is independent of the choice of i ∈ I; thus ei is ∗-cyclic for Uσ,λ

for some i ∈ I if and only if ei is ∗-cyclic for each i ∈ I. In general I partitionsinto ergodic subsets: I = ∪Iα : α ∈ A where Iα ∩ Iα′ = ∅ for α = α′, Iα isinvariant under both σ and σ−1, and the restriction of σ to Iα is ergodic for eachα ∈ A. Then the corresponding row-unitary d-tuple Uσ,λ splits as a direct sumUσ,λ = ⊕α∈AUσα,λα . Hence, for purposes of studying the problem of classificationup to unitary equivalence of the row-unitary d-tuples Uσ,λ, there is no loss ofgenerality in assuming that I is ergodic with respect to σ, i.e., the only subsetsI ′′ of I satisfying (4.10) are ∅ and the whole set I.

Thus throughout the rest of this section we assume that σ is ergodic on I.Let us now fix an i ∈ 1, . . . , d; then ei is ∗-cyclic with respect to Uσ,λ. Our nextgoal is to compute the symbol W〈ei〉(z, ζ) of U = Uσ,λ with respect to the cyclicsubspace 〈ei〉 = span ei. The coefficients of W〈ei〉 by definition are operators onthe one-dimensional space 〈ei〉; we identify the element cei of 〈ei〉 with the complexnumber c ∈ C and an operator cei → d · cei on 〈ei〉 with the complex number d.Under these conventions, we compute

W〈ei〉(z, ζ) =∑

v,w∈Fd

(P〈ei〉UwU∗v⊤ |〈ei〉

)zv⊤

P〈ei〉

i e(σ−1)v⊤ (i)

)zv⊤

v,w : σw((σ−1)v⊤ (i))=i

λw(σ−1)v⊤ (i)

λv⊤

i zv⊤

(i′,v,w)∈Sλw

i′ λv⊤

i zv⊤

ζw (4.11)

where we have set

S := (i′, v, w) : i′ ∈ 1, . . . , d, v, w ∈ Fd with σv(i′) = σw(i′) = i. (4.12)

If ρ(i) = (k1, i1), (k2, i2), . . . , (kn, in), . . . , we let x be the infinite word x =ρ1(i) = gk1gk2 · · · gkn · · · (see (4.7) and(4.8)). For each m = 0, 1, 2, . . . , let xm bethe finite word of length m given by

xm = gk1 · · · gkm

(with x0 = ∅) for each m = 0, 1, 2, . . . . To complete the notation, set i0 = i.Suppose that i′ ∈ 1, . . . , d and that v and w are words in Fd with |w| ≥ |v|.From the definitions we see that (i′, v, w) ∈ S if and only if there are non-negativeintegers j and N so that

v = xj , w = xj+N =: xjy, ij = ij+N = i′. (4.13)

Note that (4.13) is trivially true if we take N = 0 and j = 0, 1, 2, . . . arbitrary;from this observation we see that

D := (im, xm, xm) : m = 0, 1, 2, . . . ⊂ S. (4.14)

Suppose next that (i′, v, w) = (i′, xj , xj+N ) := (i′, xj , xjy) is in S for some N > 0.Then by definition we have

i = σxj (i′) = σxjy(i′)

from which it follows that σy(i′) = i′. From the rule defining x = ρ1(i), we seenext that

x = xjyyy · · · =: xjy∞.

In the statements below, we shall use the following notation. Given a triple ofparameters (x, y, λ) with x, y ∈ Fd and λ a complex number of modulus 1 (λ ∈ T),

we let W x,y,λ(z, ζ) denote the symbol

W x,y,λ(z, ζ) =∑

α : x=αα′,α′ =∅ζαzα⊤

β : ββ′=y,β′ =∅

m,m≥0

λmλm

z(xymβ)⊤ζxymβ

(4.15)

and W x,y,λ = [W x,y,λv,w;α,β ] the associated [∗]-Haplitz operator as in (2.29) in Proposi-

tion 2.6. In case x = ∅, we shall abbreviate W ∅,y,λ(z, ζ) to W y,λ(z, ζ), and similarlyW ∅,y,λ to W y,λ; thus

W y,λ(z, ζ) =∑

β : ββ′=y,β′ =∅

m,m≥0

λmλm

z(ymβ)⊤ζymβ . (4.16)

Similarly, if x = x1x2x3 . . . xn . . . is an infinite word (with letters xj ∈ g1, . . . , gdfor j ∈ N), we let W x(z, ζ) denote the symbol

W x(z, ζ) =

∞∑

zx⊤mζxm (4.17)

with associated [∗]-Haplitz operator W x. We are well on our way to proving thefollowing result.

Proposition 4.1. Suppose that (σ, λ) is an iterated function system with the actionof σ = (σ1, . . . , σd) on the index set I ergodic as above. Fix an index i ∈ I andthe associated ∗-cyclic vector ei for the row-unitary d-tuple Uσ,λ and let

ρ(i) = (k1, i1), (k2, i2), . . . , (kn, in), . . .

with x equal to the infinite word

x = ρ1(i) = gk1gk2 · · · gkn · · ·as in (4.7) and (4.8) and xm = gk1 · · · gkm ∈ Fd as above. Then:

Case 1: The orbit-eventually-periodic case: Suppose that there are integers j ≥ 0and N > 0 so that

(kj′+N , ij′+N ) = (kj′ , ij′) for all j′ ≥ j.

Assume also that j and N are chosen to be minimal with these properties. Thenx has the form

x = xjy∞

for a word y ∈ Fd of length N such that y has no common tail with xj:

x = x′v and y = y′v =⇒ v = ∅,the set S defined by (4.12) is given by

S =(i|α|, α, α) : xj = αα′ where α′ = ∅⋃(i|xj|+|β|, xjy

mβ, xjymβ) : m, m = 0, 1, 2, . . . , and y = ββ′ with β′ = ∅,

and the symbol W〈ei〉(z, ζ) for the row-unitary Uσ,λ with respect to the cyclic sub-space 〈ei〉 has the form

W〈ei〉(z, ζ) = W xj ,y,λ(z, ζ) (4.18)

where λ = λyσxj (i)

and W x,y,λ(z, ζ) in general is as in (4.15). Hence Uσ,λ is unitar-

ily equivalent to the model row-unitary UW x,y,λ on the space LW x,y,λ (with x = xj,λ = λy

i and ρ1(i) = xmy∞ as above).

Case 2: The orbit-non-periodic case: Suppose that there is no integer N > 0 as inCase 1. Then the S in (4.12) is given by

S = D := (im, xm, xm) : m = 0, 1, 2, . . .

and the symbol W〈ei〉(z, ζ) for the row-unitary Uσ,λ with respect to the cyclic sub-space 〈ei〉 has the form

W〈ei〉(z, ζ) = W x(z, ζ). (4.19)

with W x(z, λ) as in (4.17). Hence in this case Uσ,λ is unitarily equivalent to themodel row-unitary UW x on the space LW x (where x = ρ1(i) as above).

Proof. The description of the set S for the two cases follows from the discussionimmediately preceding the statement of Proposition 4.1. One can check from thedefinitions that the lack of a common tail between xj and y in the orbit-eventually-periodic case is forced by the assumed minimality in the choice of j and N . Theformulas for W〈e〉(z, ζ) then follow by plugging into (4.11) and using the properties(4.5) of the λ-functional calculus. The statements on unitary equivalence thenfollow from Theorem 2.10.

Remark 4.2. In the orbit-eventually-periodic case, we see that x = ρ1(i) has theform x = xjy

∞ for some j ≥ 0 and some word y ∈ Fd. Here the length of yis chosen equal to the smallest possible eventual period for ρ(i) (see (4.7)). Thesmallest possible eventual period for x = ρ1(i) (see (4.8)) may well be smaller: inthis case y has the form y = un for some shorter word u (and then |y| = n|u|). Aswe shall see below and is obtained in [9], in this case Uσ,λ is reducible, despite theassumption that (σ, λ) is ergodic. Indeed, if (σ, λ) is ergodic, then Uσ,λ can haveno non-trivial reducing subspaces spanned by a subset of the shift orthonormalbasis ei : i ∈ I; non-trivial reducing subspaces not in the span of any subset ofei : i ∈ I can occur.

Similarly, in the orbit-non-periodic case, the assumption is that ρ(i) (as in(4.7)) is never eventually periodic. This does not eliminate the possibility thatx = ρ1(i) (as in (4.8)) is eventually periodic; in this case, as we shall see belowand is obtained in [9], Uσ,λ is a direct integral of orbit-periodic representations,and hence has many non-trivial reducing subspaces. If (σ, λ) is ergodic, then noneof these non-trivial reducing subspaces can have an orthonormal basis consistingof a subset of the shift basis ei : i ∈ I.

A consequence of Proposition 4.1 is that, for purposes of classifying Uσ,λ up tounitary equivalence, we need study only the model operators UW x,y,λ (for the orbit-eventually-periodic case) and UW x (for the non-periodic case). In particular, we seethat the unitary-equivalence class of Uσ,λ depends only on the three parameters,x, y, λ (where x, y ∈ Fd and λ ∈ T) in the orbit-eventually-periodic case, or thesingle parameter x (where x is an infinite word) in the orbit-non-periodic case.Conversely, given any parameters (x, y, λ) with x, y ∈ Fd and λ ∈ T, there existsan ergodic iterated function system (σ, λ) on an index set I with ∗-cyclic vectorei so that ρ1(i) = xy∞, λ = λy

σx(i) and such that ρ(i) is periodic (with period

|y| once the index j satisfies j > |x|); indeed, one can see that the representationσu,λ written down in [9] (with an appropriate choice of ∗-cyclic vector) is of thistype, or one can use the details of the model row-operator UW x,y,λ on the modelspace LW x,y,λ (see Remark 4.3 below for a brief sketch). Similarly, given an infiniteword x = gk1gk2 · · · gkn · · · , there is an ergodic iterated function system (σ, 1) withρ1(i) = x and with ρ(i) aperiodic; one can check that a representation of theform πx as defined in [9] has this property, or one can work with the model UW x

on the model space LW x as outlined in Remark 4.3 below. Let us denote anyrow-unitary of the first type as Ux,y,λ (abbreviated to Uy,λ if x = ∅) and of thesecond type as Ux.

Remark 4.3. If W = W x,y,λ (with x and y equal to finite words and λ ∈ T)

or if W = W x (with x equal to an infinite word), then it is a direct verifica-tion that there exists a formal power series Y (z) (with coefficients equal to infi-nite block-rows) in the single set of non-commuting variables z = (z1, . . . , zd) so

that W (z, ζ) = Y (ζ)Y (z)∗, and that W (z, ζ) has zero Cuntz defect (W (z, ζ) =∑dk=1 z−1

k W (z, ζ)ζ−1k ). Hence, UW on LW defines a row-unitary d-tuple. A further

direct verification shows that UW on LW is atomic, i.e.,

UwWU∗v

W (W [1]) = W (zwζv)

is an orthonormal basis, up to having many zero vectors and having many rep-etitions up to a unimodular factor. Thus the index set for a shift basis for UW

is I = (Fd × Fd)/ ∼=, where the rule ∼= drops elements (v, w) of Fd × Fd forwhich W [zvζw] = 0 and identifies elements (v, w) and (v′, w′) in Fd×Fd for which

W [zvζw] and W [zv′

ζw′

] are the same up to a unimodular factor. The row-unitaryUW then induces an iterated function system (σ, λ) on the index set I/ ∼=, fromwhich we recover

ρ((∅, ∅)) periodic with period |y| once n > |x|, ρ1((∅, ∅)) = xy∞, λ = λyσx((∅,∅))

if W = W x,y,λ,

ρ((∅, ∅)) non-periodic, ρ1(∅, ∅) = x if W = W x.

In this way, starting with parameters (x, y, λ) or x, we produce a “functionalmodel” for the free-atomic representations of Od given in [7, 9].

To sort out how these model operators are related, we first need to understandhow these various symbols are related. The following proposition sorts these issuesout.

Proposition 4.4. The following relations among symbols of the form W x,y,λ(z, ζ),

W y,λ(z, ζ) and W x(z, ζ) as in (4.15), (4.16) and (4.17) hold:

1. Let x, y ∈ Fd and λ ∈ T given. Assume that x does not share a tail with y,i.e.,

x = x′γ and y = y′γ for some γ ∈ Fd =⇒ γ = ∅.If we set T (z) = zx⊤

, then

W x,y,λ(z, ζ) = [L[∗]T W y,λLT ]∧(z, ζ). (4.20)

2. For x′ ∈ Fd and an infinite word x = x1x2x3 . . . xn . . . with xj ∈ g1, . . . , gdfor each j ∈ N given, if we set T (z) = zx′⊤

, then

W x′x(z, λ) = [L[∗]T W xLT ]∧(z, ζ). (4.21)

3. For y′, y ∈ Fd and λ ∈ T with y′ a cyclic permutation of y (so y = γγ′ and

y′ = γ′γ for some γ ∈ Fd), if we set T (z) = zγ⊤

, then

W y,λ(z, ζ) = [L[∗]T W y′,λLT ]∧(z, ζ). (4.22)

4. For y ∈ Fd given of the form y = un for some u ∈ Fd, then

W y,λ(z, ζ) =1

µ : µn=λ

Wu,µ(z, ζ). (4.23)

5. Given an infinite word x = x1x2x3 · · ·xn · · · of the form x = uuu · · · =: u∞

for some u ∈ Fd, then

W x(z, ζ) =

Wu,µ(z, ζ) dm(µ) (4.24)

where m is normalized Lebesgue measure on the unit circle T.

Proof of (4.20). Consider (4.20). Note that

W y,λ(0, ζ) =∑

λmζym

, W y,λ(z, 0) =∑

z(ym)⊤ .

Since x does not share a tail with y, one easily checks that

W y,λ(0, ζ)kper(z, ζ)zx⊤

= 0 and(W y,λ(0, z−1)− W y,λ

∅,∅

)zx⊤

Similarly, we see that

ζxkper(z, ζ)[λm

z(ym)⊤zx⊤

] = 0 for 0 = m

and hence

ζxkper(z, ζ)

⎡⎣∑

z(ym)⊤zx⊤

⎤⎦ = ζxkper(z, ζ)zx⊤

We now have all the pieces to plug into the formula (3.13) for [L[∗]T W y,λLT ]∧(z, ζ);

the result is

[L[∗]T W y,λLT ]∧(z, ζ) = ζxW y,λ(z, ζ)zx⊤

+ ζx[W y,λ(0, ζ)kper(z, ζ)zx⊤

+ ζxkper(z, ζ)[W y,λ(z, 0)zx⊤

+ ζxkper(z, ζ)[(

W y,λ(0, z−1)− W y,λ∅,∅

)zx⊤]

= ζxW y,λ(z, ζ)zx⊤

+ ζxkper(z, ζ)[W y,λ(z, 0)zx⊤

+ ζxkper(z, ζ)

⎡⎣∑

z(ym)⊤zx⊤

⎤⎦

= W x,y,λ(z, ζ)

as asserted, and (4.20) follows.

Proof of (4.21). We next consider (4.21). Note that W x is a [∗]-Haplitz extensionof the identity. Hence, we may apply (3.17) to compute

[L[∗]T W xLT ]∧(z, ζ) = ζx′

W x(z, ζ)zx′⊤

+ ζx′

kper(z, ζ)zx′⊤

= ζx′

W x(z, ζ)ζx′⊤

α : αα′=x′,α′ =∅ζαzα⊤

= W x′x(z, ζ)

and (4.21) follows as wanted.

Proof of (4.22). Now suppose that y = γγ′, y′ = γ′γ in Fd and we wish to check

(4.22) as follows. By (3.13) applied to the case where W∗ = W y′,λ and T (z) = zγ⊤

,we know that

[L[∗]T W y′,λLT ]∧(z, ζ) = k1(z, ζ) + k2(z, ζ) + k3(z, ζ) + k4(z, ζ) (4.25)

k1(z, ζ) = ζγW y′,λ(z, ζ)zγ⊤

k2(z, ζ) = ζγ[W y′,λ(0, ζ)kper(z, ζ)zγ⊤

k3(z, ζ) = ζγkper(z, ζ)[W y′,λ(z, 0)zγ⊤

k4(z, ζ) = ζγkper(z, ζ)[(

W y′,λ(0, z−1)−W y′,λ∅,∅

)zγ⊤].

Plugging in the definition (4.16) of W y′,λ(z, ζ) into the formula for k1(z, ζ) thengives

k1(z, ζ) = ζγ

⎡⎣ ∑

α : y′=αα′,α′ =∅

m,m≥0

λmλm

z(y′mα)⊤ζy′mα

⎤⎦ zγ⊤

α : y′=αα′,α′ =∅

mm≥0

λmλm

z(γy′mα)⊤ζγy′mα

α : y′=αα′,α′ =∅

m,m≥0

λmλm

z(ymγα)⊤ζymγα (4.26)

where we used the identity γy′m = ymγ,

= k11(z, ζ) + k12(z, ζ) (4.27)

k11(z, ζ) =∑

α : α=γ′u,uu′=γ,u′ =∅

m,m≥0

λmλm

z(ymγα)⊤ζymγα

u : uu′=γ,u′ =∅

mm≥0

λmλm

z(ym+1u)⊤ζym+1u

u : uu′=γ,u′ =∅

m,m≥0

λm+1λm+1

z(ym+1u)⊤ζym+1u

where we use that λλ = 1,

(β,m,m)∈S11

λmλm

z(ymβ)⊤ζymβ (4.28)

S11 = (β, m, m) : ββ′ = y where β′ = u′γ′ for some u′ = ∅, m, m ≥ 1,while

k12(z, ζ) =∑

α : αv=γ′

mm≥0

λmλm

z(ymγα)⊤ζymγα

(β,m,m)∈S12

λmλm

S22 = (β, m, m) : ββ′ = y with β = γα for some α, m, m ≥ 0.To analyze k2(z, ζ) we first compute

W y′,λ(0, ζ)kper(z, ζ)zγ⊤

⎛⎝∑

λmζy′m

⎞⎠ · kper(z, ζ)zγ⊤

⎛⎝∑

λmζy′m

⎞⎠ ·

α : αα′=γ,α′ =∅(ζ−1)α′⊤

zα⊤

⎛⎝∑

λm+1ζy′m

⎞⎠ ζγ′

⎛⎝ ∑

α : αα′=γ,α′ =∅ζαzα⊤

⎞⎠

and hence

k2(z, ζ) = ζγ ·

⎛⎝∑

λm+1ζy′m

⎞⎠ ζγ′

⎛⎝ ∑

α : αα′=γ,α′ =∅ζαzα⊤

⎞⎠

⎛⎝∑

λm+1ζγy′mγ′

⎞⎠ ·

⎛⎝ ∑

α : αα′=γ

ζαzα⊤

⎞⎠

⎛⎝∑

λm+1ζym+1γ′

⎞⎠ ·

⎛⎝ ∑

α : αα′=γ

ζαzα⊤

⎞⎠

(β,m,m)∈S2

λmλm

S2 = (β, m, m) : ββ′ = γ with β′ = α′γ′ and α′ = ∅, m ≥ 1, m = 0.Similarly, to analyze k3(z, ζ) we first note that

W y′,λ(z, 0)zγ⊤

⎛⎝∑

z(y′m)⊤

⎞⎠ · zγ⊤

z(γy′m)⊤ ,

and hence

k3(z, ζ) = ζγ · kper(z, ζ) ·[W y′,λ(z, 0)zγ⊤

⎛⎝ ∑

u,uu′=γ,u′ =∅ζu(z−1)u

⎞⎠ ·

⎛⎝∑

z(y′(m)⊤zγ⊤

⎞⎠

= k31(z, ζ) + k32(z, ζ) (4.31)

k31(z, ζ) =∑

u,uu′=γ,u′ =∅ζu(z−1)u′

zγ⊤

u,uu′=γ,u′ =∅ζuzu⊤

(β,m,m)∈S31

λmλm

S31 = (β, m, m) : m = m = 0, ββ′ = y with β′ = u′γ′ and u′ = ∅,while

k32(z, ζ) =

⎛⎝ ∑

u,uu′=γ,u′ =∅ζu(z−1)u

⎞⎠ ·

⎛⎝∑

z(y′(m)⊤zγ⊤

⎞⎠

⎛⎝ ∑

u,uu′=γ,u′ =∅ζu(z−1)u′

⎞⎠ zγ⊤

zγ′⊤

⎛⎝∑

z(γy′m)⊤

⎞⎠

⎛⎝ ∑

⎞⎠ ·(λ

m+1z(γy′mγ′)⊤

⎛⎝ ∑

⎞⎠ ·

⎛⎝∑

z(ym+1)⊤

⎞⎠

(β,m,m)∈S32

λmλm

S32 = (β, m, m) : m = 0, m ≥ 1, ββ′ = y with β′ = u′γ′ for some u′ = ∅.

We next analyze k4(z, ζ). First note that

W y′,λ(0, z−1)−W y′,λ∅,∅ =

λm(z−1)ym

As y = γγ′ with γ′ = ∅ by assumption, we see that (z−1)ym

zγ⊤

= 0 for all m ≥ 0from which we see that

k4(z, ζ) = ζγkper(z, ζ)

⎡⎣⎛⎝∑

λm(z−1)ym

⎞⎠ · zγ⊤

⎤⎦

= ζγkper(z, ζ) [0] = 0. (4.34)

Combining (4.27), (4.28), (4.29), (4.30), (4.31), (4.32), (4.33) and (4.34) we seethat

[L[∗]T W y′,λLT ]∧(z, ζ) =

(β,m,m)∈S11∪S12∪S2∪S31∪S32

λmλm

z(ymβ)⊤ζymβ . (4.35)

On the other hand, by definition

W y,λ(z, ζ) =∑

(β,m,m)∈Sλmλ

mz(ymβ)⊤ζymβ (4.36)

S = (β, m, m) : m ≥ 0, m ≥ 0, ββ′ = y for some β′ = ∅.Now it is a simple matter to check that S11, S12, S2, S31 and S32 forms a parti-tioning of S:

S = S11 ∪ S12 ∪ S2 ∪ S31 ∪ S32 with S11,S12,S2,S31,S32 pairwise disjoint.

This combined with (4.35) and (4.36) immediately gives (4.22) as wanted.

Proof of (4.23). To verify (4.23) we shall use the character formula for the dualof finite cyclic group: given µ ∈ T,

µ : µn=λ

µrµr =

0, r = r

1, r = r(4.37)

for r, r = 0, 1, . . . , n− 1. Assuming now that y = un in Fd, we compute

µ : µn=λ

Wu,µ(z, ζ) =1

µ : µn=1

α : αα′=u,α′ =∅

k,k≥0

µkµkz(ukα)⊤ζukα.

Write k = mn+r and k = mn+r, where r, r = 0, 1, . . . , n−1 and m, m = 0, 1, 2, . . . .Then, continuing the computation above gives

µ : µn=λ

Wu,µ(z, ζ)

µ : µn=λ

α:αα′=u,α′ =∅

n−1∑

m,m≥0

[µnm+rµnm+rz(unm+rα)⊤ζunm+rα

α : αα′=u,α′ =∅

n−1∑

m,m≥0

⎡⎣⎛⎝ 1

µ : µn=λ

µrµr

⎞⎠ · λmλ

mz(ymurα)⊤ζymurα

⎤⎦

α : αα′=u,α′ =∅

n−1∑

m,m≥0

λmλm

z(ymurα)⊤ζymurα

where we used the character formula (4.37)

β : ββ′=y,β′ =∅

m,m≥0

λmλm

z(ymβ)⊤ζymβ

where we made the change of variable β = urα

W y,λ(z, ζ)

and (4.23) follows.

Proof of (4.24). This time we use the orthogonality relations for the circle group:

µmµm dm(µ) =

0, m = m

1, m = mfor m, m = 0,±1,±2, . . . . (4.38)

We then compute

Wu,µ(z, ζ) dm(µ) =

⎡⎣ ∑

α : αα′=u,α′ =∅

m,m≥0

µmµmz(umα)⊤ζumα

⎤⎦ dm(µ)

α : αα′=u,α′ =∅

⎡⎣ ∑

m,m≥0

µmµm dm(µ)

⎤⎦ z(umα)⊤ζumα

α : αα′=u,α′ =∅

z(umα)⊤ζumα where we used (4.38)

w : w=umα,m≥0,αα′=u,α′ =∅zw⊤

= W x(z, ζ)

and (4.24) follows as claimed. This completes the proof of all parts of Proposi-tion 4.4.

The following two theorems collect some results on the unitary classificationof the operators Ux,y,λ and Ux from [9]. We present alternate proofs from those in[9] as an application of our model theory and symbol calculus.

Theorem 4.5. For x, y ∈ Fd and λ ∈ T, let Ux,y,λ be the associated row-unitary.Without loss of generality (see Proposition 4.1) we assume in addition that x andy share no common tail. Then:

1. Ux,y,λ is unitarily equivalent to Uy,λ.2. If y′ is a cyclic permutation of y (so y = γγ′ and y′ = γ′γ for some γ, γ′ ∈ Fd,

then Uy,λ and Uy′,λ are unitarily equivalent.3. If y has the form y = un for some u ∈ Fd, then Uy,λ is unitarily equivalent

to the direct sum row-unitary ⊕µ : µn=λUu,µ.

For the basic definitions and constructions concerning direct integral spaceswhich come up in the statement of the next theorem, we refer to [11].

Theorem 4.6. For x = gk1gk2 · · · gkn · · · an infinite word, let U = Ux be the asso-ciated row-unitary. Then:

1. If x′ = gk′1gk′

2· · · gk′

n· · · is another infinite word which is tail equivalent to x

in the sense that there is an infinite word v so that

x = αv, x′ = α′v

for some finite words α, α′ ∈ Fd (not necessarily of the same length), then

Ux is unitarily equivalent to Ux′

.2. If x is a periodic infinite word, say x = uuu · · · =: u∞ for some non-empty

u ∈ Fd, then Ux is unitarily equivalent to the direct integral∫ ⊕

TUu,µ dm(µ),

where m is normalized Lebesgue measure on the unit circle T, and Uu,µ isthe row-unitary for the orbit-periodic case as in Theorem 4.5.

Remark 4.7. Additional results in [9] are that Uy,λ is irreducible if y is primitive(i.e., not the power of a word of shorter length), and that Ux is irreducible if xdoes not have a periodic tail. We have nothing new to say about these results, asour calculus techniques here are tailored to exhibiting the reducibility of a givenmodel rather than proving that a given model is irreducible.

Proof of Theorem 4.5. As we have seen in Proposition 4.1, it suffices to studythe models UW x,y,λ on the model space LW x,y,λ . For the proof of the first twostatements, we combine the third statement of Proposition 3.3 with the formulas(4.20) and (4.22), respectively, in Proposition 4.4. To see that LT is unitary rather

than merely isometric (where T (z) = zx⊤

and T (z) = zγ⊤

respectively), it sufficesto note that, for any scalar Cuntz weight W (i.e., W = [Wv,w;α,β ] with Wv,w;α,β ∈C for v, w, α, β ∈ Fd) any vector of the form W [zvζw ] ∈ LW is ∗-cyclic for UW .

The validity of the third statement is suggested by the identity (4.23) inProposition 4.4, but there is still more work to do. The operator-theoretic content

of (4.23) is as follows: the map ι given by

ι :⊕

µ : µn=λ

fµ(z, λ) →∑

µ : µn=λ

fµ(z, λ) (4.39)

is a coisometry from⊕

µ : µn=λ L 1n W u,µ onto LW y,λ with initial space D equal to

the closure of the linear manifold

⎧⎨⎩⊕

µ : µn=λ

nWu,µ[p] : p ∈ P(Fd ×Fd, C)

⎫⎬⎭ (4.40)

µ : µn=λ L 1n W u,µ . To see this, note that

ι :⊕

µ : µn=λ

nWu,µ[p] →

µ : µn=λ

nWu,µ[p] = W y,λ[p]

with preservation of norm∥∥∥∥∥∥⊕

µ : µn=λ

nWu,µ[p]

∥∥∥∥∥∥

⊕µ : µn=λ L 1

nWu,µ

µ : µn=λ

nWu,µ[p], p

=⟨W y,λ[p], p

⟩L2 (by (4.23))

=∥∥W y,λ[p]

∥∥2L

Wy,λ.

This calculation shows that, indeed, ι is isometric from D0 into LW y,λ with imagecontaining the dense subset W y,λ[p] : p ∈ P(Fd×Fd, C) of LW y,λ . Therefore, byapproximation, the same formula (4.39) defines an isometry (still denoted ι) fromD onto LW y,λ . Finally, if f =

⊕µ : µn=λ fµ is an element of

⊕µ : µn=λ L 1

n W u,µ

which is orthogonal to D0, then we have, for all p ∈ P(Fd ×Fd, E),

0 =∑

µ : µn=λ

⟨fµ,

nWu,µ[p]

LWu,µ

µ : µn=λ

〈fµ, p〉L2

⟨ ∑

µ : µn=λ

fµ, p

for all polynomials p ∈ P(Fd × Fd, C). This in turn forces∑

µ : µn=λ fµ to be the

zero power series, or f =⊕

µ : µn=λ fµ is in the kernel of ι. In this way we see

that indeed ι defines a coisometry from⊕

µ : µn=λ L 1n W u,µ onto LW y,λ with initial

space equal to D defined as the closure of D0 in (4.40) whenever (4.24) is satisfied.

It is clear from the definition of ι and the fact that the formulas for UW,j andU∗

W,j is independent of W (see (2.27) and (2.28)) that we have the intertwining

relations

⎛⎝⊕

µn=λ

U 1n W u,µ

⎞⎠ = UW y,λ,jι

⎛⎝⊕

µn=λ

U∗1n W u,µ

⎞⎠ = U∗

W y,λ,jι.

Hence, if it is the case that ι is actually unitary rather than merely coisometric,it will follow that

⊕µ : µn=λ U 1

n W u,µ and UW y,λ are unitarily equivalent (via ι),

and hence also that⊕

µ : µn=λ Uu,µ and Uy,λ are unitarily equivalent. (Note thatthe spaces L 1

n W u,µ and LW u,µ are the same after a trivial rescaling of the norm,

and hence U 1n W u,µ and UW u,µ are trivially unitarily equivalent.) This is the extra

ingredient required to complete the proof of Theorem 4.5.To this point the analysis has been relatively straightforward. It now remains

only to show that the map ι in (4.39) is an isometry, i.e., that certain overlappingspaces are trivial; as often happens in this model-theoretic approach, this requiressome additional work which we now show is feasible in this case. For each fixed νwith νn = µ, let us identify L 1

n W u,ν as a subspace of⊕

µn=λ L 1n W u,µ in the natural

way: f → ⊕µ : µn=λδµ,νf . To show that ι is an isometry, it suffices to show that ι isan isometry on each L 1

n W u,ν ; indeed, in this case the initial space D for ι contains

each of the subspaces L 1n W ν,λ and hence, since D is linear, D then contains the

sum of the subspaces L 1n W u,ν over all ν with νn = λ which is the whole space⊕

µ : µn=λ LW u,µ . To show that ι is isometric on each LW u,ν , we need the followinglemma:

Lemma 4.8. Assume that y ∈ Fd has the form y = un for some u ∈ Fd as above.For ν ∈ T with νn = λ and for α, β ∈ Fd, the polynomial

pν,α,β(z, ζ) =n−1∑

νizαζuiβ (4.41)

satisfies

Wu,ν [zαζβ ] = W y,λ[pν,α,β(z, ζ)]. (4.42)

Assuming the validity of the lemma, we prove that ι is an isometry on eachLW u,ν as follows. For an arbitrary pair of monomials zαζβ and zα′

ζβ′

, we have onthe one hand

〈Wu,ν [zαζβ ], Wu,ν [zα′

ζβ′

]〉L 1n

W u,ν= n2

nWu,ν [zαζβ ],

nWu,ν [zα′

ζβ′

nWu,ν [zαζβ ], zα′

ζβ′

= nWu,να′,β′;α,β . (4.43)

On the other hand we have

〈Wu,ν [zαζβ ], Wu,ν [zα′

ζβ′

]〉LW y,λ = 〈Wu,ν [zαζβ ], W y,λ[pν,α′,β′

]〉LW y,λ

= 〈Wu,ν [zαζβ ], pν,α′,β′〉L2

= 〈zαζβ , Wu,ν [pν,α′,β′

]〉L2

⟨zαζβ ,

v,w∈Fd

n−1∑

Wu,νv,w;α′,uiβ′ν

izvζw

n−1∑

〈zαζβ , Wu,να,β;α′,uiβ′ν

izαζβ〉L2

n−1∑

νiWu,να′,uiβ′;α,β

∑n−1i=0 νiWu,ν

(α′α−1)β⊤,uiβ′ , |α′| ≥ |α|,∑n−1

i=0 νiWu,νβ⊤,uiβ′(αα′−1)⊤

, |α′| ≤ |α| (by (2.29))

n−1∑

νiνiWu,να′,β′;α,β

= nWu,να′,β′;α,β. (4.44)

By combining (4.43) and (4.44), we see that ι is isometric on each L 1n W u,ν (for

ν ∈ T with νn = λ), and the proof of Theorem 4.5 is complete, once we completethe proof of Lemma 4.8.

Proof of Lemma 4.8. For pν,α,β(z, ζ) =:∑

ϕ,ψ∈Fdpν,α,β

ϕ,ψ zϕζψ given by (4.41), we

need to verify (4.42). In terms of coefficients, (4.42) can be expressed as

Wu,νv,w;α,β =

W y,λv,w;ϕ,ψpν,α,β

ϕ,ψ . (4.45)

Assume for the moment that |v| ≥ |α|. We start with the right-hand side of (4.45)and compute

W y,λv,w;ϕ,ψpν,α,β

ϕ,ψ =

n−1∑

W y,λv,w;α,uiβνi (by the definition of pν,α,β)

µ : µn=λ

n−1∑

Wu,µv,w;α,uiβνi (by (4.23))

µ : µn=λ

n−1∑

Wu,µ(vα−1)β⊤(ui)⊤,w

νi (by (2.29))

µ : µn=λ

n−1∑

Wu,µ(vα−1)β⊤,w

µiνi

µ : µn=λ

Wu,µv,w;α,β

n−1∑

µiνi

µ : µn=λ

Wu,µv,w;α,β(δµ,ν)

= Wu,νv,w;α,β

and (4.45) follows. This completes the proof of Lemma 4.8.

We now tackle the proof of Theorem 4.6.

Proof of Theorem 4.6. For the proof of the first statement in Theorem 4.6, use thethird statement of Proposition 3.3 combined with formula (4.21) in Proposition4.4. Again we use that W (zvζw) is ∗-cyclic for UW for any scalar Cuntz weight W .

The validity of the second statement in Theorem 4.6 is suggested by theformula (4.24) in Proposition 4.4, but, as was the case for the analogous statementin Theorem 4.5, there remains some non-trivial work to do. We are given an infiniteword x of the form x = u∞ for a finite word u ∈ Fd. By a continuous analogueof the argument in the proof of the last part of Theorem 4.5, one sees that the

operator-theoretic content of (4.24) is: the operator τ :∫ ⊕

TLW u,λ dm(λ) → LW x

given by

τ : f(·) →∫

f(λ) dm(λ)

is a coisometry from the direct integral space∫ ⊕

TLW u,λ dm(λ) onto LW x with

initial space D equal to the closure in∫ ⊕

TLW u,λ dm(λ) of the linear manifold

D0 = f : f(λ) = Wu,λ[p] : p ∈ P(Fd ×Fd, C).Again, from (2.27) and (2.28), the intertwining conditions

(∫ ⊕

UW u,λ,j

)= UW xτ, τ

(∫ ⊕

UW u,λ,j

)∗= (UW x)

∗τ for j = 1, . . . , d

are clear. If we can show that τ is actually unitary rather than merely coisomet-ric (i.e., that τ is isometric), it will then follow that UW x is unitarily equivalent

(via τ∗) with∫ ⊕

TUW u,λ dm(λ), and hence also that Ux is unitarily equivalent to∫ ⊕

TUu,λ dm(λ), and the proof of Theorem 4.6 will be complete. As in the proof of

Theorem 4.5, proof that the coisometry τ is actually unitary (i.e., that certain over-lapping spaces are trivial) is where the extra work enters in in this model-theoreticapproach.

Given a Borel subset B of T, we define a linear submanifold of∫ ⊕

TLW u,λ dm(λ) by

DB,0 = f : f(λ) = χB(λ)Wu,λ[p] : p ∈ P(Fd ×Fd, C). (4.46)

A useful fact concerning the space∫ ⊕

TLW u,λ dm(λ) is that the span of the subspaces

DB,0 over all Borel subsets B of T is dense in∫ ⊕

TLW u,λ dm(λ). To see this, we

show that any f ∈∫ ⊕

TLW u,λ dm(λ) orthogonal to DB,0 for each Borel subset B

of T is zero. Thus, suppose that

f(λ) =∑

fα,β(λ)zαζβ ∈∫ ⊕

LW u,λ dm(λ)

is orthogonal to the subspace DB,0 for B any Borel subset of T. Then, for anywords α, β ∈ Fd, we have

〈f(λ), Wu,λ[zαζβ ]〉LWu,λ

〈f(λ), zαζβ〉L2 dm(λ)

fα,β(λ) dm(λ)

for all Borel subsets B of T. Hence it follows that fα,β(λ) = 0 for m-almost every

λ ∈ T for all α, β, which in turn forces f to be zero in∫ ⊕

TLW u,λ dm(λ) as wanted.

Thus, to show that τ is isometric on∫ ⊕

TLW u,λ dm(λ), it suffices to show that τ

is isometric on the subspace DB,0 for each Borel subset B ⊂ T. Note that, forf(λ) = χB(λ)Wu,λ[p] ∈ DB,0 (with p equal to a polynomial in z and ζ), we seethat τ : f → WB[p] ∈ LW B where we have set WB equal to the positive Haplitzoperator

WBv,w;α,β =

Wu,λv,w;α,β dm(λ) with symbol WB(z, ζ) =

Wu,λ(z, ζ) dm(λ).

(4.47)Furthermore, via an interchange of integration and summation we see that

‖f‖2∫ ⊕

TLu,λ dm(λ)

= ‖τf‖2LWBif f(λ) = χB(λ)Wu,λ[p], p ∈ P(Fd ×Fd, C).

Thus, for a given Borel subset B ⊂ T, to show that τ is isometric from DB,0 intoWx is the same as to show that the inclusion map f → f is isometric from thedense subset WBP(Fd×Fd, C) of LW B into LW x . Thus the proof of Theorem 4.6is complete once we show: for each Borel subset B ⊂ T and any choice of wordsα, β, α′, β′ ∈ Fd,⟨

WB[zαζβ ], WB[zα′

ζβ′

=⟨WB[zαζβ ], WB[zα′

ζβ′

]⟩LWx

. (4.48)

For this purpose we need the following more elaborate continuous analogueof Lemma 4.8. In the statement of the lemma, for 0 < r < 1 and λ ∈ T we definethe Poisson kernel Pr(λ) by

Pr(λ) =

∞∑

k=−∞r|k|λk.

Lemma 4.9. For 0 < r < 1, µ ∈ T and α, β ∈ Fd, define the polynomial pµ,rα,β(z, ζ) ∈

P(Fd ×Fd, C) by

pµ,rα,β(z, ζ) =

⎧⎪⎨⎪⎩

∑∞k=0 rkµkzαζukβ +

∑∞k=1 rkµkzγ⊤(uk−1)⊤γ′⊤αζβ if

β has the form β = ujγ, γγ′ = u, γ′ = ∅, j = 0, 1, 2, . . .

0, otherwise.

(4.49)Then

Wu,ν [pµ,rα,β] = Pr(µ/ν)Wu,ν [zαζβ ]. (4.50)

Moreover, if x is an infinite word of the form x = uuu · · · =: u∞ and we defineWB as in (4.47), then, if we define the formal power series pB

α,β by

pBα,β[z, ζ) = lim

pµ,rα,β(z, ζ) dm(µ), (4.51)

then pBα,β is in the domain of W x and

WB[zαζβ ] = W x[pBα,β ]. (4.52)

Assuming the validity of the Lemma, we now complete the proof of Theorem4.6 as follows. A consequence of (4.24) and (4.50) in Lemma 4.9 is

〈W xpµ,rα,β , pµ′,r′

α′,β′〉L2dm(ν) =

〈Wu,ν [pµ,rα,β], pµ′,r′

α′,β′〉L2 dm(ν)

Pr(µ/ν)〈Wu,ν [zαζβ ], pµ′,r′

α′,β′〉L2 dm(ν)

Pr(µ/ν)〈zαζβ , Wu,ν [pµ′,r′

α′,β′ ]〉L2 dm(ν)

Pr(µ/ν)〈zαζβ ,Pr′(µ′/ν)Wu,ν [zα′

ζβ′

]〉L2 dm(ν)

limr→1〈W x[pµ,r

α,β], pµ′,r′

α′,β′〉L2 = Pr′(µ′/µ)〈zαζβ , Wu,µ[zα′

ζβ′

]〉L2 . (4.53)

Integration of both sides of (4.53) with respect to µ over the Borel set B thengives

〈W x[pBα,β], pµ′,r′

α′,β′〉L2 =

Pr′(µ′/µ)〈zαζβ , Wu,µ[zα′

ζβ′

]〉L2 dm(µ) (4.54)

where pBα,β is as in (4.51).

With these preliminaries out of the way, we start with the right-hand side of(4.48) and compute⟨WB[zαζβ ], WB[zα′

ζβ′

]⟩LWx

=⟨W x[pB

α,β ], W x[pBα′,β′ ]

⟩LWx

(by (4.52))

=⟨W x[pB

α,β ], pBα′,β′

= limr′→1

⟨W x[pB

α,β ], pµ′,r′

α′,β′

dm(µ′) (by definition of pBα′,β′)

= limr′→1

Pr′(µ′/µ)⟨zαζβ , Wu,µ[zα′

ζβ′

L2dm(µ)

]dm(µ′) (by (4.54))

⟨zαζβ , Wu,µ′

[zα′

ζβ′

L2dm(µ′) (since P is an approximate identity)

=⟨zαζβ , WB[zα′

ζβ′

L2=⟨WB[zαζβ ], WB[zα′

ζβ′

and (4.48) follows as wanted. The only remaining piece of the proof of Theorem4.6 is now the proof of Lemma 4.9.

Proof of Lemma 4.9. We first check the general identities

Wu,νv,w;α,ukβ

= νkWu,νv,w;α,β (4.55)

Wu,νv,w;γ⊤(uk−1)⊤γ′⊤α,ujγ

= νkWu,νv,w;α,ujγ where γγ′ = u with γ′ = ∅. (4.56)

To check (4.55), in case |v| ≥ |α|, by (2.29) we have

Wu,νv,w;α,ukβ

= Wu,ν(vα−1)β⊤(uk)⊤,w

= νkWu,ν(vα−1)β⊤,w

= νkWu,νv,w;α,β ,

and similarly, if |v| < |α|, we have

Wu,νv,w;α,ukβ

= Wu,νβ⊤(uk)⊤,w(αv−1)⊤

= νkWu,νβ⊤,w(αv−1)⊤

= νkWu,νv,w;α,β

and (4.55) follows in all cases.

To check (4.56), one must consider three special cases:

Case 1: |v| ≥ k|u|+ |α|,Case 2: |α| ≤ |v| < k|u|+ |α|, andCase 3: |v| < |α|.

Each of these cases is handled by using (2.29) to reduce the issue to coefficients

of the symbol Wϕ,ψ and then using the special structure of Wu,νϕ,ψ apparent from

the definition (4.16). As the computations are somewhat tedious, we will not gothrough all the details here, but simply accept the validity of (4.56).

We next verify that pBα,β is in the domain of W x for each α, β ∈ Fd and

Borel subset B ⊂ T. It is not difficult to see that the non-zero coefficients of PBα,β ,

although infinite in number, are square summable in modulus, since they roughlycorrespond to the Fourier coefficients for the L2-function χB on the circle T. SinceW x is supported on the symmetrized diagonal (v⊤, v) : v ∈ Fd with non-zero

diagonal entries all equal to 1, the square summability of the coefficients of pBα,β

implies that pBα,β is in the domain of W x as claimed.

We are now ready to verify (4.50). Consider first the case where β is not of theform β = ujγ for some j ∈ 0, 1, 2, . . . and γ with γγ′ = u for some γ′ = ∅. Then

pµ,rα,β = 0 by definition while, by (2.29) and the definition of Wu,ν(z, ζ) we have

Wu,νv,w;α,β = Wu,v

(vα−1)β⊤,w= 0 in case |v| ≥ |α|, and

Wu,νv,w;α,β = Wu,ν

β⊤,w(αv−1)⊤= 0 in case |v| < |α|.

We conclude that (4.50) holds (in the form 0 = 0) for the case where β is not ofthe form ujγ for some j ∈ 0, 1, 2, . . . and γ with γγ′ = u for some γ′ = ∅.

We next consider the case where β does have the special form β = ujγ (wherej ∈ 0, 1, 2, . . . and γγ′ = u for some γ′ = ∅). We compute

Wu,ν [pµ,rα,β ] = Wu,ν

[ ∞∑

rkµkzαζukβ +

∞∑

rkµkzγ⊤(uk−1)⊤γ′⊤αζβ

∞∑

rkµkWu,ν [zαζukβ ] +

∞∑

rkµkWu,ν [zγ⊤(uk−1)⊤γ′⊤αζβ ]

∞∑

rkµkνkWu,ν [zαζβ ] +

∞∑

rkµkνkWu,ν [zαζβ ] (by (4.55) and (4.56))

= Pr(µ/ν)Wu,ν [zαζβ ]

and (4.50) follows.To verify (4.52), we observe

WB [zαζβ ]−∫

W x[pµ,rα,β ] dm(µ)=WB [zαζβ ]−

Wu,ν [pµ,rα,β] dm(ν)

]dm(µ)

Wu,µ[zαζβ ] dm(µ)−∫

Pr(µ/ν)Wu,ν [zαζβ ] dm(ν)

]dm(µ) (by (4.50))

[Wu,µ[zαζβ ]−

Pr(µ/ν)Wu,ν [zαζβ ] dm(ν)

]dm(µ)

where the last expression tends to zero as r → 1 (coefficientwise) since the Poissonkernel Pr(λ) is an approximate identity. This completes the proof of (4.52) and ofLemma 4.9.

References

[1] W. Arveson, Subalgebras of C∗-algebras III: multivariable operator theory, ActaMath. 181 (1998), 159–228.

[2] J.A. Ball, Linear systems, operator model theory and scattering: multivariable gen-eralizations, in Operator Theory and Its Applications (Winnipeg, MB, 1998) (Ed.

A.G. Ramm, P.N. Shivakumar and A.V. Strauss), Fields Institute CommunicationsVol. 25, Amer. Math. Soc., Providence, 2000, pp. 151–178.

[3] J.A. Ball and V. Vinnikov, Lax-Phillips scattering and conservative linear systems:a Cuntz-algebra multidimensional setting, Memoirs of the American MathematicalSociety, to appear.

[4] J.A. Ball and V. Vinnikov, Formal reproducing kernel Hilbert spaces: the commuta-tive and noncommutative settings, in Reproducing Kernel Spaces and Applications(Ed. D. Alpay), pp. 77–134, OT143, Birkhauser-Verlag, Basel-Boston, 2003.

[5] S.S. Boiko, V.K. Dubovoy and A.Ja. Kheifets, Measure Schur complements andspectral functions of unitary operators with respect to different scales, in OperatorTheory, System Theory and Related Topics: The Moshe Livsic Anniversary Volume,pp. 89–138, OT 123, Birkhauser (Basel-Boston), 2000.

[6] L. de Branges and J. Rovnyak, Canonical models in quantum scattering theory, inPerturbation Theory and its Applications in Quantum Mechanics (Ed. C.H. Wilcox),pp. 295–392, Wilcox, New York-London-Sidney, 1966.

[7] O. Bratteli and P.E.T. Jorgensen, Iterated function systems and permutation rep-resentations of the Cuntz algebra, Memoirs of the American Mathematical SocietyVolume 139, Number 663 (second of 5 numbers), 1999.

[8] K.R. Davidson, C∗-Algebras by Example, Fields Institute Monograph 6, AmericanMathematical Society, Providence, 1996.

[9] K.R. Davidson and D.R. Pitts, Invariant subspaces and hyper-reflexivity for freesemigroup algebras, Proc. London Math. Soc. 78 (1999), 401–430.

[10] K.R. Davidson and D.R. Pitts, The algebraic structure of non-commutative analyticToeplitz algebras, Math. Ann. 311 (1998), 275–303.

[11] J. Dixmier, Les algebres d’operateurs dans l’espace Hilbertien (Algebres de von Neu-mann), Gauthier-Villars, Paris, 1969.

[12] G. Popescu, Models for infinite sequences of noncommuting operators, Acta Sci.Math. 53 (1989), 355–368.

[13] G. Popescu, Characteristic functions for infinite sequences of noncommuting opera-tors, J. Operator Theory 22 (1989), 51–71.

[14] G. Popescu, Isometric dilations for infinite sequences of noncommuting operators,Trans. Amer. Math. Soc. 316 (1989), 523–536.

[15] G. Popescu, Multi-analytic operators on Fock spaces, Math. Ann. 303 (1995), 31–46.

Joseph A. BallDepartment of MathematicsVirginia TechBlacksburg, Virginia 24061e-mail: ball@math.vt.edu

Victor VinnikovDepartment of MathematicsBen Gurion University of the NegevBeer-Sheva 84105, Israele-mail: vinnikov@math.bgu.ac.il

Relations on Non-commutative Variablesand Associated Orthogonal Polynomials

T. Banks, T. Constantinescu and J.L. Johnson

Abstract. This semi-expository paper surveys results concerning three classesof orthogonal polynomials: in one non-hermitian variable, in several isometricnon-commuting variables, and in several hermitian non-commuting variables.The emphasis is on some dilation theoretic techniques that are also describedin some details.

Mathematics Subject Classification (2000). Primary 47A20; Secondary 42C05.

Keywords. Orthogonal polynomials in several variables, recurrence relations,asymptotic properties.

1. Introduction

In this semi-expository paper we deal with a few classes of orthogonal polynomialsassociated to polynomial relations on several non-commuting variables. Our initialinterest in this subject was motivated by the need of more examples related to [18].On the other hand there are transparent connections with interpolation problemsin several variables as well as with the modeling of various classes of non-stationarysystems (see [1] for a list of recent references), which guided our choice of topics.Thus we do not relate to more traditional studies on orthogonal polynomials ofseveral variables associated to a finite reflection group on an Euclidean space orother types of special functions of several variables, for which a recent presentationcould be found in [21], instead we are more focused on results connected withvarious dilation theoretic aspects and Szego kernels.

Our aim is to give an introduction to this point of view. We begin our presen-tation with a familiar setting for algebras given by polynomial defining relationsand then we introduce families of orthonormal polynomials associated to somepositive functionals on these algebras. Section 3 contains a discussion of the firstclass of orthogonal polynomials considered in this paper, namely polynomials inone variable on which there is no relation. This choice is motivated mainly by

62 T. Banks, T. Constantinescu and J.L. Johnson

the fact that we have an opportunity to introduce some of the basic dilation the-oretic techniques that we are using. First, we discuss (in a particular case thatis sufficient for our goals) the structure of positive definite kernels and their tri-angular factorization. Then these results are used to obtain recurrence relationsfor orthogonal polynomials in one variable with no additional relations, as well asasymptotic properties of these polynomials. All of these extend well-known resultsof G. Szego. We conclude this section with the introduction of a Szego type kernelwhich appears to be relevant to our setting.

In Section 4 we discuss the example of orthogonal polynomials of severalisometric variables. Most of the results are just particular cases of the correspond-ing results discussed in Section 3, but there is an interesting new point aboutthe Szego kernel that appears in the proof of Theorem 4.1. We also use a certainexplicit structure of the Kolmogorov decomposition of a positive definite kernelon the set of non-negative integers in order to produce examples of families ofoperators satisfying Cuntz-Toeplitz and Cuntz relations.

The final section contains a discussion of orthogonal polynomials of severalnon-commuting hermitian variables. This time, some of the techniques describedin Section 3 are not so relevant and instead we obtain three-terms recursions inthe traditional way, and we introduce families of Jacobi matrices associated tothese recursions. Many of these results can be proved by adapting the classicalproofs from the one scalar variable case. However, much of the classical functiontheory is no longer available so we present some proofs illustrating how classicaltechniques have to be changed or replaced. Also some results are not presented inthe most general form in the hope that the consequent simplifications in notationwould make the paper more readable.

2. Orthogonal polynomials associated to polynomial relations

In this section we introduce some classes of orthogonal polynomials in several vari-ables. We begin with the algebra PN of polynomials in N non-commuting variablesX1, . . . , XN with complex coefficients. Let F+

N be the unital free semigroup on Ngenerators 1, . . . , N . The empty word is the identity element and the length of theword σ is denoted by |σ|. It is convenient to use the notation Xσ = Xi1 . . . Xik

for σ = i1 . . . ik ∈ F+N . Thus, each element P ∈ PN can be uniquely written in the

form P =∑

σ∈F+N

cσXσ, with cσ = 0 for finitely many σ’s.

We notice that PN is isomorphic with the tensor algebra over CN . Let (CN )⊗k

denote the k-fold tensor product of CN with itself. The tensor algebra over CN isdefined by the algebraic direct sum

T (CN ) = ⊕k≥0(CN )⊗k.

If e1, . . . , eN is the standard basis of CN , then the set

1 ∪ ei1 ⊗ · · · ⊗ eik| 1 ≤ i1, . . . , ik ≤ N, k ≥ 1

Non-commutative Variables and Orthogonal Polynomials 63

is a basis of T (CN ). For σ = i1 . . . ik we write eσ instead of ei1 ⊗ · · · ⊗ eik, and

the mapping Xσ → eσ, σ ∈ F+N , extends to an isomorphism from PN to T (CN ),

hence PN ≃ T (CN ).It is useful to introduce a natural involution on P2N as follows:

X+k = XN+k, k = 1, . . . , N,

X+l = Xl−N , l = N + 1, . . . , 2N ;

on monomials,(Xi1 . . .Xik

)+ = X+ik

. . .X+i1

and finally, if Q =∑

σ∈F+2N

cσXσ, then Q+ =∑

σ∈F+2N

cσX+σ . Thus, P2N is a unital,

associative, ∗-algebra over C.We say that A ⊂ P2N is symmetric if P ∈ A implies cP+ ∈ A for some

c ∈ C − 0. Then the quotient of P2N by the two-sided ideal generated by A isan associative algebra R(A). Letting π = πA : P2N → R(A) denote the quotientmap then the formula

π(P )+ = π(P+) (2.1)

gives a well-defined involution on R(A). Usually the elements of A are called thedefining relations of the algebra R(A). For instance, R(∅) = P2N ,

R(Xk −X+k | k = 1, . . . , N) = PN ;

also,R(XkXl −XlXk | k, l = 1, . . . , 2N)

is the symmetric algebra over C2N and

R(XkXl + XlXk | k, l = 1, . . . , 2N)is the exterior algebra over C2N . Examples abound in the literature (for instance,see [20], [21], [29]).

There are many well-known difficulties in the study of orthogonal polynomialsin several variables. The first one concerns the choice of an ordering of F+

N . In thispaper we consider only the lexicographic order ≺, but due to the canonical gradingof F+

N it is possible to develop a basis free approach to orthogonal polynomials. Inthe case of orthogonal polynomials on several commuting variables this is presentedin [21]. A second difficulty concerns the choice of the moments. In this paperwe adopt the following terminology. A linear functional φ on R(A) is called q-positive (q comes from quarter) if φ(π(P )+π(P )) ≥ 0 for all P ∈ PN . In this case,

φ(π(P )+) = φ(π(P )) for P ∈ PN and

|φ(π(P1)+π(P2))|2 ≤ φ(π(P1)

+π(P1))φ(π(P2)+π(P2))

for P1, P2 ∈ PN . Next we introduce

〈π(P1), π(P2)〉φ = φ(π(P2)+π(P1)), P1, P2 ∈ PN . (2.2)

By factoring out the subspace Nφ = π(P ) | P ∈ PN , 〈π(P ), π(P )〉φ = 0 andcompleting this quotient with respect to the norm induced by (2.2) we obtain aHilbert space Hφ.

The index set G(A) ⊂ F+N of A is chosen as follows: if α ∈ G(A), choose the

next element in G(A) to be the least β ∈ F+N with the property that the elements

π(Xα′), α′ α, and π(Xβ) are linearly independent. We will avoid the degeneratesituation in which π(1) = 0; if we do so, then ∅ ∈ G(A). Define Fα = π(Xα) forα ∈ G(A). For instance, G(∅) = F+

N , in which case Fα = Xα, α ∈ F+N . Also,

G(XkXl −XlXk | k, l = 1, . . . , 2N) = i1 . . . ik ∈ F+N | i1 ≤ · · · ≤ ik, k ≥ 0,

G(XkXl +XlXk | k, l = 1, . . . , 2N) = i1 . . . ik ∈ F+N | i1 < · · · < ik, 0 ≤ k ≤ N

(we use the convention that for k = 0, i1 . . . ik is the empty word).

We consider the moments of φ to be the numbers

sα,β = φ(F+α Fβ) = 〈Fβ , Fα〉φ, α, β ∈ G(A). (2.3)

The kernel of moments is given by Kφ(α, β) = sα,β , α, β ∈ G(A). We notice thatφ is a q-positive functional on R(A) if and only if Kφ is a positive definite kernelon G(A). However, Kφ does not determine φ uniquely. One typical situation when

Kφ determines φ is Xk − X+k | k = 1, . . . , N ⊂ A; a more general example is

provided by the Wick polynomials,

XkX+l − ak,lδk,l −

cm,nk,l X+

mXn, k, l = 1, . . . , N,

where ak,l, cm,nk,l are complex numbers and δk,l is the Kronecker symbol.

The moment problem is trivial in this framework since it is obvious that thenumbers sα,β , α, β ∈ G(A), are the moments of a q-positive functional on R(A) ifand only if the kernel K(α, β) = sα,β, α, β ∈ G(A), is positive definite.

We now introduce orthogonal polynomials in R(A). Assume that φ is strictlyq-positive onR(A), that is, φ(π(P )+π(P )) > 0 for π(P ) = 0. In this caseNφ = 0and π(PN ) can be viewed as a subspace of Hφ. Also, Fαα∈G(A) is a linearly inde-pendent family in Hφ and the Gram-Schmidt procedure gives a family ϕαα∈G(A)

of elements in π(PN ) such that

ϕα =∑

aα,βFβ , aα,α > 0; (2.4)

〈ϕα, ϕβ〉φ = δα,β , α, β ∈ G(A). (2.5)

The elements ϕα, α ∈ G(A), will be called the orthonormal polynomials associatedto φ. An explicit formula for the orthonormal polynomials can be obtained in thesame manner as in the classical, one scalar variable case. Thus, set

Dα = det [sα′,β′ ]α′,β′α > 0, α ∈ G(A), (2.6)

and from now on τ−1 denotes the predecessor of τ with respect to the lexicographicorder on F+

N , while σ + 1 denotes the successor of σ.

We have: ϕ∅ = s−1/2∅,∅ and for ∅ ≺ α,

ϕα =1√

Dα−1Dα

⎡⎣

[sα′,β′]α′≺α;β′α

F∅ . . . Fα

⎤⎦, (2.7)

with an obvious interpretation of the determinant. In the following sections wewill discuss in more details orthonormal polynomials associated to some simpledefining relations.

3. No relation in one variable

This simple case allows us to illustrate some general techniques that can be used inthe study of orthonormal polynomials. We have A = ∅ and N = 1, so R(A) = P2.The index set is N0, the set of non-negative integers, and Fn = Xn

1 , n ∈ N0.The moment kernel of a q-positive functional on P2 is Kφ(n, m) = φ((Xn

1 )+Xm1 ),

n, m ∈ N0, and we notice that there is no restriction on Kφ other than beingpositive definite. We now discuss some tools that can be used in this situation.

3.1. Positive definite kernels on N0

We discuss a certain structure (and parametrization) of positive definite kernels onN0. The nature of this structure is revealed by looking at the simplest examples.First, we consider a strictly positive matrix

[1 aa 1

], a ∈ R.

This matrix gives a new inner product on R2 by the formula

〈x, y〉S = 〈Sx, y〉, x, y ∈ R2,

where 〈·, ·〉 denotes the Euclidean inner product on R2. Let e1, e2 be the standardbasis of R2. By renorming R2 with 〈·, ·〉S the angle between e1 and e2 was modifiedto the new angle θ = θ(e1, e2) such that

cos θ(e1, e2) =〈e1, e2〉S‖e1‖S‖e2‖S

= a. (3.1)

We can visualize the renormalization process by giving a map TS : R2 → R2 withthe property that 〈TSx, TSy〉 = 〈x, y〉S for x, y ∈ R2, and it is easily seen that wecan choose

[1 cos θ0 sin θ

We can also notice that TSe1 = e1 and TSe2 = f2 = J(cos θ)e1, whereJ(cos θ) is the Julia operator,

J(cos θ) =

[cos θ sin θsin θ − cos θ

Figure 1. Renormalization in R2

which is the composition of a reflection about the x-axis followed by the counter-clockwise rotation Rθ through angle θ. We deduce that

a = cos θ = 〈e1, f2〉 = 〈e1, J(cos θ)e1〉 = 〈J(cos θ)e1, e1〉.

The discussion extends naturally to the 3× 3 case. Thus let

⎡⎣

1 a ba 1 cb c 1

⎤⎦ , a, b, c ∈ R,

be a strictly positive matrix. A new inner product is induced by S on R3,

〈x, y〉S = 〈Sx, y〉, x, y ∈ R3,

and let e1, e2, e3 be the standard basis of R3. With respect to this new innerproduct the vectors e1, e2, e3 still belong to the unit sphere, but they are no longerorthogonal. Thus,

a = cos θ(e1, e2) = cos θ12,

c = cos θ(e2, e3) = cos θ23,

b = cos θ(e1, e3) = cos θ13.

This time, the law of cosines in spherical geometry gives a relation between thenumbers a, b, and c,

b = cos θ13 = cos θ12 cos θ23 + sin θ12 sin θ23 cos θ, (3.2)

where θ is the dihedral angle formed by the planes generated by e1, e2 and, re-spectively, e2, e3 (see, for instance, [26]). Thus, the number b belongs to a disk ofcenter cos θ12 cos θ23 and radius sin θ12 sin θ23. Once again the renormalization canbe visualized by a map TS : R3 → R3 such that 〈TSx, TSy〉 = 〈x, y〉S . In this casewe can choose

⎡⎣

1 cos θ12 cos θ12 cos θ23 + sin θ12 sin θ23 cos θ0 sin θ12 sin θ12 cos θ23 − cos θ12 sin θ23 cos θ0 0 sin θ23 sin θ

⎤⎦ ,

and we see that

TSe1 = e1,

TSe2 = f2 = (J(cos θ1,2)⊕ 1)e1,

TSe3 = f3 = (J(cos θ12)⊕ 1)(1⊕ J(cos θ))(J(cos θ23)⊕ 1)e1.

In particular,

b = cos θ13 = 〈(J(cos θ1,2)⊕ 1)(1⊕ J(cos θ))(J(cos θ2,3)⊕ 1)e1, e1〉, (3.3)

which can be viewed as a dilation formula.

Now both (3.2) and (3.3) extend to a strictly positive n × n matrix andprovide a parametrization and therefore a structure for positive definite kernels onN0 (for general results and details see [13], [16]). We apply this result to a kernelKφ associated to a strictly q-positive functional φ and obtain that Kφ is uniquelydetermined by a family γk,j0≤k<j of complex numbers with the property that

|γk,j | < 1 for all 0 ≤ k < j. Define dk,j = (1 − |γk,j |2)1/2. The extension of (3.3)mentioned above gives

sk,j = s1/2k,k s

1/2j,j 〈Uk,je1, e1〉, k < j, (3.4)

where Uk,j is a (j − k + 1) × (j − k + 1) unitary matrix defined recursively by:Uk,k = 1 for k ≥ 0; for k < j (with n = j − k),

Uk,j = (J(γk,k+1)⊕ 1n−1)(1 ⊕ J(γk,k+2)⊕ 1n−2) . . . (1n−1 ⊕ J(γk,j))(Uk+1,j ⊕ 1),

where J(γl,m) is the Julia operator associated to γl,m,

J(γl,m) =

[γl,m dl,m

dl,m −γl,m

and 1m denotes the m × m identity matrix. For instance, we deduce from (3.4)that:

s01 = s1/200 s

] [ γ01 d01

d01 −γ01

s02 = s1/200 s

[1 0 0

⎡⎣

γ01 d01 0d01 −γ01 00 0 1

⎤⎦⎡⎣

1 0 00 γ02 d02

0 d02 −γ02

⎤⎦⎡⎣

γ12 d12 0d12 −γ12 00 0 1

⎤⎦⎡⎣

⎤⎦ ,

s03 = s1/200 s

[1 0 0 0

⎡⎢⎢⎣

γ01 d01 0 0d01 −γ01 0 00 0 1 00 0 0 1

⎤⎥⎥⎦

⎡⎢⎢⎣

1 0 0 00 γ02 d02 00 d02 −γ02 00 0 0 1

⎤⎥⎥⎦

⎡⎢⎢⎣

1 0 0 00 1 0 00 0 γ03 d03

0 0 d03 −γ03

⎤⎥⎥⎦

⎡⎢⎢⎣

γ12 d12 0 0d12 −γ12 0 00 0 1 00 0 0 1

⎤⎥⎥⎦

⎡⎢⎢⎣

1 0 0 00 γ13 d13 00 d13 −γ13 00 0 0 1

⎤⎥⎥⎦

⎡⎢⎢⎣

γ23 d23 0 0d23 −γ23 0 00 0 1 00 0 0 1

⎤⎥⎥⎦

⎡⎢⎢⎣

⎤⎥⎥⎦ .

In particular, we deduce that s01 = s1/200 s

1/211 γ01. The next formula is the complex

version of (3.2), s02 = s1/200 s

1/222 γ01γ12 + s

1/200 s

1/222 d01γ02d12. Then,

s03 = s1/200 s

1/233 γ01γ12γ23 + s

1/200 s

1/233 γ01d12γ13d23

+s1/200 s

1/233 d01γ02d12γ23 − s

1/200 s

1/233 d01γ02γ12γ13d23

+s1/200 s

1/233 d01d02γ03d13d23.

Explicit formulae of this type can be obtained for each sk,j , as well as inversealgorithms allowing to calculate γk,j from the kernel of moments, see [16] fordetails.

A natural combinatorial question would be to calculate the number N(sk,j)of additive terms in the expression of sk,j . We give here some details since thecalculation of N(sk,j) involves another useful interpretation of formula (3.4). Wenotice that for k ≥ 0,

N(s01) = N(sk,k+1) = 1,

N(s02) = N(sk,k+2) = 2,

N(s03) = N(sk,k+3) = 5.

The general formula is given by the following result.

Theorem 3.1. N(sk,k+l) is given by the Catalan number Cl =1

Proof. The first step of the proof considers the realization of sk,j through a timevarying transmission line (or lattice). For illustration we consider the case of s03

in Figure 2.

Figure 2. Lattice representation for s03

Each box in Figure 2 represents the action of a Julia operator. As a whole, thediagram represents the action of the unitary operator U0,3. We see in this figurethat the number of additive terms in the formula of s03 is given by the numberof paths from A to B. In it clear that to each path from A to B in Figure 2 itcorresponds a Dyck path from C to D in Figure 3, that is, a path that never stepsbelow the diagonal and goes only to the right or downward.

Figure 3. A Dyck path from C to D

Thus, each box in Figure 2 corresponds to a point strictly above the diagonalin Figure 3. Once this one-to-one correspondence is established, we can use thewell-known fact that the number of Dyck paths like the one in Figure 3 is givenexactly by the Catalan numbers.

3.2. Spectral factorization

The classical theory of orthogonal polynomials is intimately related to the so-called spectral factorization. Its prototype would be the Fejer-Riesz factorizationof a positive trigonometric polynomial P in the form P = |Q|2, where Q is apolynomial with no zeros in the unit disk. This is generalized by Szego to theSzego class of those measures on the unit circle T with log µ′ ∈ L1, and verygeneral results along this line can be found in [31] and [33].

Here we briefly review the spectral factorization of positive definite kernels onthe set N0 described in [14]. For two positive definite kernels K1 and K2 we writeK1 ≤ K2 if K2 −K1 is a positive definite kernel. Consider a family F = Fnn≥0

of at most one-dimensional vector spaces (this restriction is made only for the

purpose of this paper) and call lower triangular array a family Θ = Θk,jk,j≥0

of complex numbers Θk,j with the following two properties: Θk,j = 0 for k < jand each column cj(Θ) = [Θk,j ]k≥0, j ≥ 0, belongs to the Hilbert space ⊕k≥jFk.Denote by H2

0(F) the set of all lower triangular arrays as above. An element ofH2

0(F) is called outer if the set cj(Θ) | j ≥ k is total in ⊕j≥kFj for each k ≥ 0.It is easily seen that if Θ is an outer triangular array, then the formula

KΘ(k, j) = ck(Θ)∗cj(Θ)

gives a positive definite kernel on N0. The following result extends the abovementioned Szego factorization and at the same time it contains the Cholesky fac-torization of positive matrices.

Theorem 3.2. Let K be an positive definite kernel on N0. Then there exists a familyF = Fnn≥0 of at most one-dimensional vector spaces and an outer triangulararray Θ ∈ H2

0(F), referred to as a spectral factor of the kernel K, such that

(1) KΘ ≤ K.(2) For any other family F ′ = F ′

nn≥0 of at most one-dimensional vector spacesand any outer triangular array Θ′ ∈ H2

0(E ,F ′) such that KΘ′ ≤ K, we haveKΘ′ ≤ KΘ.

(3) Θ is uniquely determined by (a) and (b) up to a left unitary diagonal factor.

It follows from (3) above that the spectral factor Θ can be uniquely deter-mined by the condition that Θn,n ≥ 0 for all n ≥ 0. We say that the kernel Kbelongs to the Szego class if infn≥0 Θn,n > 0. If γk,j are the parameters of Kintroduced in Subsection 3.1 then it follows that the kernel K belongs to the Szegoclass if and only if

infk≥0

s1/2k,k

dk,n > 0. (3.5)

This implies that Fn = C for all n ≥ 0 (for details see [14] or [16]).

3.3. Recurrence relations

Formula (2.7) is not very useful in calculations involving the orthogonal polyno-mials. Instead there are used recurrence formulae. In our case, A = ∅ and N = 1,we consider the moment kernel Kφ of a strictly q-positive functional on P2 andalso, the parameters γk,j of Kφ as in Subsection 3.1. It can be shown that theorthonormal polynomials associated to φ obey the following recurrence relations

ϕ0(X1, l) = ϕ♯0(X1, l) = s

−1/2l,l , l ∈ N0, (3.6)

and for n ≥ 1, l ∈ N0,

ϕn(X1, l) =1

dl,n+l

(X1ϕn−1(X1, l + 1)− γl,n+lϕ

♯n−1(X1, l)

), (3.7)

ϕ♯n(X1, l) =

dl,n+l

(−γl,n+lX1ϕn−1(X1, l + 1) + ϕ♯

n−1(X1, l))

, (3.8)

where ϕn(X1) = ϕn(X1, 0) and ϕ♯n(X1) = ϕ♯

n(X1, 0).

Somewhat similar polynomials are considered in [19], but the form of therecurrence relations as above is noticed in [17]. It should be mentioned thatϕn(X1, l)n≥0 is the family of orthonormal polynomials associated to a q-positivefunctional on P2 with moment kernel K l(α, β) = sα+l,β+l, α, β ∈ N0. Also, theabove recurrence relations provide us with a tool to recover the parameters γk,jfrom the orthonormal polynomials.

Theorem 3.3. Let kln be the leading coefficient of ϕn(X1, l). For l ∈ N0 and n ≥ 1,

γl,n+l = −ϕn(0, l)kl+10 . . . kl+1

kl0 . . . kl

Proof. We reproduce here the proof from [8] in order to illustrate these conceptsand to introduce one more property of the parameters γk,j. First, we deducefrom (3.7) that

ϕn(0, l) = −γl,n+l

dl,n+lϕ♯

n−1(0, l),

while formula (3.8) gives

ϕ♯n(0, l) =

dl,n+lϕ♯

n−1(0, l) = . . . = s−1/2l,l

dl,p+l,

ϕn(0, l) = −s−1/2l,l γl,n+l

dl,p+l.

Now we can use another useful feature of the parameters γk,j, namely the factthat they give simple formulae for determinants. Let Dm,l denote the determinantof the matrix [sk,j ]l≤k,j≤m. By Proposition 1.7 in [13],

Dl,m =

sk,k ×∏

l≤j<p≤m

d2j,p. (3.9)

One simple application of this formula is that it reveals the equality behind Fisher-Hadamard inequality. Thus, for l ≤ n ≤ n′ ≤ m, we have

Dl,m =Dl,n′Dn,m

Dn′,n

(k,j)∈Λ

d2k,j ,

where Λ = (k, j) | l ≤ k < n ≤ n′ < j ≤ m. Some other applications of (3.9)can be found in [16], Chapter 8. Returning to our proof we deduce from (3.9) that

d2l,p+l = s−1

Dl,l+n

Dl+1,l+n

γl,n+l = −ϕn(0, l)

√Dl,l+n

Dl+1,l+n. (3.10)

We can now relate this formula to the leading coefficients kln. From (3.7) we deduce

kln = s

−1/2l+n,l+n

n−1∏

dl+p,l+n, n ≥ 1,

and using once again (3.9), we deduce

√Dl,l+n−1

Dl,l+n, n ≥ 1,

which concludes the proof.

3.4. Some examples

We consider some examples, especially in order to clarify the connection withclassical orthogonal polynomials. Thus, consider firstA = 1−X+

1 X1. In this casethe index set is still N0 and if φ is a linear functional on R(A), then the kernel ofmoments is Toeplitz, Kφ(n+k, m+k) = Kφ(n, m), m, n, k ∈ N0. Let φ be a strictlyq-positive functional on R(A) and let γk,j be the parameters associated to Kφ.We deduce that these parameters also satisfy the Toeplitz condition, γn+k,m+k =

γn,m, n < m, k ≥ 1. Setting γn = γk,n+k, n ≥ 1, k ≥ 0, and dn = (1−|γn|2)1/2, therecurrence relations (3.7), (3.8) collapse to the classical Szego recursions obeyedby the orthogonal polynomials on the unit circle,

ϕn+1(z) =1

dn+1(zϕn(z)− γn+1ϕ

♯n(z)),

ϕ♯n+1(z) =

dn+1(−γn+1zϕn(z) + ϕ♯

n(z)).

Therefore γn, n ≥ 1 are the usual Szego coefficients, [32].Another example is given by A = X1−X+

1 . In this case the index set is stillN0 and the moment kernel of a strictly q-positive functional on R(A) will have theHankel property, Kφ(n, m + k) = Kφ(n + k, m), m, n, k ∈ N0. Orthogonal polyno-mials associated to functionals on R(A) correspond to orthogonal polynomials onthe real line. This time, the parameters γk,j associated to moment kernels haveno classical analogue. Instead there are so-called canonical moments which areused as a counterpart of the Szego coefficients (see [27]). Also, recurrence relationsof type (3.7), (3.8) are replaced by a three term recurrence equation,

xϕn(x) = bnϕn+1(x) + anϕn(x) + bn−1ϕn−1(x), (3.11)

with initial conditions ϕ−1 = 0, ϕ0 = 1 ([32]). Definitely, these objects are moreuseful (for instance, it appears that no simple characterization of those γk,j corre-sponding to Hankel kernels is known). Still, computations involving the parametersγk,j might be of interest. For instance, we show here how to calculate the pa-rameters for Gegenbauer polynomials. For a number λ > − 1

2 , these are orthogonal

polynomials associated to the weight function w(x) = B(12 , λ + 1

2 )−1(1 − x2)λ− 12

on (−1, 1) (B denotes the beta function). We use the normalization constants from[21], thus the Gegenbauer polynomials are

Pλn (x) =

(−1)n

2n(λ + 12 )n

(1 − x2)12−λ dn

dxn(1− x2)n+λ− 1

where (x)n is the Pochhammer symbol, (x)0 = 1 and (x)n =∏n

k=1(x + k − 1) forn ≥ 1. We have:

hλn =

B(12 , λ + 1

n (x))2

(1− x2)λ− 12 dx =

n!(n + 2λ)

2(2λ + 1)n(n + λ)

and the three term recurrence is:

Pλn+1(x) =

2(n + λ)

n + 2λxPλ

n (x)− n

n + 2λPλ

n−1(x)

(see [21], Ch. 1). We now let ϕλn(x, 0) denote the orthonormal polynomials associ-

ated to the weight function w, hence ϕλn(x, 0) = 1√

Pλn (x). From the three term

relation we deduce

ϕλn(0, 0) = (−1)n+1

√2(2λ + 1)n(n + λ)

n!(n + 2λ)×

k − 1

k − 1 + 2λ,

and also, the leading coefficient of ϕλn(x, 0) is

kλ,0n =

(n + 2λ)n

2n(λ + 1

√2(2λ + 1)n(n + λ)

n!(n + 2λ).

In order to compute the parameters γλk,j of the weight function w we use

Theorem 3.3. Therefore we need to calculate the values ϕλn(0, l) and kλ,l

n , n ≥ 1,l ≥ 0, where kλ,l

n denotes the leading coefficient of ϕλn(0, l). The main point for these

calculations is to notice that ϕλn(x, l)n≥0 is the family of orthonormal polynomi-

als associated to the weight function x2lw(x). These polynomials are also classicalobjects and they can be found for instance in [21] under the name of modifiedclassical polynomials. A calculation of the modified Gegenbauer polynomials canbe obtained in terms of Jacobi polynomials. These are orthogonal polynomialsassociated to parameters α, β > 1 and weight function

2−α−β−1B(α + 1, β + 1)−1(1− x)α(1 + x)β

on (−1, 1) by the formula

P (α,β)n (x) =

(−1)n

2nn!(1 − x)−α(1 + x)−β dn

dxn(1− x)α+n(1 + x)β+n.

According to [21], Sect. 1.5.2, we have

ϕλ2n(x, l) = c2nP

λ− 12 ,l− 1

2n (2x2 − 1)

ϕλ2n+1(x, l) = c2n+1xP

λ− 12 ,l+ 1

2n (2x2 − 1),

where cn is a constant that remains to be determined. But first we can alreadynotice that the above formulae give ϕλ

2n+1(0, l) = 0, so that γλl,2n+1+l = 0.

Theorem 3.4. For n, l ≥ 1,

ϕλ2n(0, l) = (−1)n+1

√(λ + 1)l(12

)lhλ,l

×n∏

λ + l + k − 1

kλ,l2n =

(λ + l)2n(l + 1

√(λ + 1)l(12

)lhλ,l

kλ,l2n+1 =

(λ + l)2n+1(l + 1

√(λ + 1)l(12

)lhλ,l

hλ,l2n =

(λ + 1

(λ + l)n(λ + l)

n!(l + 1

(λ + l + 2n)

hλ,l2n+1 =

(λ + 1

(λ + l)n+1(λ + l)

n!(l + 1

(λ + l + 2n + 1).

Proof. It is more convenient to introduce the polynomials

C(λ,l)2n (x) =

(λ + l)n(l + 1

P(λ− 1

2 ,l− 12 )

n (2x2 − 1),

C(λ,l)2n+1(x) =

(λ + l)n+1

(l + 12 )n+1

xP(λ− 1

2 ,l+ 12 )

n (2x2 − 1),

and again by classical results that can be found in [21], we deduce

x2l(ϕλ

2n(x, l))2

w(x)dx

= c22n

((l+ 1

(λ+l)n

)2B(l+ 1

2 ,λ+ 12 )

B( 12 ,λ+ 1

x2l(Cλ,l

2n (x))2 1

B(l + 12 , λ + 1

2 )(1− x2)λ− 1

= c22n

((l+ 1

(λ+l)n

)2 B(l+ 12 ,λ+ 1

B( 12 ,λ+ 1

2 )hλ,l

hλ,l2n =

(λ + 12 )n(λ + l)n(λ + l)

n!(l + 1

(λ + l + 2n).

Using thatB(l+ 1

2 ,λ+ 12 )

B( 12 ,λ+ 1

( 12 )l

(λ+1)l, we deduce

ϕλ2n(x, l) =

√(λ + 1)l(12

)lhλ,l

× Cλ,l2n (x).

The calculation of ϕλ2n(0, l) reduces to the calculation of Cλ,l

2n (0) which can beeasily done due to the three term relation

Cλ,l2n+2(x) =

λ + l + 2n + 1

n + 1xCλ,l

2n+1(x) − λ + l + n

n + 1Cλ,l

2n (x).

Thus we deduce

Cλ,l2n+2(0) = −λ + l + n

n + 1Cλ,l

2n (0),

and by iterating this relation and using that Cλ,l0 (0) = 1, we get

ϕλ2n(0, l) = (−1)n+1

√(λ + 1)l(12

)lhλ,l

×n∏

λ + l + k − 1

The leading coefficient of ϕλ2n(x, l) can be obtained from the corresponding formula

in [21]. Thus,

kλ,l2n =

(λ + l)2n(l + 1

√(λ + 1)l(12

)lhλ,l

kλ,l2n+1 =

(λ + l)2n+1(l + 1

√(λ + 1)l(12

)lhλ,l

hλ,l2n+1 =

(λ + 1

(λ + l)n+1(λ + l)

n!(l + 1

(λ + l + 2n + 1).

Now the parameters γλk,j can be easily calculated by using Theorem 3.4.

Of course, the explicit formulae look too complicated to be recorded here.

3.5. Asymptotic properties

In the classical setting of orthogonal polynomials on the unit circle there are severalremarkable asymptotic results given by G. Szego, see [25]. Let µ be a measure in theSzego class, and let ϕnn≥0 be the family of orthonormal polynomials associatedto µ. Then, the orthonormal polynomials have the following asymptotic properties:

ϕn → 0 (3.12)

ϕ♯n

→ Θµ, (3.13)

where Θµ is the spectral factor of µ and the convergence is uniform on the compactsubsets of the unit disk D. The second limit (3.13) is related to the so-called Szegolimit theorems concerning the asymptotic behavior of Toeplitz determinants. Thus,

detTn−1=

|ϕ♯n(0)|2

where Tn = [si−j ]ni,j=0 and skk∈Z is the set of the Fourier coefficients of µ. As

a consequence of the previous relation and (3.13) we deduce Szego’s first limittheorem,

limn→∞

detTn−1= |Θµ(0)|2 = exp(

∫ 2π

log µ′(t)dt). (3.14)

The second (strong) Szego limit theorem improves (3.14) by showing that

limn→∞

gn+1(µ)= exp

∫ ∫

|z|≤1

|Θ′µ(z)/Θµ(z)|2dσ(z)

), (3.15)

where g(µ) is the limit in formula (3.14) and σ is the planar Lebesgue measure.These two limits (3.14) and (3.15) have an useful interpretation in terms of asymp-totics of angles in the geometry of a stochastic process associated to µ (see [25])and many important applications. We show how these results can be extendedto orthogonal polynomials on P1. The formulae (3.7) and (3.8) suggest that it ismore convenient to work in a larger algebra. This is related to the so-called Toeplitzembedding, see [19], [23]. Thus, we consider the set L of lower triangular arraysa = [ak,j ]k,j≥0 with complex entries. No boundedness assumption is made on thesearrays. The addition in L is defined by entry-wise addition and the multiplicationis the matrix multiplication: for a = [ak,j ]k≥j , b = [bk,j ]k,j≥0 two elements of L,

(ab)k,j =∑

ak,lbl,j ,

which is well defined since the sum is finite. Thus, L becomes an associative, unitalalgebra.

Next we associate the element Φn of L to the polynomials ϕn(X1, l) =∑nk=0 al

n,kXk1 , n, l ≥ 0, by the formula

(Φn)k,j =

n,k−j k ≥ j

0 k < j;(3.16)

similarly, the element Φ♯n of L is associated to the family of polynomials ϕ♯

n(X1, l) =∑nk=0 bl

n,kXk1 , n, l ≥ 0, by the formula

(Φ♯n)k,j =

bjn,k−j k ≥ j

0 k < j.(3.17)

We notice that the spectral factor Θφ of Kφ is an element of L and we assumethat Θφ belongs to the Szego class. This implies that Φ♯

n is invertible in L for alln ≥ 0. Finally, we say that a sequence an ⊂ L converges to a ∈ L if (an)k,jconverges to ak,j for all k, j ≥ 0 (and we write an → a).

Theorem 3.5. Let φ belong to the Szego class. Then

Φn → 0 (3.18)

and(Φ♯

n)−1 → Θφ. (3.19)

We now briefly discuss the geometric setting for the kernel Kφ. By a classicalresult of Kolmogorov (see [30]), Kφ is the covariance kernel of a stochastic processfnn≥0 ⊂ L2(µ) for some probability space (X,M, µ). That is,

Kφ(m, n) =

fnfmdµ.

The operator angle between two spaces E1 and E2 of L2(µ) is defined by

B(E1, E2) = PE1PE2PE1 ,

where PE1 is the orthogonal projection of L2(µ) onto E1. Also define

∆(E1, E2) = I −B(E1, E2).

We can assume, without loss of generality, that fnn≥0 is total in L2(µ) andwe associate to the process fnn≥0 a family of subspaces Hr,q of L2(µ) such thatHr,q is the closure of the linear space generated by fk, r ≤ k ≤ q. We consider ascale of limits:

s− limr→∞

∆(H0,n,Hn+1,r) = ∆(H0,n,Hn+1,∞) (3.20)

for n ≥ 0, and then we let n→∞ and deduce

s− limn→∞

∆(H0,n,Hn+1,∞) = ∆(H0,∞,∩n≥0Hn,∞), (3.21)

where s− lim denotes the strong operatorial limit.

We then deduce analogues of the Szego limit theorems (3.14) and (3.15) byexpressing the above limits of angles in terms of determinants. This is possible dueto (3.9).

Theorem 3.6. Let φ belong to the Szego class. Then

Dr+1,q= sr,r det∆(Hr,r ,Hr+1,q) =

|ϕ♯q−r(0, r)|2

(3.22)

limq→∞

Dr+1,q= sr,r det∆(Hr,r,Hr+1,∞) = |Θφ(r, r)|2 = sr,r

d2r,r+j . (3.23)

If we denote the above limit by gr and

L = limn→∞

0≤k<n<j

d2k,j > 0,

limn→∞

D0,n∏nl=0 gl

det∆(H0,∞,∩n≥0Hn,∞)=

L. (3.24)

Details of the proofs can be found in [8].

3.6. Szego kernels

The classical theory of orthogonal polynomials is intimately related to some classesof analytic functions. Much of this interplay is realized by Szego kernels. Here weexpand this idea by providing a Szego type kernel for (a slight modification of) thespaceH2

0(F) which is viewed as an analogue of the Hardy class H2 on the unit disk.We mention that another version of this idea was developed in [2] (and recentlyapplied to a setting of stochastic processes indexed by vertices of homogeneoustrees in [3]); see also [4], [6]. The difference is that H2

0(F) is larger than the spaceU2 of [2] and also, the Szego kernel that we consider is positive definite.

We return now to the spaceH20(F) introduced in Subsection 3.2. Its definition

involves a family of Hilbertian conditions therefore its natural structure shouldbe that of a Hilbert module (we use the terminology of [28]). Assume Fn = Cfor all n ≥ 0 and consider the C∗-algebra D of bounded diagonal operators on⊕n≥0Fn = l2(N0). For a sequence dnn≥0 we use the notation

diag (dnn≥0) =

⎡⎢⎢⎢⎢⎢⎢⎣

d0 0 . . .0 d1 . . . 0...

... d2

0. . .

⎤⎥⎥⎥⎥⎥⎥⎦

so that diag (dnn≥0) belongs to D if and only if supn≥0 |dn| <∞. We now definethe vector space

H2(F) = Θ ∈ H20(F) | diag (cn(Θ)∗cn(Θ)n≥0) ∈ D

and notice that D acts linearly on H2(F) by ΘD = [Θk,jdj ]k,j≥0, therefore H2(F)

is a rightD-module. Also, if Θ, Ψ belong toH2(F), then diag (cn(Ψ)∗cn(Θ)n≥0)belongs to D, which allows us to define

〈Θ, Ψ〉 = diag (cn(Ψ)∗cn(Θ)n≥0) ,

turning H2(F) into a Hilbert D-module. As a Banach space with norm ‖Θ‖ =‖〈Θ, Θ〉‖1/2, the space H2(F) coincides with l∞(N0, l

2(N0)), the Banach space ofbounded sequences of elements in l2(N0). A similar construction is used in [11] fora setting of orthogonalization with invertible squares.

Next we introduce a Szego kernel for H2(F). Consider the set

B1 = znn≥0 ⊂ C | supn≥0|zn| < 1

and for z = znn≥0 ∈ B1, Θ ∈ H2(F), notice that

Θ(z) = diag

(Θn,n +

Θk,nzk−1 . . . znn≥0

is a well-defined element of D. Also, for z ∈ B1, we define

⎡⎢⎢⎢⎢⎢⎣

1 0 0 . . .z0 1 0 . . .

z0z1 z1 1 . . .z0z1z2 z1z2 z2 . . .

......

.... . .

⎤⎥⎥⎥⎥⎥⎦

which is an element ofH2(F) and the Szego kernel is defined on B1 by the formula:

S(z, w) = 〈Sw, Sz〉, z, w ∈ B1. (3.25)

Theorem 3.7. S is a positive definite kernel on B1 with the properties:

(1) Θ(z) = 〈Θ, Sz〉, Θ ∈ H2(F), z ∈ B1.

(2) The set SzD | z ∈ B1, D ∈ D is total in H2(F).

Proof. Take z1, . . . zm ∈ B1 and after reshuffling the matrix [S(zj , zl)]mj,l=1 can be

written in the form

⊕n≥0

[cn(Szj )

∗cn(Szl)]mj,l=1

Since each matrix[cn(Szj )

∗cn(Szl)]mj,l=1

⎡⎢⎣

cn(Sz1)...

cn(Szm)

⎤⎥⎦

∗ ⎡⎢⎣

cn(Sz1)...

cn(Szm)

⎤⎥⎦ is positive,

we conclude that S is a positive definite kernel on B1. Property (1) of S followsdirectly from definitions. For (2) we use approximation theory in L∞ spaces inorder to reduce the proof to the following statement: for every element Θ ∈ H2(F)for which there exists A ⊂ N such that cn(Θ) = h ∈ l2(N0) for n ∈ A andcn(Θ) = 0 for n /∈ A, and for every ǫ > 0, there exists a linear combination L ofelements SzD, z ∈ B1, D ∈ D, such that supn≥0 ‖cn(Θ − L)‖ < ǫ. This can be

achieved as follows. Since the set φw(z) = 11−zw | w ∈ B1 is total in the Hardy

space H2 on the unit disk, we deduce that there exist complex numbers c1, . . .,cm and w1, . . ., wm, |wk| < 1 for all k = 1, . . .m, such that

‖h−m∑

⎡⎢⎢⎢⎣

w2k...

⎤⎥⎥⎥⎦ ‖ < ǫ.

Then define zk = wk,nn≥0 for k = 1, . . . , m, where wk,n = wk for all n ≥ 0. Sozk ∈ B1. Also define dA

n = 1 for n ∈ A and dAn = 0 for n /∈ A, and consider

ckSzkdiag

We deduce that L ∈ H2(F), cn(L) = 0 for n /∈ A, and cn(L) =∑m

k=1 ck

⎡⎢⎢⎢⎣

w2k...

⎤⎥⎥⎥⎦

for n ∈ A, so that

supn≥0‖cn(Θ − L) = sup

n≥0‖cn(Θ)− cn(L)

= maxsupn∈A‖h− cn(L)‖, sup

n/∈A

‖cn(L)‖

= ‖h−m∑

⎡⎢⎢⎢⎣

w2k...

⎤⎥⎥⎥⎦ ‖ < ǫ.

4. Several isometric variables

In this section we discuss an example of a defining relation in several variables.More precisely, we consider orthogonal polynomials in several variables satisfyingthe isometric relations X+

k Xk = 1, k = 1, . . . , N . We set A = 1 −X+k Xk | k =

1, . . . , N and notice that the index set of A is F+N . Also if φ is a linear functional

on R(A) then its kernel of moments is invariant under the action of F+N on itself

by juxtaposition, that is,

Kφ(τσ, τσ′) = Kφ(σ, σ′), τ, σ, σ′ ∈ F+N . (4.1)

In fact, a kernel K satisfies (4.1) if and only if K = Kφ for some linear functionalon R(A). Positive definite kernels satisfying (4.1) have been already studied, seefor instance [9] and references therein. In particular, the class of isotropic processeson homogeneous trees give rise to positive definite kernels for which a theory oforthogonal polynomials (Levinson recursions) was developed in [10]. Here we dis-cuss in more details another class of kernels satisfying (4.1) which was considered,for instance, in [24].

4.1. Cuntz-Toeplitz relations

Consider the class of positive definite kernels with property (4.1) and such that

K(σ, τ) = 0 if there is no α ∈ F+N such that σ = τα or τ = σα. (4.2)

We showed in [17] that K has properties (4.1) and (4.2) if and only if K =Kφ for some q-positive functional on R(ACT ), where ACT = 1 − X+

k Xk | k =

1, . . . , N ∪ X+k Xl, k, l = 1, . . . , N, k = l. The relations in ACT are defining the

Cuntz-Toeplitz algebra (see [22] for details). The property (4.2) shows that K isquite sparse, therefore it is expected to be easy to analyze such a kernel. Still,

there are some interesting aspects related to this class of kernels, some of whichwe discuss here.

Let φ be a strictly q-positive kernel on R(ACT ) and let Kφ be the associatedkernel of moments. Since the index set ofACT is still F+

N , and this is totally orderedby the lexicographic order, we can use the results described in Subsection 3.1 andassociate to Kφ a family γσ,τσ≺τ of complex numbers with |γσ,τ | < 1, uniquelydetermining Kφ by relations of type (3.4). It was noticed in [17] that Kφ hasproperties (4.1) and (4.2) if and only if γτσ,τσ′ = γσ,σ′ and γσ,τ = 0 if thereis no α ∈ F+

N such that σ = τα or τ = σα. The main consequence of these

relations is that Kφ is uniquely determined by γσ = γ∅,σ, σ ∈ F+N − ∅. We

define dσ = (1 − |γσ|2)1/2. The orthogonal polynomials associated to φ satisfythe following recurrence relations which follow easily from (3.7), (3.8) (see [17] for

details): ϕ∅ = ϕ♯∅ = s

−1/2∅,∅ and for k ∈ 1, . . . , N, σ ∈ F+

ϕkσ =1

dkσ(Xkϕσ − γkσϕ♯

kσ−1), (4.3)

ϕ♯kσ =

dkσ(−γkσXkϕσ + ϕ♯

kσ−1). (4.4)

The results corresponding to Theorem 3.5 and Theorem 3.6 can be also easilyobtained (see [8]), but the constructions around the Szego kernel are more inter-esting in this situation. Thus, there is only one Hilbertian condition involved inthe definition of H0(F) in this case. In fact, it is easy to see that H0(F) can beidentified with the full Fock space l2(F+

N ), the l2 space over F+N . Now, concerning

evaluation of elements of H0(F), if we are going to be consistent with the point ofview that the “points for evaluation” come from the unital homomorphisms of thepolynomial algebra inside H0(F), then we have to consider an infinite-dimensionalHilbert space E and the set

B1(E) = Z = (Z1, . . . , ZN) ∈ L(E)N |N∑

ZkZ∗k < I.

For σ = i1 . . . ik ∈ F+N we write Zσ instead of Zi1 . . . Zik

. Then we define for

Θ ∈ l2(F+N )⊗ E and Z ∈ B1(E),

Θ(Z) =∑

σ∈F+N

ZσΘσ,

which is an element of the set L(E) of bounded linear operators on the Hilbertspace E . Next, for Z ∈ B1(E) we define SZ : E → l2(F+

N )⊗ E by the formula:

SZf =∑

σ∈F+N

eσ ⊗ (Zσ)∗f, f ∈ E .

Then SZ ∈ L(E , l2(F+N ) ⊗ E) and we can finally introduce the Szego kernel on

B1(E) by the formula:

S(Z, W ) = S∗ZSW , Z, W ∈ B1(E).

Theorem 4.1. S is a positive definite kernel on B1(E) with the properties:

(1) Θ(z) = S∗ZΘ, Θ ∈ l2(F+

N )⊗ E , Z ∈ B1(E).(2) The set SZf | Z ∈ B1(E), f ∈ E is total in l2(F+

N )⊗ E.Proof. The fact that S is positive definite and (1) are immediate. More interestingis (2) and we reproduce here the proof given in [17]. Let f = fσσ∈F

element of l2(F+N ) ⊗ E orthogonal to the linear space generated by SZf | Z ∈

B1(E), f ∈ E. Taking Z = 0, we deduce that f∅ = 0. Next, we claim that for eachσ ∈ F+

N − ∅ there exist

Zl = (Zl,1, . . . , Zl,N) ∈ B1(E), l = 1, . . . , 2|σ|,such that

range[

Z∗1,σ . . . Z∗

2|σ|,σ]

= E ,and

Zl,τ = 0 for all τ = σ, |τ | ≥ |σ|, l = 1, . . . , 2|σ|.Once this claim is proved, a simple inductive argument gives f = 0. In order

to prove the claim we need the following construction. Let enijni,j=1 be the matrix

units of the algebra Mn of n× n matrices. Each enij is an n× n matrix consisting

of 1 in the (i, j)th entry and zeros elsewhere. For a Hilbert space E1 we defineEn

ij = enij⊗IE1 and we notice that En

ijEnkl = δjkEn

il and E∗nji = En

ij . Let σ = i1 . . . ik

so that E = E⊕2|σ|1 for some Hilbert space E1 (here we use in an essential way the

assumption that the space E is of infinite dimension). Also, for s = 1, . . . , N , wedefine Js = l ∈ 1, . . . , k | ik+1−l = s and

Z∗p,s =

r∈Js

E2|σ|r+p−1,r+p, s = 1, . . . , N, p = 1, . . . , |σ|.

We can show that for each p ∈ 1, . . . , |σ|,

Z∗p,σ =

1√2k

E2|σ|p,k+p (4.5)

Zp,τ = 0 for τ = σ, |τ | ≥ |σ|. (4.6)

We deduce∑N

s=1 Zp,sZ∗p,s = 1

∑Ns=1

∑r∈Js

E2|σ|r+p,r+p−1E

2|σ|r+p−1,r+p

∑Ns=1

∑r∈Js

E2|σ|r+p,r+p = 1

∑kr=1 E

2|σ|r+p,r+p < I,

hence Zp ∈ B1(E) for each p = 1, . . . , |σ|. For every word τ = j1 . . . jk ∈ F+N − ∅

we deduce by induction that

Z∗p,jk

. . . Z∗p,j1 =

1√2k

r∈Aτ

E2|σ|r+p−1,r+p+k−1, (4.7)

where Aτ = ∩k−1p=0(Jjk−p

− p) ⊂ 1, . . . , N and Jjk−p− p = l − p | l ∈ Jik−p

We show that Aσ = 1 and Aτ = ∅ for τ = σ. Let q ∈ Aτ . Therefore, forevery p ∈ 0, . . . , k − 1 we must have q + p ∈ Jjk−p

or ik+1−q−p = jk−p. Forp = k − 1 we deduce j1 = i2−q and since 2 − q ≥ 1, it follows that q ≤ 1. Alsoq ≥ 1, therefore the only element that can be in Aτ is q = 1, in which case we musthave τ = σ. Since l ∈ Jik+1−l

for every l = 1, . . . , k − 1, we deduce that Aσ = 1and Aτ = ∅ for τ = σ. Formula (4.7) implies (4.5). In a similar manner we canconstruct elements Zp, p = |σ|+ 1, . . . , 2|σ|, such that

Z∗p,σ =

1√2k

E2|σ|p+k,p

Zp,τ = 0 for τ = σ, |τ | ≥ |σ|.Thus, for s = 1, . . . , N , we define Ks = l ∈ 1, . . . , k | ik = s and

Z∗p,s =

r∈Ks

E2|σ|r+p−k,r+p−k−1, s = 1, . . . , N, p = |σ|+ 1, . . . , 2|σ|.

Z∗1,σ . . . Z∗

2|σ|,σ]

=1√2k

2|σ|1,k+1 . . . E

2|σ|k,2k E

2|σ|k+1,1 . . . E

2|σ|2k,k

whose range is E . This concludes the proof.

It is worth noticing that property (2) of S is no longer true if E is finite-dimensional. In fact, for dimE = 1 the set SZf | Z ∈ B1(E), f ∈ E is total inthe symmetric Fock space of CN (see [5]).

4.2. Kolmogorov decompositions and Cuntz relations

This is a short detour from orthogonal polynomials, in order to show a constructionof bounded operators satisfying the Cuntz-Toeplitz and Cuntz relations, based onparameters γσσ∈F

+N−∅ associated to a positive definite kernel with properties

(4.1) and (4.2).

First we deal with the Kolmogorov decomposition of a positive definite kernel.This is a more abstract version of the result of Kolmogorov already alluded to inSubsection 3.5. For a presentation of the general result and some applications,see [22], [30]. Here we consider K : N0 × N0 → C a positive definite kernel andlet γk,j be the family of parameters associated to K as in Subsection 3.1. Inaddition we assume K(j, j) = 1 for j ≥ 0. This is not a real loss of generality andit simplifies some calculations. We also assume |γk,j | < 1 for all k < j. Then weintroduce for 0 ≤ k < j the operator Vk,j on l2(N0) defined by the formula

Vk,j = (J(γk,k+1)⊕ 1n−1)(1⊕ J(γk,k+2)⊕ 1n−2) . . . (1n−1 ⊕ J(γk,j))⊕ 0

and we notice that

Wk = s− limj→∞

Vk,j (4.8)

is a well-defined isometric operator on l2(N0) for every k ≥ 0. If we define V (0) =I/C and V (k) = W0W1 . . .Wk−1/C for k ≥ 1, then we obtain the following resultfrom [13].

Theorem 4.2. The map V : N0 → l2(N0) is the Kolmogorov decomposition of thekernel K in the sense that

(1) K(j, l) = 〈V (l), V (j)〉, j, l ∈ N0.

(2) The set V (k) | k ∈ N0 is total in l2(N0).

It is worth noticing that we can write explicitly the matrix of Wk:⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

γk,k+1 dk,k+1γk,k+2 dk,k+1dk,k+2γk,k+3 . . .dk,k+1 −γk,k+1γk,k+2 −γk,k+1dk,k+2γk,k+3 . . .

0 dk,k+2 −γk,k+2γk,k+3 . . .... 0 dk,k+3

... 0. . .

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

We also see that Theorem 3.7 and Theorem 4.1 produce various kinds of Kol-mogorov decompositions for the corresponding Szego kernels. Based on a remarkin [12], we use Theorem 4.2 in order to obtain some large families of boundedoperators satisfying Cuntz-Toeplitz and Cuntz relations. Thus, we begin with apositive definite kernel K with properties (4.1) and (4.2). For simplicity we alsoassume K(∅, ∅) = 1 and let γσσ∈F

+N−∅ be the family of corresponding parame-

ters. In order to be in tune with the setting of this paper, we assume |γσ| < 1 forall σ ∈ F+

N − ∅. Motivated by the construction in Theorem 4.2, we denote by

cσ(W0), σ ∈ F+N − ∅, the columns of the operator W0. Thus,

c1(W0) =

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦ and cσ(W0) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

d1 . . . dσ−1γσ

−γ1d2 . . . dσ−1γσ

...−γσ−1γσ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

for 1 ≺ σ.

We now define the isometry U(k) on l2(F+N ) by the formula:

U(k) = [ckτ (W0)]τ∈F+N

, k = 1, . . . , N.

Theorem 4.3.

(a) The family U1, . . . , UN satisfies the Cuntz-Toeplitz relations: U∗kUl = δk,lI,

k, l = 1, . . . , N .

(b) The family U1, . . . , UN satisfies the Cuntz relations U∗kUl = δk,lI, k, l =

1, . . . , N and∑N

k=1 UkU∗k = I, if and only if

σ∈F+N−∅

dσ = 0. (4.9)

Proof. (a) follows from the fact that W0 is an isometry. In order to prove (b) weneed a characterization of those W0 which are unitary. Using Proposition 1.4.5 in[16], we deduce that W0 is unitary if and only if (4.9) holds.

It is worth mentioning that if we define V (∅) = I/C and V (σ) = U(σ)/C,σ ∈ F+

N − ∅, where U(σ) = U(i1) . . . U(ik) provided that σ = i1 . . . ik, then V isthe Kolmogorov decomposition of the kernel K. A second remark here is that thecondition (4.9) is exactly the opposite of the condition for K being in the Szegoclass. Indeed, it is easily seen that for a positive definite kernel with properties(4.1) and (4.2), the condition (3.5) is equivalent to

∏σ∈F

+N−∅ dσ > 0.

5. Several hermitian variables

In this section we discuss another example of defining relations in several variables.The theory corresponding to this case might be viewed as an analogue of the theoryof orthogonal polynomials on the real line. We set A = Yk − Y +

k | k = 1, . . . , Nand A′ = A ∪ YkYl − YlYk | k, l = 1, . . . , N, and notice that R(A) = PN . Also,R(A′) is isomorphic to the symmetric algebra over CN . Orthogonal polynomialsassociated to R(A′), that is, orthogonal polynomials in several commuting vari-ables were studied intensively in recent years, see [21]. In this section we analyzethe non-commutative case. The presentation follows [15].

Let φ be a strictly q-positive functional on TN (A2) and assume for somesimplicity that φ is unital, φ(1) = 1. The index set of A is F+

N and let ϕσσ∈F+N

be the orthonormal polynomials associated to φ. We notice that for any P, Q ∈ PN ,

〈XkP, Q〉φ = φ(Q+XkP )

= φ(Q+X+k P )

= 〈P, XkQ〉φ,

which implies that the kernel of moments satisfies the relation sασ,τ = sσ,I(α)τ for

α, σ, τ ∈ F+N , where I denotes the involution on F+

N given by I(i1 . . . ik) = ik . . . i1.This can be viewed as a Hankel type condition, and we already noticed that evenin the one-dimensional case the parameters γk.j of the kernel of moments of aHankel type are more difficult to be used. Therefore, we try to deduce three-termsrelations for the orthonormal polynomials. A matrix-vector notation already usedin the commutative case, turns out to be quite useful. Thus, for n ≥ 0, we definePn = [ϕσ]|σ|=n , n ≥ 0, and P−1 = 0.

Theorem 5.1. There exist matrices An,k and Bn,k such that

XkPn = Pn+1Bn,k + PnAn,k + Pn−1B∗n−1,k, k = 1, . . . , N, n ≥ 0.

Each matrix An,k is a selfadjoint Nn × Nn matrix, while each Bn,k is anNn+1 ×Nn matrix such that

Bn,1 . . . Bn,N

is an upper triangular invertible matrix with strictly positive diagonal for everyn ≥ 0. For n = −1, B−1,k = 0, k = 1, . . . , N . The fact that Bn is upper triangularcomes from the order that we use on F+

N . The invertibility of Bn is a consequenceof the fact that φ is strictly q-positive and appears to be a basic translation of thisinformation. It turns out that there are no other restrictions on the matrices An,k,Bn,k as shown by the following Favard type result.

Theorem 5.2. Let ϕσ =∑

τσ aσ,τXτ , σ ∈ F+N , be elements in PN such that ϕ∅ = 1

and aσ,σ > 0. Assume that there exists a family An,k, Bn,k | n ≥ 0, k = 1, . . . , N,of matrices such that A∗

n,k = An,k and Bn =[

Bn,1 . . . Bn,N

]is an upper

triangular invertible matrix with strictly positive diagonal for every n ≥ 0. Alsoassume that for k = 1, . . . , N and n ≥ 0,

Xk [ϕσ]|σ|=n = [ϕσ]|σ|=n+1 Bn,k + [ϕσ]|σ|=n An,k + [ϕσ]|σ|=n−1 B∗n−1,k,

where [ϕσ]|σ|=−1 = 0 and B−1,k = 0 for k = 1, . . . , N . Then there exists a unique

strictly positive functional φ on R(A) such that ϕσσ∈F+N

is the family of or-

thonormal polynomials associated to φ.

There is a family of Jacobi matrices associated to the three-term relation inthe following way. For P ∈ R(A)(= PN ), define

Ψφ(P )ϕσ = Pϕσ.

Since the kernel of moments has the Hankel type structure mentioned above, itfollows that each Ψφ(P ) is a symmetric operator on the Hilbert space Hφ withdense domain D, the linear space generated by the polynomials ϕσ, σ ∈ F+

N .Moreover, for P, Q ∈ PN ,

Ψφ(PQ) = Ψφ(P )Ψφ(Q),

and Ψφ(P )D ⊂ D, hence Ψφ is an unbounded representation of Pn (the GNSrepresentation associated to φ). Also, φ(P ) = 〈Ψφ(P )1, 1〉φ for P ∈ PN . We dis-tinguish the operators Ψk = Ψφ(Yk), k = 1, . . . , N , since Ψφ(

∑σ∈F

cσYσ) =∑σ∈F

cσΨφ,σ, where we use the notation Ψφ,σ = Ψi1 . . . Ψikfor σ = i1 . . . ik. Let

e1, . . . , eN be the standard basis of CN and define the unitary operator W froml2(F+

N ) onto Hφ such that W (eσ) = ϕσ, σ ∈ F+N . We see that W−1D is the linear

space D0 generated by eσ, σ ∈ F+N , so that we can define

Jk = W−1Ψφ,kW, k = 1, . . . , N,

on D0. Each Jk is a symmetric operator on D0 and by Theorem 5.1, the matrix of(the closure of) Jk with respect to the orthonormal basis eσσ∈F

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

A0,k B∗0,k 0 . . .

B0,k A1,k B∗1,k

0 B1,k A2,k. . .

.... . .

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

We call (J1, . . . , JN ) a Jacobi N -family on D0. It is somewhat unexpected that theusual conditions on An,k and Bn,k insure a joint model of a Jacobi family in thefollowing sense.

Theorem 5.3. Let (J1, . . . , JN ) a Jacobi N -family on D0 such that A∗n,k = An,k and

Bn,1 . . . Bn,N

]is an upper triangular invertible matrix with strictly

positive diagonal for every n ≥ 0. Then there exists a unique strictly q-positivefunctional φ on PN with associated orthonormal polynomials ϕσσ∈F

such that

the map

W (eσ) = ϕσ, σ ∈ F+N ,

extends to a unitary operator from l2(F+N ) onto Hφ and

Jk = W−1Ψφ,kW, k = 1, . . . , N.

Proof. First the Favard type Theorem 5.2 gives a unique strictly q-positive func-tional φ on PN such that its orthonormal polynomials satisfy the three-term re-lation associated to the given Jacobi family, and then the GNS construction willproduce the required W and Ψφ,k, as explained above.

One possible application of these families of Jacobi matrices involves someclasses of random walks on F+

N . Figure 4 illustrates an example for N = 2 andmore details are planned to be presented in [7].

Figure 4. Random walks associated to a Jacobi family, N = 2

We conclude our discussion of orthogonal polynomials on hermitian variablesby introducing a Szego kernel that should be related to orthogonal polynomialson PN . Thus, we consider the Siegel upper half-space of a Hilbert space E by

H+(E) = (W1 . . . WN ) ∈ L(E)N |W1W∗1 + . . . + WN−1W

∗N−1 <

2i(WN −W ∗

We can establish a connection between B1(E) and H+(E) similar to the well-knownconnection between the unit disk and the upper half plane of the complex plane.Thus, we define the Cayley transform by the formula

C(Z) = ((I + ZN)−1Z1, . . . , (I + ZN )−1ZN−1, i(I + ZN )−1(I − ZN )),

which is well defined for Z = (Z1, . . . , ZN ) ∈ B1(E) since Zk must be a strictcontraction (‖Zk‖ < 1) for every k = 1, . . . , N . In addition, C establishes a one-to-one correspondence from B1(E) onto H+(E). The Szego kernel on B1(E) canbe transported on H+(E) by the Cayley transform. Thus, we introduce the Szegokernel on H+(E) by the formula:

S(W, W ′) = F ∗W FW ′ , W, W ′ ∈ H+(E),

where FW = 2diag((−i + W ∗N ))SC−1(W ). Much more remains to be done in this

direction. For instance, some classes of orthogonal polynomials of Jacobi type andtheir generating functions are considered in [7].

Finally, we mention that there are examples of polynomial relations for whichthere are no orthogonal polynomials. Thus, consider

A = X+k −Xk | k = 1, . . . , N ∪ XkXl + XlXk | k, l = 1, . . . , 2N,

then R(A) ≃ Λ(CN ), the exterior algebra over CN . If φ is a unital q-positivedefinite functional on R(A), then φ(X2

k) = 0 for k = 1, . . . , N . This and theq-positivity of φ force φ(Xσ) = 0, σ = ∅, therefore there is only one q-positivefunctional on R(A) which is not strictly q-positive. Therefore there is no theoryof orthogonal polynomials over this A. However, the situation is different for A =XkXl + XlXk | k, l = 1, . . . , 2N. This and other polynomial relations will beanalyzed elsewhere.

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T. Banks and T. ConstantinescuDepartment of MathematicsUniversity of Texas at DallasRichardson, TX 75083, USAe-mail: banks@utdallas.edue-mail: tiberiu@utdallas.edu

J.L. JohnsonDepartment of Mathematics and Computer ScienceWagner CollegeStaten Island, NY 10301, USAe-mail: joeljohn@wagner.edu

Functions of Several Variables in theTheory of Finite Linear StructuresPart I: Analysis

M. Bessmertnyı

Abstract. The notion of a finite linear structure is introduced which gener-alizes the notion of a linear electrical network. A finite linear structure is aKirchhoff graph to which several pairs of external vertices (“terminals”) arecoupled. A Kirchhoff graph is a finite linear graph g to whose edges p thequantities Ug and Ig are related. The values of Ug and Ig are complex num-bers. It is assumed that the circuit Kirchhoff law holds for Ug and the nodalKirchhoff law holds for Ig. It is also assumed that for the quantities Ug (gener-alized voltages) and the quantities Ig (generalized currents) corresponding tothe edges of the Kirchhoff graph the generalized Ohm law Ug = zg · Ig holds.The generalized impedances zg are complex numbers which are considered asfree variables. The Kirchhoff laws and the state equations Ug = zg · Ig lead tolinear relations for the values Ug and Ig corresponding to the “external” edgesof the finite linear structures, i.e., the edges incident to the terminals. Theselinear relations between external voltages and currents can be expressed eitherin terms of the impedance matrix Z if the considered finite linear structureis k-port or in terms of the transmission matrix A if the considered finitelinear structure is 2k-port. The properties of the impedance and transmis-sion matrices Z and A as functions of the complex variables zg are studied.The consideration of the present paper served as a natural motivation for thestudy of the class of matrix functions which was introduced in the previouspaper MR2002589 of the author.

Mathematics Subject Classification (2000). Primary: 47B50; Secondary: 30A9694A20.

Keywords. System theory, electrical networks, impedance matrices, transfermatrices, functions of several complex variables.

This paper is a translation, prepared by V. Katsnelson, of a part from the third chapter fromthe author’s Ph.D. thesis entitled “Functions of Several Variables in the Theory of Finite LinearStructures”, Kharkov, 1982.

92 M. Bessmertnyı

1. Summary of the graph theory

1 In graph theory the terminology and symbolism is not standardized. Below we

follow mainly [SeRe] and [Zy].

• A finite graph g = gX, P is a finite set X = x = ∅ of points (calledvertices) connected by a finite set P = p of line segments (called edges).Every edge connects two and only two vertices of the graph g; several edgescan be connected to one vertex.• A vertex x and an edge p are said to be incident if the vertex x is an endpoint

of the edge p. The notation x ∈ p means that the vertex x and the edge pare incident.• Let g = gX, P be a graph with the vertices set X and the edges set P , and

let P ′ be a subset of the set P . The subgraph of the graph g generated by thesubset P ′ ⊆ P is the finite graph g′ = g′X ′, P ′, where the set X ′ consistsof all those vertices xj ∈ X of the graph g which are incident at least to onep′ ∈ P ′.• An orientation of an edge p is a triple x1, p, x2, where x1, x2 are the ver-

tices incident to the edge p. The orientation x2, p, x1 is said to be oppositeto the orientation x1, p, x2.• The edge set P of the graph gX, P is said to be oriented if a certain

orientation is assigned to every edge p ∈ P . The oriented edges are denotedby arrows. The direction of the arrow corresponds to the orientation of theedge.• A path in the graph g=gX,P is a finite sequence x1p1x2p2x3 . . . xk pk xk+1

such that each edge pl is incident to the vertices xl and xl+1 (1 ≤ l ≤ k).• If xk+1 = x1, and the edges and the vertices (except x1) are not repeated,

then the path is said to be a simple circuit.• The set of all simple circuits of the graph gX, P is denoted by F.• An edge p ∈ P is incident to a circuit f ∈ F if the circuit f contains the

edge p.• We distinguish the simple circuits f = x1 p1 x2 p2 x3 . . . xk pk x1 and f ′ =

x1 pk xk . . . x3 p2 x2 p1 x1 by the orientations of their edges plkl=1.• A simple circuit f ∈ F is said to be an oriented simple circuit, or a contour

of the graph gX, P if one of its two possible orientations is chosen.• The orientations of a contour f and of an edge p which is incident to the

contour, p ∈ f , are– equal if the orientation of the edge p coincides with the chosen orientation

of the contour;– opposite if the orientation of the edge p is opposite to the chosen orientation

of the contour.• A graph g is said to be connected if every two vertices can be connected by

some path.• A connected component of the graph g is a connected subgraph of g which is

not contained in another connected subgraph of g.

Functions of Several Variables in Finite Linear Structures 93

• An isolated vertex of the graph g (i.e., the vertex which does not connectto another vertices by edges) is considered as a connected component of thegraph g.

• A tree of a connected graph g is a connected subgraph of g which containsall the vertices but no circuit of the graph g.

• If a tree of a graph is fixed, then the edge belonging to the tree is said to bethe branch of the tree, and the edge which does not belong to the tree is saidto be the chord of the tree.

• The forest of the graph g is the set of all the trees of all its connected com-ponents.

• A simple cut-set of the graph g is the set P0 of its edges such that

a) the subgraph g0 obtained from the graph g by removing the set P0 hasmore connected components than the original graph g;

b) no proper subset of the set P0 satisfies the condition a).

It is clear that every simple cut-set of the graph g is a simple cut-set of someconnected component of g. Removing the set P0 of the edges of some simple cut-setincreases the number of connected components of the graph by one.

In what follows we need to consider oriented simple cut-sets. Let P0 be asimple cut-set of a connected graph g. Then the subgraph g0 obtained from thegraph g by removing the set P0 of the edges has two connected components. Thecut-set P0 of the connected graph g is oriented if each edge p of the cut-set P0 isassigned the orientation x1 p x2, where x1, x2 are vertices which are incident tothe edge p, and x1 ∈ X1, x2 ∈ X2.

• We say that the orientations of the cut-set P0 and of the edge p which belongsto P0

– are equal if the directions on the edge corresponding to the orientations ofthe cut-set and of the edge coincide;

– are opposite if the directions on the edge corresponding to the orientationsof the cut-set and of the edge are opposite.

Definition 1.1. A graph gX, P is said to be an oriented finite graph if

1. gX, P is a finite graph.

2. The edges, the simple circuits and the simple cut-sets of the graph are enu-merated in such a manner that:(a) pj (j = 1, 2, . . . , e) are all the edges of the graph gX, P;(b) fk (k = 1, 2, . . . , c) are all the simple circuits of the graph gX, P.(c) sν (ν = 1, 2, . . . , s) are all the simple cut-sets of the graph gX, P;

3. The edges, the simple circuits and the simple cut-sets of the graph gX, Pare oriented.

94 M. Bessmertnyı

For oriented graphs, the notions of the circuit matrix Ba and the cut-setmatrix Qa are introduced:

⎧⎪⎨⎪⎩

1, if pj ∈ fk, and their orientations are the same;

0, if pj ∈ fk;

−1, if pj ∈ fk, and their orientations are opposite;

qνj =

⎧⎪⎨⎪⎩

1, if pj ∈ sν , and their orientations are the same;

0, if pj ∈ sν ;

−1, if pj ∈ sν , and their orientations are opposite.

It is known that the circuit matrix is of rank e−v+r, and the cut-set matrix is of therank v− r. (Here e, v and r are the total number of edges, vertices and connectedcomponents, respectively.) As usual, B and Q are submatrices of the exhaustivecircuit matrix Ba and the cut-set matrix Qa, respectively, consisting of some e−v+r and v− r linearly independent rows. One of the main properties of the matricesB and Q is their orthogonality: Q ·B′ = 0. The proof can be found in [SeRe].

2 Let us define the notion of a Kirchhoff graph.

Definition 1.2. A graph g = gX, P is said to be a Kirchhoff graph if

1. gX, P is a finite oriented graph.2. To each oriented edge α, pj , β (α, β are the vertices incident to the edge

pj) correspond some quantities U(j)αβ and I

(j)αβ . To the opposite oriented edge

β, pj, α correspond the quantities U(j)βα = −U

(j)αβ ; I

(j)βα = −I

(j)αβ . (We do not

concretize the nature of the quantities U and V : they can be numbers orfunctions, real or complex, or, more general, can belong to abelian groups,

the quantities U(j)αβ to one group, the quantities I

(j)αβ to another.)

Moreover the following assumptions are satisfied:(a) The first Kirchhoff law

pj∈S

I(j)αβ = 0 (1.1)

holds for every cut-off S of the graph g. (The summation runs over thedirections on the edges pj ∈ S which are determined by the orientationof the cut-off S.)

(b) The second Kirchhoff law∑

pj∈f

U(j)αβ = 0 (1.2)

holds for every contour f of the graph g. (The summation runs over thedirections on the edges pj ∈ f which are determined by the orientationof the contour f .)

3. The multiplication U(j)αβ ·I

(j)αβ of the quantities U

(j)αβ and I

(j)αβ (not commutative

in general, but distributive with respect to addition) is defined.

Remark 1.3. Since the graph gX, P is oriented, to each edge pj corresponds one

distinguished direction. Introducing the vector-columns I and U with entries I(j)αβ

and U(j)αβ , one can represent the Kirchhoff laws in the form

Q · I = 0, B · U = 0,

where Q and B are the above-described submatrices of the exhaustive cut-off andcircuit matrices Qa and Ba.

Remark 1.4. Usually one formulates the first Kirchhoff law for special cut-offs ofthe graph g, namely for the so-called nodes. A node is the set of edges which areincident to one vertex. In [SeRe] is shown that this “restricted” formulation of thefirst Kirchhoff law is equivalent to the general one.

Langevin’s Theorem. For every Kirchhoff graph gX, P the relations∑

I(j)αβ · U

(j)αβ = 0,

(j)αβ · I

(j)αβ = 0 (1.3)

hold. (The summation is taken over all edges directed according to given orienta-tion.)

Proof. Without loss of generality we may assume that the graph is connected. Letus fix some vertex α0 of the graph g and introduce the “potential” Φ(αk) of anarbitrary vertex αk:

Φ(αk) =∑

Π(α0,β0)

α′,β′ ,

whereΠ(α0, αk) = α0 p0 α1 p1 α2 . . . αk−1 pk−1 αk

is an arbitrary path.The summation is taken according to the directions αj , pj αj+1 on the

edges pj belonging to the path Π(α0, αk). In view of the second Kirchhoff law,Φ(αk) does not depend on the choice of the path leading from α0 to αk. Then

∑I jαβ · U

(j)αβ =

I(j)αβ ·

[Φ(β)− Φ(α)

However, ∑

I(j)αβ · Φ(β) =

I(j)αβ

)· Φ(β) = 0

since the set of all edges incident to the vertex α is a simple cut-off set (or theunion of simple cut-off sets), and according to the first Kirchhoff law, the sumin the square brackets is equal to zero. The same holds for the second summand∑α,β

I(j)αβ ·Φ(α).

In textbooks on theoretical electrical engineering one formulates Langevin’stheorem (in the setting of electrical networks) in the following manner: the totalactive (reactive) power produced in all energy sources is equal to the total active(reactive) power discharged in all energy collectors.

96 M. Bessmertnyı

In [EfPo] there was mentioned that Langevin’s Theorem is the immediateconsequence of Kirchhoff’s laws. Hence this theorem holds for every system inwhich Kirchhoff’s laws are postulated.

If the quantities U(j)α,β and I

(j)α,β are functions of some parameter t and if

Kirchhoff’s laws hold for every t, then, according to Langevin’s theorem,∑

I(j)α,β(t′) · U (j)

α,β(t′′) = 0

for every two values t′ and t′′ of the parameter t.

In what follows we omit the subindices α, β in the expressions U(j)

α,β, I(j)α,β

for the sake of simplicity specifying to what directions of the edges correspond“positive” values of the quantities Uj and Ij . In pictures we draw the arrow nearthe notation Uj and Vj . The direction of the arrow denotes the direction to whichthe “positive” value of the quantity corresponds.

2. Finite linear structures

1 Electrical networks will be the progenitor of the abstract objects considered in

the paper. Therefore, the terminology taken from the theory of electrical networksis used.

Definition 2.1. A finite linear structure is a Kirchhoff graph with an additionalweighting structure:

1. The set of the edges of the Kirchhoff graph is decomposed into three subsets

P0, P and P , where the subsets P and P have the same number of edges.2. The edges from P0 are said to be the elementary 2-ports. (Figure 1).

1 1′

Figure 1. Elementary 2-port and its graph.

The complex numbers Uj and Ij , which are concordant to the directionon the edge pj ∈ P0 corresponding to the orientation of pj, are related by

Uj = δj zj Ij (δj = ±1),

where zj is a complex number.

The complex number zj is said to be the impedance of the elementary2-port pj . In what follows these impedances zj are considered as independentvariables1.

If δj = +1 for all elementary 2-ports, the finite linear structure is saidto be passive.

3. The edges from P0 and P are paired: (pj , pj). Each pair (pj , pj) forms anideal 2× 2 transformer (see Figure 2).

The complex numbers Uj,Ij and Uj,−Ij , concordant 2 to the directionscorresponding to the orientations of the edges pj , pj , are related by

[tj 00 t−1

]·[Uj

where transmission coefficients tj are assumed to be non-zero real numbers.

U UP P

Figure 2. Ideal 2× 2-transformer and its graph.

2 Let us define the operation of opening coupling channels in a finite linear

structure.Let g be a finite linear structure; let x1 and x2 be two different vertices of the

graph belonging to the same connected component. Let us connect these verticesby the edge p of the ideal 2× 2 transformers (see Figure 2). The pair 1 and 1′ ofthe vertices incident to the edge p is left free and is said to be the exterior pair ofvertices (or terminals) of the new finite linear structure g′ which is derived fromthe initial structure g and the ideal 2× 2 transformer.

The above-described method of obtaining the pair of exterior vertices is saidto be opening coupling channels in the structure g. The considered finite linear

1It was not ruled out that some different elementary 2-ports of the considered linear structurehave the same impedances, that is, some different edges from P0 are weighted by the samecomplex variable.2The reason why the current Ij carries a negative sign is that most transmission engineers liketo regard their output current as coming out of the output point instead of going into the portas per standard usage.

98 M. Bessmertnyı

structure with the open coupling channels (interacting with other finite linearstructures by means of these coupling channels) is an open system (see [Liv]).

3 In what follows we consider finite linear structures with open coupling channels.

The structures interact with each other by means of these channels.

Definition 2.2. A finite linear structure with k open coupling channels (i.e., with2k exterior terminals) is said to be 2k-port.

Definition 2.3. A finite linear structure with open coupling channels is said to be2×2k-port if there are k coupling channels forming the input of the structure andthere are k coupling channels forming the output of the structure.

Let us connect each pair of the exterior vertices by the oriented exterior edgeand relate to these exterior edges the quantities3 Uj , Ij : U in

j , Iinj to the input edges

and Uoutj , −Iout

j to the output edges 4. Thus the state of the coupling channels of2k-port is characterized by two k-dimensional vector-columns

U =[U1, . . . , Uk

]′; I =

[I1, . . . , Ik

and the state of the coupling channels of 2 × 2k-port is characterized by two 2k-dimensional vector-columns

f in =[U in

1 , . . . , U ink ; I in

1 , . . . , I ink

fout =[Uout

1 , . . . , Uoutk ; Iout

1 , . . . , Ioutk

2k-ports and 2× 2k-ports will be depicted as it is shown in Figure 3.

Figure 3

In what follows we use often the simplest 2 × 2k-port – the so-called ideal2× 2k-transformer (see Figure 4).

3The values of these quantities are established after the transition to the steady state.4See Footnote 2.

•k′ (2k)′

•k 2k

Ik Ik...

.•1′ (k + 1)′

•1 (k + 1)I1 I1

k′ (2k)′

1′ (k + 1)′

1 (k + 1)

Figure 4. Ideal 2× 2k transformer and its graph.

For an ideal 2× 2k-transformer the relation[U

[t′ 0

0 t−1

holds, where U, I, U , U are vector-columns of the dimension k composed of the

quantities Uj , Ij and Uj , Ij , respectively.

The matrix t = tpqkp,q=1 of the transmission numbers of the transformeris assumed to be real and non-singular. The last requirement is equivalent to the

following: choosing the appropriate values of the quantities Uj, Ij , one can obtainarbitrary prescribed values of the quantities Uj , Ij .

100 M. Bessmertnyı

Definition 2.4. A 2×2k-port (2k-port) is said to be passive if for all 2-ports whichare interior elements of the 2× 2k-port (2k-port) the relations

Uj = δj zj Ij , where δj = +1,

4 Before we consider interaction characteristics of a finite linear structure, we

carry out a simple calculation.The number of independent equations of the first Kirchhoff law (i.e., the rank

of the cut-off matrix) is equal to v−r; the number of independent equations of thesecond Kirchhoff law (i.e., the rank of the circuit matrix) is equal to e+2k−v+ r.The number of the equations related the values of Uj and Ij (the relations on theinner edges) is equal to e. Here v is the number of the vertices, e is the number ofthe interior, and 2k is the number of the exterior edges of the 2× 2k-port; r is thenumber of connected components of the Kirchhoff graph. The total number of theequations is equal to 2(e + k). They relate 2(e + 2k) variables Uj and Ij .

If these equations are compatible and the variables U inj and Iin

j , j = 1, . . . , k,can be chosen as “free” variables, then, solving the obtained system with respectto unknown Uout

j and Ioutj , we obtain the relation

fout = A(z1, . . . , zn) · fin,

where A(z1, . . . , zn) is a matrix function of dimension 2k × 2k depending onthe variables z1, . . . , zn which are the impedances of the inner components of thestructure. The matrix function A(z1, . . . , zn) is said to be the transmission matrixof the 2× 2k-port.

Analogously, if the relation

U = Z(z1, . . . , zn) · Iholds for a 2k-port, where

U =[U1, . . . , Uk

]′, I =

[I1, . . . , Ik

are vector columns characterizing the states of the exterior edges of the 2k-port,then the matrix function Z(z1, . . . , zn) of dimension k × k is said to be theimpedance matrix of the 2k port.

Analogously, the admittance matrix of an 2k-port can be defined.

Since the considered linear structures are finite, the entries of the transmis-sion matrix, the impedance matrix and the admittance matrix are expressed fromthe coefficients of the appropriate linear systems rationally. Therefore, the trans-mission matrix, the impedance matrix and the admittance matrix are rationalfunctions of the variables z1, . . . , zn which are the impedances of the interioredges of the multiport. In the next sections we obtain necessary conditions whichthese matrices must satisfy if they exist. Remark that real physical problems leadto such finite linear structures for which at least one of the interaction characteris-tics (the transmission matrix, the impedance matrix and the admittance matrix)exists, and very often all three matrices exist.

3. Characteristic matrices of finite linear structures

The main goal of this section is to obtain a necessary condition which matricesmust satisfy for to be the transmission matrix, the impedance or the admittancematrix of a finite linear structure.

1 Since for all Kirchhoff graphs considered below the values of Uj and Ij are

complex numbers, Langevin’s theorem can be presented in the form∑

Ij · Uj = 0 ,∑

U j · Ij = 0.

These equalities imply that∑(

U jIj + IjUj

)= 0; i

∑(U jIj − IjUj

)= 0 . (L)

Characteristic properties of the matrices of multiports will be obtained asthe consequences of Langevin’s theorem. This approach was outlined in the paper[Ef] by A.V. Efimov. There apparently firstly Langevin’s theorem was applied forderiving characteristic properties of the matrices of multiports as functions of thecomplex frequency λ. (See also [EfPo].) We apply Langevin’s theorem for obtainingcharacteristic properties of the matrices of finite linear structures as functions ofseveral complex variables.

To formulate the result about the properties of the impedance matrix of apassive 2k-port, we recall the definition of a positive matrix function of severalcomplex variables.

To formulate this definition, we need the following subsets of Cn:

D+R = z : z ∈ Cn, Re z1 > 0, . . . ,Re zn > 0 ,D−

R = z : z ∈ Cn, Re z1 < 0, . . . ,Re zn < 0 ,D+

I = z : z ∈ Cn, Im z1 > 0, . . . , Im zn > 0 ,D−

I = z : z ∈ Cn, Im z1 < 0, . . . , Im zn < 0 .

Definition 3.1. A rational k×k matrix function M(z) = M(z1, . . . , zn) of complexvariables z1, . . . , zn is said to be positive if the following positivity conditions hold:

M(z) + M(z)∗ ≥ 0 for z ∈ D+R,

M(z) + M(z)∗ ≤ 0 for z ∈ D−R,

i (M(z)∗ −M(z)) ≥ 0 for z ∈ D+I ,

i (M(z)∗ −M(z)) ≤ 0 for z ∈ D−I .

The class of positive functions was introduced in the author’s PhD thesis[Be1]. See [Be2] for an English translation of the part of this thesis containingthe definition of the class of positive functions. Definition 3.1 of the present paperappears in [Be1] as Definition 0.3.

102 M. Bessmertnyı

We recall

Definition 3.2. A rational matrix function M(z1, . . . , zn) of complex variables

z1, . . . , zn is said to be real if it satisfies the condition M(z1, . . . , zn) ≡M(z1, . . . , zn).

Theorem 3.3. Assume that the impedance matrix Z(z1, . . . , zn) of some 2k-portexists. Then Z(z1, . . . , zn) it is a k × k rational matrix function of the variablesz1, . . . , zn which possesses the properties:

1. Z is real (in the sense of Definition 3.2).2. Z(z1, . . . , zn) is a homogeneous function of homogeneity degree 1, i.e.,

Z(λz1, . . . , λzn) = λZ(z1, . . . , zn) ∀λ ∈ C .

3. Z is a symmetric matrix function:

Z(z1, . . . , zn) ≡ Z ′(z1, . . . , zn) .

If moreover the 2k-port is passive, then the matrix function Z(z1, . . . , zn) is pos-itive in the sense of Definition 3.1.

Proof. Since the impedance matrix of the 2k port exists, the appropriate system oflinear equations for the values Uj , Ij is compatible, and we can take the variablesIj corresponding to the exterior edges as “free variables”. Solving this system ofequations, we obtain the relations

U = Z(z1, . . . , zn) · I,

I(e) = Ψ(z1, . . . , zn) · I,

where U, I and I(e) are vector-columns composed from the values Uj, Ij (j =

1, . . . , k) and I(e)ν (ν = 1, . . . , e) corresponding respectively to the interior and

exterior edges of the 2k-port. Since the total numbers of the linear equations isfinite, the entries of the matrix functions Z and Ψ are rational functions of the co-efficients of the system, thus, rational matrix functions of the variables z1, . . . , zn.If all z1, . . . , zn take real values, then all the coefficients of the system are real.Hence, the entries of the matrices Z(z1, . . . , zn), Ψ(z1, . . . , zn) take real valuesfor real z1, . . . , zn. Therefore, the matrix function Z(z1, . . . , zn) satisfies the con-dition

Z(z1, . . . , zn) ≡ Z(z1, . . . , zn).

The same is true for the matrix function Ψ(z1, . . . , zn).

Let λ ∈ C. Changing variables

zν → λzν , Iν → Iν , Uν → λUν , Z → λZ

does not affect the equalities in the considered system of linear equations.

Therefore,

Ψ(λz1, . . . , λzn) = Ψ(z1, . . . , zn) ,

Z(λz1, . . . , λzn) = λZ(z1, . . . , zn) .

Let us apply Langevin’s theorem in the form L to the Kirchhoff graph. Takinginto account that the values corresponding to the directions distinguished by theorientations on exterior edges and the output edges of the ideal transformers arerespectively Uj,−Ij (j = 1, . . . , k) and Uµ,−Iµ , we can rewrite L in the form

1≤j≤k

(U jIj ± IjUj

1≤ν≤e

ν I(e)ν ± I

ν U (e)ν

−∑

1≤µ≤t

[(I µUµ − IµUµ

)±(U µIµ − UµIµ

)], (3.3)

where the sums correspond to the exterior edges j, to the elementary 2-ports (ν)and to the ideal transformers (µ). After transformation, the sums in (3.3) take theform ∑

1≤j≤k

(U jIj ± IjUj

)= I∗

(Z∗(z)± Z(z)

)I, (3.4)

1≤ν≤e

ν I(e)ν ± I

ν U (e)ν

1≤ν≤e

(Iν(zν ± zν)Iν

1≤j≤n

(zj ± zj) I∗ψ∗(z)Λνj ψ(z)I, (3.5)

where Λνj are diagonal matrices whose diagonal entries are either 1, −1 or 0. (Therank of Λνj is equal to the total number of the components with impedance zj .

See Footnote 1.)

The summands corresponding to the ideal transformer vanish. Indeed, for theµth transformer,

(I µUµ − IµUµ

)+(U µIµ − UµIµ

)=[U µ, I µ

] (JΠ − T ∗

µJΠTµ

) [Uµ

(I µUµ − IµUµ

)−(U µIµ − UµIµ

)= −i ·

[U µ, I µ

] (JH − T ∗

µJHTµ

) [Uµ

JΠ − T ∗µJΠTµ = 0 , JH − T ∗

µJHTµ = 0.

[0 11 0

]; JH =

[0 −ii 0

]; Tµ =

[tµ 00 t−1

is the transmission matrix of the µth transformer.

104 M. Bessmertnyı

Taking into account (3.4) and (3.5), the equalities (3.3) can be rewritten as

I∗[Z∗(z) + Z(z)

1≤j≤n

(zj + zj)I∗Ψ∗(z)ΛjΨ(z)I;

iI∗[Z∗(z)− Z(z)

1≤j≤n

i(zj − zj)I∗Ψ∗(z)ΛjΨ(z)I.

Since the entries of I can be arbitrary, we have the equalities[Z∗(z) + Z(z)

1≤j≤n

(zj + zj)Ψ∗(z)ΛjΨ(z); (3.6)

i[Z∗(z)− Z(z)

1≤j≤n

i(zj − zj)Ψ∗(z)ΛjΨ(z). (3.7)

If all zj are real, the right-hand side of (3.7) vanishes. Thus, Z∗(x) = Z(x) for

real x. However, Z(x) = Z(x) for real x. Hence, Z ′(x) = Z(x) for real x. From theuniqueness theorem, Z ′(z) = Z(z) for all complex z.

If the considered 2k-port is passive, then all the matrices Λj are non-negative,and the inequalities (3.2) hold for the matrix function Z.

Let us turn to consider the transmission matrices.

Recall that a square rational 2k-matrix function W (z) = w(z1, , . . . , zn)of the complex variables z1, , . . . , zn is said to be J-expanding if the followingJ-expandability conditions are satisfied:

W ∗(z)JΠW (z)− JΠ ≥ 0 for z ∈ D+R,

W ∗(z)JΠW (z)− JΠ ≤ 0 for z ∈ D−R,

W ∗(z)JHW (z)− JH ≥ 0 for z ∈ D+I ,

W ∗(z)JHW (z)− JH ≤ 0 for z ∈ D−I ,

], JH =

[0 −iIm

and the domains D+R, D−

R,D+I , D−

I were defined in (3.1).

Theorem 3.4. Assume that the transmission matrix A(z1, . . . , zn) of a passive 2×2k-port exists. Then A(z1, . . . , zn) is a real J-expanding rational matrix functionof the variables z1, . . . , zn.

As before, elementary reasoning leads us to the relations

fout = A(z1, . . . , zn) fin,

I(e) = Ψ(z1, . . . , zn) fin.

Langevin’s theorem in the form L results in the equalities[I∗outUout − I∗inUin

]±[U∗

outIout − U∗inIin

1≤ν≤e

ν Ieν ± I

ν Ueν

where the summands in the left-hand side of the equality correspond to the exterioredges of the 2× 2k-port, and the summands in the right-hand side correspond tothe elementary 2-ports. The summands corresponding to the ideal transformersare annihilated.

Carrying out calculations and using the matrices

]and JH =

[0 −iIk

we obtain

f∗in

[A∗(z)JΠA(z)− JΠ

]fin =

1≤j≤n

(zj + zj) f∗in Ψ∗(z)ΛjΨ(z) fin, (3.8)

f∗in

[A∗(z)JHA(z)− JH

]fin =

1≤j≤n

i(zj − zj) f∗in Ψ∗(z)ΛjΨ(z) fin, (3.9)

where Λj ≥ 0 since 2×2k-port is passive. Since the vector-column fin is arbitrary,these inequalities result in J-expandability of the matrix function A(z1, . . . , zn).

Putting zj = iτj (τj , j = 1, . . . , n, are real) in (3.8), and zj = xj (xj , j =1, . . . , n, are real) in (3.9), we obtain:

A′(−iτ)JΠA(iτ) − JΠ ≡ 0,

A′( x )JHA(x ) − JH ≡ 0.

Since A(z) is rational,

A′(−z)JΠA(z)− JΠ ≡ 0, (3.10)

A′( z )JHA(z)− JH ≡ 0. (3.11)

Deriving the identities (3.10), (3.11), we did not use that the 2×2k-port is passive,that is the matrices Λj are non-negative.

Thus, for arbitrary (not necessary passive) 2× 2k ports the following state-ment holds:

Theorem 3.5. Assume that the transmission matrix A(z1, . . . , zn) of a 2×2k-portexists. Then A(z1, . . . , zn) is a real rational matrix function of variables z1, . . . ,zn satisfying the symmetry conditions (3.10) and (3.11).

Editorial Remark. From (3.6), (3.7) it follows that

Z(z) =∑

1≤j≤n

zjΨ∗(z)ΛjΨ(z) .

The left-hand side is analytic with respect to z, the factor Ψ(z) in the right-handside also is analytic with respect to z, and the factor Ψ∗(z) in the right-hand sideis anti-analytic with respect to z. Therefore,

Z(z) =∑

1≤j≤n

zjΨ∗(ζ)ΛjΨ(z) ∀z, ζ . (3.12)

106 M. Bessmertnyı

References

[Be1] Bessmertnyi, M.F.: Funkcii neskolьkih kompleksnyh peremennyh v teo-rii koneqnyh lineinyh struktur. Kandidacka dissertaci, Harьkovskiiuniversitet. Harьkov, 1982. 143 ss. (in Russian).

[Bessmertnyı,M.F. Func-

tions of several complex variables in the theory of finite linear structures. PhDthesis, Kharkov University. Kharkov, 1982. 143 pp.

[Be2] Bessmertnyı,M.F. On realizations of rational matrix functions of several com-plex variables. Translated from the Russian by D. Alpay and V. Katsnelson. Oper.Theory Adv. Appl., 134, Interpolation theory, systems theory and related topics(Tel Aviv/Rehovot, 1999), 157–185, Birkhauser, Basel, 2002.

[Ef] Efimov, A.V.: Ob odnom primenenii teoremy Lanжevena v teorii cepei.DAN Armnskoi SSR, 49:3 (1969), 118–123 (in Russian).[Efimov, A.V.: Onone application of Langevin’s theorem in the theory of electrical networks. DANArmyansk. SSR, 49:3 (1969), 118–123.]

[EfPo] Efimov, A.V. i V.P. Potapov: J - rastgivawie matricy-funkcii i ihrolь v analitiqeskoi teorii зlektriqeskih cepei. Uspehi matem. nauk,28:1 (1973), 65–130 (in Russian). English transl.:Efimov, A.V. and V.P.Potapov: J-expanding matrix functions and their rolein the analytical theory of electrical circuits. Russ. Math. Surveys, 28:1 (1973),pp. 69–140.

[Liv] Livxic, M.S.: Operatory, Kolebani, Volny (otkrytye sistemy).Nauka. Moskva, 1966. 298 ss. [In Russian]. English transl.:Livsic,M.S.(=Livshits,M.S.): Operators, oscillations, waves (open systems).(Translations of Mathematical Monographs, Vol. 34.) American MathematicalSociety, Providence, R.I., 1973. vi+274 pp.

[SeRe] Seshu, S. and M.B.Reed: Linear Graphs and Electrical Networks. (Addison-Wesley series in the engineering sciences. Electrical and control systems). Addison-Wesley, Reading, MA, 1961. 315 p. Russian transl.:Sexu, S., i M.B.Rid: Lineinye Grafy i Зlektriqeskie Cepi. Izda-telьstvo “Vysxa Xkola”. Moskva, 1971. 448 s.

[Zy] Zykov, A.A.: Teori Koneqnyh Grafov. Izdatelьstvo “Nauka”, SibirskoeOtdelenie. Novosibirsk, 1969. 543 s. [In Russian: Zykov, A.A.: Theory ofFinite Graphs.]

M. BessmertnyıSvody Square, 4Department of Mathematics, Faculty of PhysicsKharkov National University61077, Kharkov, Ukraine

Operator Method for Solution of PDEsBased on Their Symmetries

Samuil D. Eidelman and Yakov Krasnov

Abstract. We touch upon “operator analytic function theory” as the solutionof frequent classes of the partial differential equations (PDEs).

Mathematics Subject Classification (2000). 17A35, 30G30, 30G35, 35C05.

Keywords. Linear partial differential equations; Second-order constant coeffi-cient PDEs; Cauchy problem; Explicit solutions; Symmetry operator.

Introduction

Definition 0.1. A symmetry of PDEs is a transformation that maps any solutionto another.

It is well known that solving any problem in PDE theory can be substantiallyfacilitated by appropriate use of the symmetries inherit in the problem. For a givensolution of the DE, knowledge of an admitted symmetry leads to the generationof another solution. Thus the following main question arises:

Question 0.2. Given a PDE Qu = 0. When does it have a property, that if onestarts with a simple solution and supplied symmetries, one can generate all solu-tions?

We start with several motivation examples.

Example. The explicit (d’Alembert) form of the solution to two-dimensional ellip-tic and hyperbolic equations Qu ≡ uxx + εuyy = 0. Determining the formulas oftheir solutions is now straightforward:

u(x, y) = f(K1) + g(K2); K1 := y +√

εx, K2 := y −√εx, (0.1)

with (complex) f(z), g(z).

108 S.D. Eidelman and Y. Krasnov

Example. The operator form [6], [29], [33] of the heat polynomial solution to theparabolic equation Qu ≡ ut − uxx = 0. It can be written as:

u(x, t) = pm(K)[1] :=

Ki[ai], pm(x) =

aixi, K := x + 2x∂x. (0.2)

By K[ϕ] in (0.2) we denote the action K on ϕ and Ki[ϕ] := K[Ki−1[ϕ]].

Example. The polynomial-exponential solution of the Helmholz equation in R2:Qu ≡ uxx + uyy − εu = 0. Namely, in [29] solutions obtained as

u(x, y) = pm(K)[eαx+βy], K = x∂y − y∂x, α2 + β2 = ε (0.3)

with any complex polynomial pm(z).

Surprisingly, although types of the PDEs in Examples 1–3 are essentiallydifferent, their solutions are represented in unified (unconventional) form:

u(x1, x2, . . . , xn) =

fi(K1, K2, . . . , Ks)[u0i(x1, x2, . . . , xn)], (0.4)

The common factor in representations of the solutions to various type PDEsin the form (0.4) is the existence of an symmetry operators K. We start with

Definition 0.3. Given PDE Q(x, ∂x)u = 0. We could say that Q have enoughsymmetries if (0.4) holds. For all f(K), u(x) is a solution to PDE Qu = 0 in Rn.

In turn, the question whether or not a given PDO Q have enough symmetriescan not be answered without explicit calculations.

Definition 0.4. Let PDO Q have enough symmetries. Recall u(x) is a regularsolution to PDE Qu = 0 in Rn iff (0.4) holds with some entire functions fi(z).

Example. The heat equation has enough symmetries because (0.2) hold.

1. Let K be taken from (0.2). Obviously, ez is an entire function in C. Henceex+t := eK [1] is a regular solution.

2. The fundamental solution of the heat equation E(x, t) = 12√

πte−

4t is singu-

lar. The straightforward computations show that E(x, t) belongs to the kernelof K, (KE = 0) and therefore no analytic function F (z) fulfills the represen-tation E(x, t) := F (K)[1]. Only the Dirac delta function F (x) := δ(x) maybe considered as the reliable representative in this special case!

3. The function f(z) := 1/z is not analytic; therefore the function v(x, t)

v(x, t) := K−1[u(x, t)] =1

ey2−x2

4t u(y, t)dy (0.5)

is neither necessarily regular nor is it the solution to heat equation for u(x, t)in (0.2). Nevertheless, v0(x, t) ≡ K−1[1] is a singular solution to (0.2):

v0(x, t) := K−1[1] =1

√πe

4t erfi(x

Here erfi(z) is an imaginary error function erf(iz)/i and K[v(x, t)] = 1.

Symmetry Operator Method 109

Definition 0.5. The minimal possible set of a PDOs K1, . . . , Ks in Definition 0.3will be called an operator-valued characteristics or an operator indeterminates.

Remark 0.6. The linear ODE’s do not allow non-trivial (non-constant) operatorindeterminates since their solution space is always finite-dimensional.

Claim 0.7. Given PDE Q(x, ∂x)u = 0 in Rn. Suppose Q have enough symmetries(see Definition 0.3). Then Q admit an operator-valued characteristics.

Proof. If such PDOs K1, K2, . . . , Ks exists and u(x) fulfills (0.4) then Q[u(x)] = 0implies Q[Ki[u(x)]] = 0 for i = 1, . . . , s. Therefore K1, K2, . . . , Ks in (0.4) mustbe the symmetry operators to PDO Q and Qu0i(x) = 0. We call functions u0i(x)a “simplest” solution (see [4], [24], [20]).

The above cited PDEs possess, for arbitrary initial data, a regular solution, forone may use, at least formally, the theory of entire functions in operator variables(cf. [32], [18]). Moreover, their symmetries operators form a finite generated algebraLie (see [20]). However, a priori not all PDEs allow to write their solution in theform (0.4). For a more refined use of (0.4), a deeper knowledge of symmetries tothe given PDE is indispensable.

The objectives of this paper is to show, for a given PDE Qu = 0, how toconstruct its solution space SolQ(s, p). Namely, how to

– find their p “simplest” solutions,– construct s operator-valued characteristics in order to obtain (0.4),– study the conditions under which u(x) in (0.4) is a genuine function.

The following question is also of great importance:

Question 0.8. What is a “correct” definition of the singular solution to PDEs andhow can one construct these singular solutions using their symmetries?

Mathematical background

In the 19th century literature on differential equations, one can find extensivediscussions of the relation between “singular” and “general” (regular) solutions.However, no rigorous definitions of these terms emerged.

The basic idea is that a general solution is a family of solutions with the“expected” properties. It stands to reason that singular solutions, conversely, dif-fer from a general one by a specialization of its non-smoothness. For example,Euler effectively showed that exponential-polynomial functions, i.e., functions be-ing represented in the form pm(x)e〈λ,x〉 where pm(x) is polynomial of order m,forms the basis for a general solution to a linear constant coefficient system. Wefollow the results of L. Ehrenpreis [9] and V.P. Palomodov [25] in this direction.The natural and important problem of finding a polynomial basis in the solutionspace was studied in [19], [26], [29]. The method presented in this paper establishes(formally) regular solutions, mainly exponential-polynomial solutions (cf. [19]).

Here and throughout the sequel, C[x] = C[x1, . . . , xn] denote the space of(complex) exponential-polynomial functions and Q stands for a linear system ofPDO acting on the functions f ∈ C[x1, . . . , xn].

Our main observation is that by knowing a symmetry operators, it is possibleto study the solution space

SolQ(Rn) := f ∈ C[x1, . . . , xn] : Q[f ] = 0, SolQ(Rn) ⊂ ker Q. (0.6)

For a more detailed introduction we refer to L.V. Ovsiannikov [24], P.J. Olver [21],[22], G.W. Bluman and J.D. Cole [3] and to many other texts that have followedsince the fundamental insight to symmetries of PDE due to Lie appeared.

One of the most widely accepted definitions of symmetry of PDO Q is:

Definition 0.9. (cf. [20],[21]) The linear PDO L is said to be a symmetry operatorof the operator Q in C[x] if L[u] ∈ SolQ for all u ∈ SolQ.

The second definition, also accepted and equivalent to the one stated above,requires the existence of an additional operator to be in convolution with a givenoperator, as follows:

Proposition 0.10. (cf. [20],[21]) The linear PDO L is a symmetry operator of theoperator Q in C[x] if (i) L : C[x] → C[x] and (ii) QL = L′Q for some lineardifferential operator L′.

The rather trivial case in which L is a symmetry of Q is of the form RQ forsome linear operator R. In order to avoid such trivialities we adopt the conventionthat L1 ∼ L2 if L1 − L2 = RQ for some R. Denote by Sym(Q) the set of allsymmetries operators for Q factorized by the above equivalence. Thus, it is nothard to prove

Proposition 0.11. Sym(Q) is an algebra with composition as a basic operation.

If the symmetry operator L of PDO Q is a mth-order PDO with exponential-polynomial coefficients (possible with values in certain associative algebra A), thenwe call the operator L the m-symmetry of Q.

Definition 0.12. An operator Q is said to be of finite type if its symmetry algebraSym(Q) is finitely generated by m-symmetries operators.

In [20], [21] it was proved that the symmetry algebra of PDO Q and finitelygenerated by 1-symmetry operators, may be infinite-dimensional only in R2. Weshow in Section 5 that symmetries of the second order, linear, constant coefficientPDE are (algebraically) isomorphic to the pseudo-conformal Lie algebra so(p, m),p ≥ m. We finish this section with

Definition 0.13. so(p, m) is the N = (p+m)(p+m−1)/2-dimensional Lie algebraof all real (p + m)× (p + m) matrices A such that AGp,m + Gp,mAT = 0 where

Gp,m =

Eii −m∑

E(p+i)(p+i).

Here Eij is the (p + m)× (p + m) matrices with one in row i, column j, and zeroesotherwise:

Eij = εijkl

p+mk,l=1, εij

1, if k = i, l = j,0, otherwise.

A basis in so(p, m) provided by N matrices Γij :

Γij = −Γji = Eij − Eji, if 1 ≤ i, j ≤ p, or q + 1 ≤ i, j ≤ p + m,

and Γij = Γji = Eij + Eji otherwise.

Moreover, Γij fulfills the following commutation relations

[Γij , Γrs] = δjrΓis + δisΓjr + δriΓsj + δsjΓri,

[Γin, Γrs] = −δisΓrn + δirΓsn, [Γin, Γjn] = Γij

where δmn-Kronecker delta.

Notations

We shall be working on the Euclidean space Rn or in Cn, in which case we willdenote the partial differentiation ∂i := ∂xi as differentiation in coordinates. Insome cases we shall adopt the standard convention that 〈a, b〉 is a scalar prod-uct and multi-indices are merely markers, serving to identify the differentiation:uα(x) := ∂α1

x1· · · ∂αn

xnu(x). Let C[D] = C[∂1, . . . , ∂n, x1, . . . , xn] denote the set of all

PDO with complex exponential-polynomial coefficients. Actually, C[D] is a gener-alization of the well-known Weil algebra. Let X ⊂ Rn be any open neighborhoodof an origin 0 ∈ X . C∞(X) as usual denotes the ring of functions f : X → C whosepartial derivatives of all order exists on X . Let C[[x]] = C[[x1, . . . , xn]] denote thering of formal, i.e., not necessarily convergent, power series in (complex) variablesx1, . . . , xn. For analytic f we will often use the Taylor (Maclaurin) series of f at 0in the indeterminates x, otherwise for f ∈ C∞ we will use the formal power seriesas a C-algebra homomorphism C∞ → C[[x]] :

Tf(x) :=∑

α!f (α)(0).

Actually, a power series representation for the PDEs have recently developedby many authors (see [29], [9], [25]). It is applicable to ordinary equations and givesfruitful results. In particular, it is shown that solutions in the formal power seriesof constant coefficient PDEs are multisummable (see [25]). In the present paper wetreat solutions of a linear partial differential equations and we provide proofs onlyfor new results. We will often construct the solution in the formal power series inpairwise commutative operator indeterminates Ki, i = 1, . . . , n. The solution isC∞ in a neighborhood of x ∈ X and

T f(x) :=∑

α!Kα1

1 · · · · ·Kαnn [f (α)(0)].

The relation between formal solutions (in operator indeterminates) and gen-uine solutions with asymptotic expansions is an important problem. For a variety

of reasons that theory is quite technical. In this paper we describe an algorithmwhich gives a constructive, countable basis for the set of exponential-power seriessolutions to a given system of linear PDEs if they have enough symmetries asit is shown in [33]. Solutions with some growth estimates of an entire functionshave power series expansion in operator indeterminates. The existence of a gen-uine solution and it representation as functions belonging to some quasi-analyticclasses is not investigated in this paper. The multisummability of formal powerseries solutions are studied for evolution equation only.

The contents of the present paper after the introduction is as follows.

In Section 1, we introduce the basic techniques involved with symmetry groupcalculation. We use here ideas based on the Baker-Campbel-Hausdorff formula forthe successive computation of an abstract operator exponent. If the operatorsrepresented by successive commutators are of finite type, the symmetry operatorsmay be considered as perturbations of ordinary differential operators in some sense.The essential observation is that the multisummability of formal series for solutionsof evolution equation holds for commutative operators of finite type.

In Section 2, we give a definition of an analytic system of PDEs and give someimportant examples. We devote Section 3 to discussions that the understandingof a qualitative properties of the solution to system of PDEs acquire an algebraicaspects. We show that many results for the system of partial differential equationsare extensively connected with the underlying algebra. We introduce also definitionof an elliptic type system derived from the original notion of quasi-division algebrasand we connect the hyperbolicity with zero divisors structure in the underlyingalgebra. However, we do not carry out the full linkage of these connections andhope to carry it out in consequent papers.

In Section 4 our attention is focused to the power series expansion of thesolution to the Dirac operator in associative unital algebra. As a culmination ofthis section, we apply symmetry operator methods to quaternionic and Cliffordanalysis.

Most of the results of Section 5 are concerned with the linear, constant co-efficient, homogeneous PDO in Rn. We effectively show that 1-symmetries of anyhomogeneous of order two PDEs are isomorphic to the pseudo-orthogonal algebraso(p, q).

1. Evolution equations

In this Section we consider the evolution PDE in the form

Qu(t, x) := ∂tu(t, x)− P (x, ∂x)u(t, x) = 0, (1.1)

where one of the independent variables is distinguished as time and the otherindependent variables x = x1, . . . , xn ∈ X are spatial.

We consider P to be an operator on the manifold X × U where dependentvariables u are elements of U . Let G be a local one parametric group of trans-formations on X × U . In turn, we consider the first-order differential operator ofspecial type as a generator of a local group G. Our next step is

Definition 1.1. A local symmetry group of an equation (1.1) is a one-parametergroup gt of the transformation G, acting on X × U , such that if u(t, x) ∈ U isan arbitrary smooth enough solution of (1.1) and gt ∈ G then gt[u(τ, x)] is also asolution of (1.1) for all small enough t, τ > 0.

S. Lie developed a technique for computing local groups of symmetries. Hisobservations were based on the theory of jet bundles and prolongation of the vectorfields whenever u is a solution of (1.1). An explicit formula can be found in [21].

In general, one can compute symmetries of (1.1) by the Lie-Backlund (LB)method. Everything necessary can be found using explicit computations due to theBaker-Campbell-Hausdorff formula (in sequel: the BCH formula). This formula isequivalent to the LB method and is based on successive commutator calculations:

Ki = etP (x,∂)xie−tP (x,∂) =

m!tm[P (x, ∂x), xi]m. (1.2)

Here [a, b]m = [a, [a, b]m−1], [a, b]1 = ab− ba and [a, b]0 = b.

If all Ki ∈ C[D] in (1.2) are PDEs of finite order, then Q in (1.1) is of finitetype (see Definition 0.12). The above considerations lead to the correctness of

Claim 1.2. Question 0.2 is answered in the positive for an evolution equation (1.1)of finite type at least if the initial value problem u(0, x) = f(x) is well posed andif eλt for some λ is the solution of (1.1).

Proof. The solution to (1.1) may be written [12] in the form u = f(K)[eλt].

Our next considerations initiate the following result which we will use inSection 2:

Proposition 1.3. If Q is of finite type, then symmetry operators Ki defined in (1.2)and completing with identity operator, form a commutative, associative, unitalsubalgebra of an algebra Sym(Q) of all symmetries of Q.

Proof. Based on the verification of the commutator relation [P, xi]m = 0 for somem and associativity of operation in the Weil algebra C[D].

1.1. Heat equation

It is not difficult to see that the BCH formula (1.2) for the heat equation

∂tu = ∆u in x ∈ Rn (1.3)

possesses the pairwise commute operators Ki = xi + 2t∂xi (cf. (0.2)). Thereforethe heat operator (1.3) is, of course, of finite type.

Let vα(x, t) for multi-indices α, |α| = m (see [33]) be the polynomial solutionsto (1.3), (heat harmonics):

vα(x, t) := Kα[1] = m!

[m/2]∑

|β|=s

tm−2s

(m− 2s)!

β!, β = β1, . . . , βn. (1.4)

Existence of the following expansion was proved in [33]:

Theorem 1.4. A solution u(x, t) of (1.3) in R has an expansion

u(x, t) :=

∞∑

Kn[an]

∞∑

n!vn(x, t), an = ∂n

x u(0, 0), (1.5)

valid in the strip |t| < ε, −∞ < x <∞, if and only if it is equal to its Maclaurinexpansion in the strip |t| < ε, |x| < ε.

In order to satisfy the Cauchy data u(x, 0) = f(x), the following necessaryand sufficient conditions on a function f are known [33]:

Proposition 1.5. The Maclaurin expansion (1.5) leads to conditions that f(x) beentire of growth (2, 1/(4ε)). In another case (1.5) is a formal representation of thesolution only.

1.2. Constant coefficient evolution equation

Consider the general constant coefficient evolution operator

Q := ∂t − P (∂x).

From the BCH formula (1.2) follows trivially that Q is of finite type. The infini-tesimal symmetries of Q, namely:

Ki = etP (∂x)xie−tP (∂x) = xi + t[P (∂x), xi] = xi + tPi(∂x) (1.6)

are pairwise commuting operators. The symbol Pi(x) in (1.6) as usual stands forpartial derivation ∂xiP (x). Of course the function u0(t) = etP (0) is one of the“simplest” solutions meaning that Qu0(t) = 0 and u0(0) = 1. The solution of theconstant coefficient evolution equation Qu = 0 with initial data u(x, 0) = f(x),f(x) ∈ C[x] may be represented at least locally in the form

u(x, t) = f(K)[etP (0)]. (1.7)

Here K = (K1, . . . , Kn) are defined in (1.6).

Remark 1.6. Representation (1.7) forms a one-to-one correspondence between co-ordinates xi and operator indeterminates Ki.

The following theorem is a straightforward generalization of Theorem 1.4:

Theorem 1.7. A solution u(x, t) of the equation Q(∂x)u(x) = 0 has an expansion

u(x, t) :=

∞∑

|α|=m

α!Kα[

∂xαu(0, 0)], (1.8)

valid in the fiber |t| < ε, x ∈ Rn and is locally analytic if and only if u(x, 0) isan entire function of order at most m/(m − 1). In this case (1.8) is equal to itsMaclaurin expansion in the cylinder |t| < ε, ||x|| < ε.

Calculation of the symmetry operator indeterminates for some importantconstant coefficient PDE leads to the following considerations:

1.2.1. 2b-parabolic equation. In particular, the regular solution to the 2b-parabolicequation [10] in Rn

∂tu = (−1)b−1∆bu (1.9)

may be written in the form:

u(x, t) = f(K1, . . . , Kn)[1], Ki = xi + (−1)b−12bt∆b−1∂xi . (1.10)

1.3. Diffusion with inertia equation

The Kolmogorov model of diffusion with inertia based on solution of equation

Qu := ∂tu− ∂2xu− x∂yu = 0. (1.11)

Q is not a constant coefficient PDE, but nevertheless, the operator Q(x, ∂x) in(1.11) allowed the technique of successively commutators via the BCH-formula(1.2). Due to careful examination we obtain two 1-symmetry operators of Q in theform

K1 := x + 2t∂x − t2∂y, K2 := y + xt + t2∂x −1

3t3∂y. (1.12)

It is possible to verify that (1.11) is of finite type and all three operators Q, K1, K2

pairwise commute:

[K1, K2] = 0, [K1, Q] = [K2, Q] = 0.

Therefore, the solution of the Cauchy problem

Qu(x, y, t) = 0, u(x, y, 0) = f(x, y) (1.13)

with f(x, y) ∈ C[x] may be obtained in the operator form as follows: u(x, y, t) =f(K1, K2)[1].

1.4. Some other evolution equations

Next we consider two PDEs [8], [4]:The Fokker-Plank equation:

∂tu = ∂2xu + x∂xu + u; (1.14)

and the harmonic oscillator

∂tu =1

2(1− x2)u. (1.15)

Calculation by the BCH formula (1.2) for both cases suppose to use infinitelymany commutators. Nevertheless, the solution of (1.14) may be written in the formu(x, t) = f(K)[et] with 1-symmetry operator K := xet+et∂x and solution of (1.15)

may be written as u(x, t) = f(K)[e−x2/2] with operator K := xet − et∂x.

In fact there is no polynomial solution of (1.14) and (1.15). Also K in bothcases is PDO with exponential-polynomial coefficients.

1.5. Evolution equations with time-depending coefficients

Following [14] consider the Cauchy problem for higher-order linear homogeneousevolution equations with coefficients depending on the time variable:

∂tu (x, t) =∑

|α|≤m

aα(t)∂αx u(x, t), u(x, 0) = f(x). (1.16)

Here x ∈ Rn, t ∈ R, and the coefficients aα(t) are real-valued continuousfunctions of t ∈ [0, ε]. Consider the following symmetry operators of (1.16):

Ki = exp(P (t, ∂x))xi exp(−P (t, ∂x)) where P (t, ∂x) =∑

|α|≤m

aα(τ)dτ ∂αx .

Using the BCH-formula we obtain an explicit formula for K = K1, . . . , Kn:Ki = xi + Pi(t, ∂x) where Pi(t, x) = ∂xiP (t, x). (1.17)

Then, at least formally using (1.17), we can write the solution of (1.16) in the form

u(x, t) = f(K)[u0(t)], u0(t) = exp∫ t

a0(τ)dτ. (1.18)

Following [14], the formal representation (1.18) converges to an analytic functionu(x, t) in the strip 0 < t < ε iff f is a real-valued entire function on Rn of growth(ρ, τ), meaning that

|f (z)| = O(

|zi|ρ])

as z →∞. (1.19)

Equivalently, f(x) can be written in form of the power series expansion

f(x) =∑

β!xβ and lim sup

β→∞

)ρ−1

|cβ |ρ/|β| ≤ τρ.

Under these conditions on the coefficients of f(x), the solution (1.18) may beexpressed as a convergent series in operator indeterminates K. In [14] there arealso estimates on the maximal width of the strip of convergence of these series.

1.6. KdVB and non-linear equations

In the limiting cases the non-linear evolution equation reduces to the well-knownconventional Korteweg de Vries and Burgers (KdVB) equations, respectively. Itarises in various contexts as a model equation incorporating the effects of disper-sion, dissipation and non-linearity. The general form of such an equation is givenby

∂tu(x, t) = µ1u(x, t)∂xu(x, t) + µ2∂2xu(x, t) + µ3∂

3xu(x, t) (1.20)

where µ1, µ2 and µ3 are some constant coefficients.

These equations are both exactly solvable and they each have a wide rangeof symmetries. It should be pointed out that the above-described technique ofsymmetry operators is applicable to the solution of the KdVB equation.

2. System of PDE

Definition 2.1. Recall a system of PDEs for x ∈ Rn is an analytic PDE if

(i) it may be written in the form of the first-order (linear) system

L(x, D)u := Lk(x, D)unk=1 :=

akij(x)∂iu

j(x) = 0, (2.1)

(ii) the coefficients akij(x) in PDOs Lk all are entire functions.

(iii) L(x, D) is an involutive system, meaning that there exist entire functionsbkij(x) such that the commutators [Li(x, D),Lj(x, D)] fulfill the relation

[Li(x, D),Lj(x, D)] =n∑

bkij(x)Lk(x, D). (2.2)

The principal matrix symbol σ(L)(x, ξ) of (2.1) is defined [1] by

σ(L)(x, ξ)kj :=

akij(x)ξi, j, k = 1, 2, . . . , n. (2.3)

In the neighborhood of the point x0, the solution u(x) of the equation (2.1)is locally equivalent to the solution of the system L(x0, D)v(x) = 0 with constantcoefficients. The constant coefficient systems are evidently analytic. In this sectionwe study some properties of the solution spaces to analytic PDEs.

The solution space to analytic PDEs may be finite-dimensional and thereforedo not allow using operator indeterminates (K = const in Definition 0.5):

Example. Consider the following system of first-order PDEs

σi∂xiui = σj∂xj uj, ∂xiuj + ∂xj ui = 0, 1 ≤ i < j ≤ N (2.4)

for N unknown functions u1(x1, . . . , xN ), . . . , uN (x1, . . . , xN ) and σk = ±1.

If N = 2 and σ1 = σ2 then (2.4) is the well-known Cauchy-Riemann system.Although, for N > 2 the solution space to (2.4) is finite-dimensional as is shown in

Proposition 2.2. For N > 2 there are exactly 12 (N + 1)(N + 2) linear independent

solutions of (2.4) being represented in a form

um(x1, . . . , xN ) =

(2xm − xi)αiσixi +

βimσixi + γxm + δm, (2.5)

where αm, βkmγ, δm for k, m = 1, . . . , N are constants, βij + βji = 0.

Proof. Cross differentiation of (2.4) yields for pairwise indices i, j, k ∈ 1, . . . , Nnot equal one to another:

σi∂xixiui + σj∂xjxj ui ≡ ∂xi(σi∂xiui − σj∂xj uj) + σj∂xj (∂xiuj + ∂xj ui) = 0;

σi∂xixiuk − σj∂xjxjuk ≡ σi∂xixiuk + σk∂xkxkuk − (σk∂xkxk

uk + σj∂xjxjuk) = 0;

2∂xixj uk ≡ ∂xi(∂xj uk + ∂xkuj)− ∂xk

(∂xiuj + ∂xj ui) + ∂xj (∂xkui + ∂xiuk) = 0.

Recall the proof that all third-order derivatives of ui are also equal to zero:

∂xixixiui = 0, ∂xixixj ui = 0, ∂xixixj uj = 0 for i = j, i, j ∈ 1, . . . , N.But if N > 2, then there exists pairwise different i, j, k and

2∂xixixiui ≡ (σi∂xixi − σk∂xkxk)(σi∂xiui − σj∂xj uj)

+σj∂xj (σi∂xixiuj − σk∂xkxkuj) + σi∂xi(σk∂xkxk

ui + σi∂xixiui) = 0;

σiσj∂xixixj ui ≡ σj∂xj (σi∂xixiui + σk∂xkxkui)− σjσk∂xk

(∂xkxjui) = 0;

σj∂xixixj uj ≡ ∂xixi(σj∂xj uj − σi∂xiui) + σi∂xixixiui = 0

for all pairwise non-equal i, j, k.

Using all the above relations we finally may conclude that only the second-order polynomials form the common solutions of (2.4), because of ∂xixjxk

um = 0for all i, j, k, m = 1, . . . , N.

Choosing explicitly the coefficients in quadratic terms, we obtain (2.5).

Belove we present results of some classical examples of the system of PDEs.

2.1. Dirac and Laplace equations in Rn

It is well known [5], [18] that the Laplacian ∆ may be factorized as ∆ = D Dwhere D is the Dirac operator and D is conjugate to D in the Clifford algebraCl(0,n). Thus, one can choose the solution to the Laplace equation as a real partof the solution to the Dirac equation. In fact, the Dirac equation is a system ofODE with constant coefficient. In R4 one can write the Dirac equation in the form

gradu + ∂tv + curl v = 0, ∂tu− div v = 0.

Algebras of 1-symmetries of the Dirac and Laplace equation in Rn both areisomorphic to the pseudo-orthogonal Lie algebra so(n + 1, 1).

2.2. Lame equation

The Lame equation

µ∆v + (µ + λ)grad θ = 0, θ = ∂xv1 + ∂yv2 + ∂zv3 (2.6)

looks like a generalization of the Dirac equation. If we denote by u = div θ and byv = curl v, we can rewrite the Lame equation equivalently as:

λ gradu + µ curl v = 0, div v = 0. (2.7)

2.3. Linearizing Navier-Stoks equation

The following equations is also a system of inhomogeneous PDEs

∆v − grad p = 0, div v = 0 (2.8)

In fact system of constant coefficient PDEs may be considered as PDEs inalgebra, as it will be shown below.

3. PDEs in algebra

An algebra A for us will always be a real n-dimensional Euclidean vector spacethat is finitely generated by the orthonormal system of basis vectors e1, e2, . . . , en

and is equipped with a bilinear map m : A× A→ A called “multiplication”. Thesymbol , unless there is ambiguity, stands for the abbreviation m(x, y) as x yin A. Of course, knowing the tensor form am

ij of a bilinear map in an orthonormalbasis e1, e2, . . . , en one can rewrite the multiplication m(ei, ej) as follows:

ei ej =

amijem. (3.1)

We use x = x1e1 + · · ·+xnen to denote vector in Rn as well as element x ∈ A, and∂i to denote ∂/∂xi, for i ∈ 1, . . . , n. The symbol 〈, 〉 stands for the Euclideanscalar product.

Every A-valued function f(x) will be for us always locally real analytic thatis represented as:

f(x) := e1u1(x1, . . . , xn) + e2u2(x1, . . . , xn) + · · ·+ enun(x1, . . . , xn), (3.2)

where ui(x) are also real analytic functions, i = 1, 2, . . . , n.

We begin with definition of the Dirac operator in A:

D := e1∂1 + · · ·+ en∂n. (3.3)

Definition 3.1. A real analytic functionf(x) is called an A-analytic if f(x) is asolution to the system of partial differential equations

D f(x) :=

ei ej∂iuj(x1, . . . , xn) = 0. (3.4)

Comparing Definition 3.1 with the notion of analytic PDE (2.1), one caneasily verify that the solution to analytic PDE at least locally at point x0 is anAx0-analytic function where Ax0 is a local algebra with the multiplication rule ,such that

D u(x) := L(x0, D)u(x). (3.5)

In this section we will study the qualitative properties of the solution to analyticPDEs in terms of algebraic properties of the local algebras bundle (cf. (3.5)).

If A in Definition 3.1 is an algebra of complex numbers C, then (3.4) coincideswith the Cauchy-Riemann equations. This gives us a good reason to denote thespace of A-analytic functions as Hol(A) in complete agreement with the definitionof holomorphic functions in complex analysis (denoted by Hol(C)).

Remark 3.2. By A-analysis we mean the systematic study of Hol(A).

3.1. Algebraic approach to function theories

Let A be a real algebra (not necessarily commutative and/or associative). Theliterature on function theory over such algebras has been developed by many au-thors (see [11], [5], [30], [32]) and contains a range of definitions for analyticity(holomorphicity, monogenicity). Three distinct approaches in these investigationsare mentioned.

• The first one (Weierstrass approach) regards functions on A as their conver-gence in some sense power series expansions (cf. [16]).• The second (Cauchy-Riemann) approach concentrated on the solution to the

Dirac equation in algebra A (cf [17], [18]).• The third one is based on the function-theoretic properties known for complex

analytic functions, such as Cauchy’s theorem, residue theory, Cauchy integralformula etc. (cf. [26], [13] ).

All these methods look like a generalization of analytic function theory of complexvariables (cf. [30], [32]). We use the term A-analysis for such cases (cf. Clifford orquaternionic analysis [5], [30] if the algebra A is embedded into a Clifford algebra).

Claim 3.3. Totality of a functions on (in general non-commutative and/or non-associative) regular algebras are splitting into the non-equivalent classes. Theseclasses are uniquely characterized by their unital hearts. If such a heart is inaddition an associative algebra then an A-analytic function may be expanded intothe commutative operator-valued power series.

3.2. Isotopy classes

In this subsection we will be concerned with Albert isotopies [2] of algebras:

Definition 3.4. Two n-dimensional algebras A1 and A2 with multiplication and⋆ are said to be isotopic (A1 ∼ A2) if there exist non-singular linear operatorsK, L, M such that

x y = M(Kx ⋆ Ly). (3.6)

Obviously, if in addition, K ≡ L ≡ M−1, then two algebras A1 and A2 areisomorphic (A1 ≃ A2).

Definition 3.5. If for given two algebras A1 and A2 there exist non-singular linearoperators P, Q such that for every g(x) ∈ Hol(A2) the function f(x) = Pg(Qx)belongs to Hol(A1) and vice versa, we would say that two function theories areequivalent and write Hol(A1) ≃ Hol(A2).

Using these results, we obtain the important

Theorem 3.6. Two function theories are equivalent iff the corresponding algebrasare isotopic.

Definition 3.7. If akij(x0) in (2.1) for all fixed x0 ∈ Ω ⊂ Rn forms a set of isotopic

algebra then we will say that L is a unique defined type PDO in Ω, otherwise L isa mixed type PDO. An algebra A0 with multiplication tensor ak

ij(x0) (see. (3.1))is called a local algebra for L in x0 ∈ Ω.

Obviously, if the coefficients akij(x) are considered as constants, then the

operator L in (2.1) coincides with the Dirac operator D defined by (3.3) and onecan obtain the solution to the homogeneous equation Lf = 0 as a (left) A-analyticfunction for the operator D.

Claim 3.8. Decomposition of algebras into the isotopy classes in turn is a powerfulclassification tool for the corresponding PDO.

3.3. Classification of the first-order PDE

Here we study many essential notions of PDE theory by treating it in algebraicterms. We begin by examining the conditions that the Dirac operator in A (3.3)is a well-defined system of PDE.

3.4. Under- and overdetermined system

Let P (D)u(x) = f(x) be a system of partial differential equations, where P (D) isa given k × l matrix of differential operators with constant coefficients, the givenf(x) (respectively, unknown u(x)) being k- (l-)tuples of functions or distributionsin x ∈ Rm. Many authors (cf. [25]) assume usually that the system is under-(over-) determined, if the rank of P (ξ) (cf. of its transpose P ′(ξ)) is less than l forall (cf. for some) non-zero ξ ∈ Rm.

The algebraic formulation of the fact that PDE with the constant coefficient(3.4) is under- (over-) determined can be roughly described as follows.

Definition 3.9. A real n-dimensional algebra A is called left (right) regular if thereexists v ∈ A, such that the linear operators Lv, Rv : Rn → Rn defined by x→ v x(resp. x→ x v) are both invertible. Otherwise, A is called a left (right) singularalgebra. In other words, A is regular iff A ⊂ A2

Recall elements u, v ∈ A a left (cf. right annihilator) if u x = 0, (x v = 0)for all x ∈ A.

Theorem 3.10. The Dirac operator D in algebra A is underdetermined iff A issingular and is overdetermined iff A is regular and contains an annihilator.

Proof. For a given Dirac operator D in the corresponding algebra A define left(right) multiplication operators Lv, Rv : Rn → Rn by x → v x (resp. x →x v) as in Definition 3.9. If Lξ, Rξ are both invertible for some ξ, then D is welldetermined. Conversely, let Lv (respectively, Rv) be k1 × l1 (k2 × l2) matrices of

differential operators. Then D is underdetermined if k1 < l1 and/or k2 > l2 andis overdetermined if k1 > l1 and/or k2 < l2. The only case k1 = l1 = k2 = l2 = nstands for the regular algebras A without annihilators and therefore, for a well-determined Dirac operator D in Rn.

In Definition 3.7, we consider some properties of L(x, D) to be of the sametype in a given open set x ∈ Ω. These properties were given in terms of the existenceof one common isotopy relation between the set of the corresponding algebras. Ifthere exists a point on the boundary x0 ∈ ∂Ω and such that an algebra A0 withmultiplication tensor (3.1) a(x0)

kij is not isotopic to any local algebra A1 with

multiplication tensor a(x1)kij for x1 ∈ Ω, we will say that the PDO L(x, D) is

of degenerate type in Ω. Many authors (see [1], [10]) present several interestingresults on the relationship between solutions in a neighborhood of the boundary∂Ω and the properties of the PDO (2.1) of same fixed type. The results are toocomplicated to be formulated here in detail. In the next section we will deal withconcrete (for example, an elliptic and parabolic type) PDE’s and will explain interms of corresponding algebras their exceptional algebraic meaning.

3.5. Elliptic type PDE

One of the basic concepts in PDE is ellipticity. Actually, one can reformulateellipticity of the Dirac operator in the regular algebra A as a property of A to bea division algebra.

Definition 3.11. An algebra A is a division algebra iff both operations of left andright multiplications by any non-zero element are invertible.

Proposition 3.12. The well-determined Dirac operator D in the necessary regular(by Theorem 3.10) algebra A is elliptic iff A is a division algebra.

Proof. The matrix symbol (2.3) of the well-determined elliptic partial differentialoperator σ(D)(ξ) is invertible for all ξ = 0 (cf. [11], [28]). In contrast, it followsimmediately from the definition of ellipticity that the symbol of partial differentialoperator D is invertible matrix for all ξ = 0.

Example. (D. Rusin) Let Qε be constructed from an algebra of quaternions Q

leaving the multiplication table unchanged except the square of i, i2 = −1 + εj.

Two algebras Qε and Q are non-isotopic division algebras if |ε| < 2.

The above example shows that in four-dimensional space there exists a non-equivalent elliptic function theory in the sense of Definition 3.5. In what follows, thequestion how many non-equivalent (well-defined elliptic) function theories existsmay be answered in terms of an existence of non-isotopic classes of the divisionalgebras. In particular, in R4 there exist well-defined elliptic function theoriesdifferent from the quaternionic analysis. Our focus will be on the general resultsabout (not necessary well-determined) elliptic system and therefore we continuewith some generalizations of the condition on an algebra to be a division algebra:

Let the notion A stand for a maximal ideal in A of the left annihilators (thatturn out to be maximal). Of course, the multiplication in A is trivial. Then for agiven algebra A:

Definition 3.13. Recall that an algebra A is a quasi-division algebra iff the factoralgebra A/A is a division algebra.

In turn, in quasi-division algebra the equation a x = b is soluble for all a, bexcept those a, being the left annihilators of A. Now we will be able to formulatethe algebraic analogue of ellipticity [1] for the under- (over-) determined system:

Proposition 3.14. The Dirac operator D defined in (3.3) is elliptic iff the algebraA corresponding to D (see (3.1)) is a quasi-division.

Proof. Immediately follows (cf. [11],[28]) from the definition of ellipticity [1] of thesymbol of the partial differential operator D.

Of course, if there are no other left/right annihilators except 0, every quasi-division (in the sense of Definition 3.13) algebra is the division algebra.

For regular algebras the following result of A. Albert [2] is true:

Theorem 3.15. Every regular algebra is isotopic to the algebra with unit e. Everyn ≥ 2-dimensional unital division algebra contains an imaginary unit i being asquare root of −e, (i2 = −e).

Proof. The proof of the first part is based on “Kaplansky trick” [26]. Namely,let Lx, (Ry) be operators of left (right) multiplication on elements x, (y) or x y ≡ Lxy ≡ Ryx. If A is a regular algebra, then there exist two elements a, bsuch that La and Rb are invertible. Define a new multiplication by the formulax ⋆ y = R−1

b x L−1a y and take the element e = a b. Then

e ⋆ y = R−1b (a b) L−1

a y = a L−1a y = LaL−1

a y = y;

x ⋆ e = R−1b x L−1

a (a b) = R−1b x b = RbR

−1b x = x

for all x, y. Combining Definition 3.13 with Proposition 2 yields that the samestatement as in Theorem 3.15 is true for a quasi-division algebras if codim (A) ≥ 2.Moreover, in this case in A there exists a 2-dimensional subalgebra A ∼= C.

Example. Consider two systems of PDE, the spherical and usual Dirac equation[21] in R4 for scalar and vector functions u(t, x), v(t, x):

div v = 0; gradu + curl v = 0; (3.7)

∂tu− div v = 0; gradu + ∂tv + curl v = 0. (3.8)

Both systems of PDE are elliptic, the first one is overdetermined. To see that,let us recall that the algebra of quaternion Q stands behind (3.8), and behind (3.7)stands an algebra Q′ with multiplication rule x y = 1/2(x− x) ⋆ y where “ ⋆ ” isthe quaternion multiplication and x is the quaternion conjugation to x.

In turn, both algebras are quasi-division, but only Q′ has non-trivial ideal oftheir (left) annihilators.

3.6. Parabolic and hyperbolic type PDE

Comparing the definitions of elliptic [1], hyperbolic [28] and parabolic [10] PDEwith results of Theorem 3.10, we can see that parabolic and hyperbolic type Diracoperators correspond to an algebra with zero divisors. Moreover, the property ofthe algebras to be regular and quasi-division, the number of zero divisors and thenumber of annihilators are invariants with respect to isotopy relation. This givesrise to the following:

Proposition 3.16. The Dirac operator in a regular algebra A is:

(i) parabolic [10] iff A/A contains exactly one (up to scalar factor) zero divisor.In this case A/A is isotopic to an algebra with one nilpotent and no otherzero divisors;

(ii) hyperbolic [28] iff A/A contains at least one pair or more of (left) zero divi-sors.

Below we formulate some common results about building an analytic functiontheory in Rn.

Claim 3.17. An A-analytic function theory equivalent (in the sense of Definition3.5) to the solution space of an evolution equation iff A is regular.

Proof. If A is regular then there exists an isotopy to an algebra with unit. But theDirac operator in algebra with unit 1 may be written in the form D = ∂t +P (∂x).Therefore D is of evolution type and we can apply methods of Section 1.

3.7. Applications

Let t, x, y, z be independent variables. Denote by u(t, x, y, z) the scalar and byv(t, x, y, z) vector function in R4. Many physically important models may be writ-ten as the following PDE:

α∂tu + βdiv v = 0, γ∂tv + δgradu + εcurl v = 0. (3.9)

Assume A is behind (3.9) and D is the Dirac operator in A. In turn, ifαβγδ < 0, and ε = 0 then the system of PDE (3.9) is elliptic and the algebra A isa division algebra.

If αβγδ > 0 and ε = 0, then the system of PDE (3.9) is hyperbolic andthe algebra behind Dirac operator (3.9) is an algebra with infinitely many zerodivisors.

If α = 0, γ = 0 and βδε = 0, then an algebra behind (3.9) is a quasi-divisionalgebra embedded into quaternions Q. Actually, if A is unital algebra, then onecan set α = γ = δ = 1 and therefore sign of β for ε = 0 distinguish elliptic andhyperbolic cases.

It is convenient to consider the case n = 2 separately.

3.8. Two-dimensional case

Remark 3.18. It follows from Theorem 3.15 and ([26], Proposition 1.7) that alltwo-dimensional division algebras are isotopic to C.

In particular, the function theory over any division algebras in R2 is equiv-alent to the complex analytic function theory. Many concepts and results in thetheory of analytic functions of complex variables can be extended to functionssatisfying the system (3.4)

Theorem 3.19. (cf. [26]) There are only three non-equivalent function theories overtwo-dimensional regular algebras, namely:

(i) elliptic theory over C, which corresponds to complex analytic function theory;(ii) parabolic (degenerate), f(x1, x2) = u(x2)e1 + (v(x2)− x1u

′(x2))e2 in N ande21 = e1, e1 e2 = e2 e1 = e2, e

22 = 0;

(iii) hyperbolic, if f(x1, x2) = u(x2)e1 + v(x1)e2 in R⊕R and e21 = e1, e1 e2 =

e2 e1 = 0, e22 = e2.

Proof. By the Unital Heart Proposition (cf. [26], 1.10) there exists exactly threeisotopy classes of regular two-dimensional real algebras. (The singular algebras areout of the realm of our interest since the function theory over these algebras areone-dimensional and therefore trivial.)

Remark 3.20. The algebras C, N and R ⊕ R listed in Theorem 3.19 are the onlyunital two-dimensional algebras. They are necessary commutative and associativeand have respectively 0, 1 and 2 zero divisors.

The following definition is about the construction of an analytic functiontheory using algebraic operations only.

Definition 3.21. If for every two functions f(x), g(x) ∈ Hol(A) the product f(x) g(x) as well as their linear combination αf(x) + βg(x) with real coefficients α, βbelongs to Hol(A), then we will call the A-analytic function theory pure algebraic.

Theorem 3.22. A-analytic function theory is pure algebraic iff A is a direct sumof R, C and N defined in Theorem 3.19.

Proof. The “if” part is based on results in [20] concerning existence of an infinite-dimensional symmetries in Hol(A) only in two-dimensional subalgebras and/orin their direct sum. Every regular A-analytic function theory in algebras A beingrepresented as a direct sum of the two-dimensional subalgebras is equivalent inthe sense of Definition 3.5 to the pure algebraic theory. The “only if” part followsfrom Theorem 3.19.

The operator of multiplication by a fixed A-analytic function in the algebrasA mentioned in Theorem 3.19 maps the space Hol(A) onto itself. This statement isrelated to the symmetries of the Dirac operator. We refer to [18] for results aboutsymmetries in Clifford algebras.

Claim 3.23. In R2 every A-analytic function theory is equivalent (in the sense ofDefinition 3.5) to the pure algebraic theory.

4. Power series expansions

As was proven in [26], every regular algebra is isotopic to its unital heart. Assumethat e0, e1, . . . , en forms a basis in the unital associative algebra A and e0 is itstwo-sided unit element. In order to construct an A-analytic function theory, thefollowing A-analytic variable are used:

zm = xme0 + x0em, m = 1, 2, . . . , n. (4.1)

In turn, Dzk = 0 for all k = 1, 2, . . . , n where D = ∂x0 + e1∂x1 + · · ·+ en∂xn

is the Dirac operator in the algebra A

Denote a canonical spherical homogenic polynomial solution of the Diracequation in A by the formula:

V0(x) = e0, Vm(x) = zm, m = 1, 2, . . . , n, (4.2)

Vµ(x) := Vm1,...,mk(x) =

π(m1,...,mk)

zm1zm2 · · · zmk, (4.3)

where the sum runs over all distinguishable permutations of m1, . . . , mk.

Proposition 4.1. (cf. [5]) The polynomials Vµ(x) of order k for all multi-indicesµ = m1, . . . , mk for mi ∈ 1, 2, . . . , n are both left and right A-analytic. AnyA-analytic and homogenic of order k function pk(x) may be written as

pk(x) =∑

m1,...,mk

Vm1,...,mk(x)∂xm1

· · ·∂xmkpk(x) (4.4)

where the sum runs over all possible combinations of m1, . . . , mk of k elements outof 1, 2, . . . , n repetition being allowed.

Proof. (cf. [5], Theorem 11.2.3,5) Clearly, for µ = m1, . . . , mk

k!x0DVµ(x) =∑

π(m1,...,mk)

x0ei∂xi(zm1(zm2 · · · zmk) . . .)

π(µ)

(emizm1 · · · zmi−1zmi+1 · · · zmk− zm1 · · · zmi−1emizmi+1 · · · zmk

π(µ)

(zmi(zm1 · · · zmi−1(zmi+1 · · · zmk) . . .)− zm1 · · · zmk

) = 0.

For the polynomial p1(x) in (4.4) by Euler’s formula, one can show that

p1(x) = x0∂x0p1(x) +

xi∂xip1(x) =

zi∂xip1(x)

To continue the proof one can use induction.

Theorem 4.2. (cf. [5]) The function f(x) that is A-analytic in an open neighbor-hood of the origin can be expanded into a normally convergent series of sphericalhomogenic polynomials

f(x) =∞∑

m1,...,mk

Vm1,...,mk(x)∂xm1

· · · ∂xmkf(0)

Proof. The proof is similar to the method described in ([5], Theorem 11.3.4) and isa generalization of results [5] to the general unital associative algebra if the seriesby spherical harmonics are convergent.

Claim 4.3. The polynomials Vµ(x) introduced in (4.1)–(4.3) play an analogous roleas the powers of the complex variable z = x+ iy in the theory of complex variables.

4.1. Symmetries

Let A be a unital associative algebra and D is the Dirac operator in A.

Theorem 4.4. The first-order PDO L ∈ C[x, ∂x] is a symmetry operator for Ddefined in (3.3) iff there exists a first-order PDO L′ ∈ C[x, ∂x] such that

DL = L′D. (4.5)

Proof. Sufficiency of (4.5) holds trivially. To prove necessity, without loss of gen-erality, choose a 1-symmetry operator L in the form

L = a0(x)D +

ak(x)∂xk+ b(x).

Here ak(x), k = 0, . . . , m, b(x) are C∞ functions.Let now L∗ be chosen such that the operator RL = DL − L∗D does not

depend on ∂x0 , for example, in the following form

L∗ = Da0(x) +m∑

ak(x)∂xk+ b(x).

Then, obviously

RLf(x) = D(Lf(x))− L∗(Df(x))

akl(x)∂xk∂xl

f(x) +m∑

bk(x)∂xkf(x) + c(x)f(x). (4.6)

All these functions akl(x), bk(x), c(x) for k, l = 1, 2, . . . , m. For (4.6) to be validwhen applied to an arbitrary f(x), it is necessary and sufficient that the coef-ficients of RLf ≡ 0 in Ω. That means, for f(x) ≡ 1 (every constant is a trivialmonogenic function), that c(x) ≡ 0. Taking consequently xk−x0ek, k = 1, 2, . . . , mas a monogenic function f(x) in (4.6), we get that all Clifford-valued coeffi-cients bk(x) ≡ 0, k = 1, 2, . . . , m. Now if f(x) is chosen equal to one of thefollowing functions: x2

0 − x2k + 2x0xkek, k = 1, 2, . . . , m, (actually hypercom-

plex second-order powers, that are, obviously, monogenic), we get akk(x) ≡ 0,

k = 1, 2, . . . , m. In order to complete the proof, we can substitute the follow-ing monogenic functions f(x) in (4.6): x2

1 − x22 − 2x1x2e12, x2

1 − x23 − 2x1x3e13,

. . . , x2m−1 − x2

m − 2xm−1xmem−1,m. Now all the remaining coefficients of RL areequal to 0. The theorem is proven for L′ = L∗. Moreover, since all coefficientsakl(x), bk(x), c(x) of RL(x) in (4.6) are equal to 0, this leads to the following sys-tem of partial differential equations:

elak(x)− ak(x)el + ekal(x)− al(x)ek = 0, k, l = 1, 2, . . . , m, (4.7)

Dak(x) + ekb(x)− b(x)ek = 0, k = 1, 2, . . . , m, (4.8)

Db(x) = 0. (4.9)

The last equation (4.9) means that b(x) is a monogenic function. The solutionof (4.7) which can be easily verified is

ak(x) =∑

|I| = even,k ∈ I

αI(x)eIek +∑

|I| = odd,k ∈ I

αI(x)eIek. (4.10)

Here αI(x) are real analytic functions defined for all multi-indices I and suchthat they all are independent of k.

After putting (4.10) in the remaining equations (4.8) and using the followingnotations:

The last equation (4.9) means that b(x) is a monogenic function. The solutionof (4.7) which can be easily verified is

ak(x) =∑

|I| = even,k ∈ I

αI(x)eIek +∑

|I| = odd,k ∈ I

αI(x)eIek. (4.11)

Here αI(x) are real analytic functions defined for all multi-indices I and suchthat they all are independent of k.

4.2. Quaternion analysis

From the point of view of quaternion analysis (cf. [5]) the entire smooth enough(differentiable in neighborhood of origin) solution of the Dirac equation can berepresented as a convergent series of quaternion harmonics defined in X ⊂ R4.They are the only homogeneous polynomial solutions of degree m to the Diracequation and

Y m(q) =∑

|α|=m

cαxα, m = 0, 1, . . . , D Y m(q) = 0, (4.12)

where q = x1 + x2i + x3j + x4k, α is multi-index, |α| = α1 + α2 + α3 + α4,xα = xα1

1 xα22 xα3

3 xα44 , and cα are the quaternion-valued constants.

Theorem 4.5. [17] The quaternion harmonics fulfill the relation

2m(m + 1)Y m(q) =

Ki[∂xiYm(q)], (4.13)

where K1, . . . , K4 are the generators of the “conformal group” in quaternion space

K1 = (x21 − x2

2 − x23 − x2

4)∂x1 + 2x1x2∂x2 + 2x1x3∂x3 + 2x1x4∂x4 + 2x1 + q;K2 = (x2

2 − x21 − x2

3 − x24)∂x2 + 2x2x1∂x1 + 2x2x3∂x3 + 2x2x4∂x4 + 2x2 − iq;

K3 = (x23 − x2

2 − x21 − x2

4)∂x3 + 2x3x2∂x2 + 2x3x1∂x1 + 2x3x4∂x4 + 2x3 − jq;K4 = (x2

4 − x22 − x2

3 − x21)∂x4 + 2x4x2∂x2 + 2x4x3∂x3 + 2x4x1∂x1 + 2x4 − kq.

Proof. Let r2 = x21+x2

2+x23+x2

4 and let operator r∂r be the generator of dilatationsin R4:

r∂r := x1∂x1 + x2∂x2 + x3∂x3 + x4∂x4 .

In fact, Y m(q) is the eigenfunction of the operator r∂r with eigenvalue m:r∂rY

m(q) = mY m(q). Using these definitions, we obtain

Ki[∂xiYm(q)] = 2m(m− 1)Y m(q)− r2∆Y m(q) + 4mY m(q) + qD Y m(q).

Here q quaternion conjugate to q, ∆ is the Laplace operator. If Y m(q) is a quater-nion harmonics, then ∆Y m(q) = 0, D Y m(q) = 0. The theorem is proven.

Now we will be able to prove the main result for quaternion harmonics:

Theorem 4.6. The homogeneous polynomial Y m(q) is a quaternion harmonics iff

Y m(q) := Pm0(K)[1] + Pm1(K)[i] + Pm2(K)[j] + Pm3(K)[k], (4.14)

where Pmi are homogeneous of the same order m real polynomials.

Proof. Quaternion harmonics ∂xiYm is also a quaternion harmonics of order m−1.

By induction, using (4.13) we can show that

Y m(q) =1

2m(m + 1)!

|α|=m

α!Kα[cα], cα = ∂αY m(0). (4.15)

Then (4.14) holds for polynomials

Pmi(x) =1

2m(m + 1)!

|α|=m

α!cαix

α. (4.16)

Here cα = cα0 + cα1i + cα2j + cα3k.

4.3. Clifford analysis

Define the Clifford algebra Cl0,n ∈ Alg(R2n

) as associative unital algebra freelygenerated by Rn with usual inner product 〈x, y〉 modulo the relation x2 = −||x||2for all x ∈ Rn. Equivalently, the Clifford algebra Cl0,n is generated by the or-thonormal basis e0, e1, . . . , en in Rn+1, and all theirs permutations. Here e0 isa unit element and ei satisfies the relationships eiej + ejei = −2〈ei, ej〉e0 for1 ≤ j ≤ n. More details on Clifford algebras can be found in [5], [30].

Below we present some results from [18].

There are exactly four classes of 1-symmetry operators for the Dirac operatorD in Cl0,n, namely:

• the generators of the translation group in Rn+1

∂xk, k = 0, 1, . . . , n; (4.17)

• the dilatations

R0 = x0∂x0 + x1∂x1 + · · ·+ xn∂xn +n

2; (4.18)

• the generators of the rotation group

Jij = −Jji = xj∂xi − xi∂xj +1

2eij , i, j = 1, 2, . . . , n, i = j

Ji0 = −J0i = x0∂xi − xi∂x0 +1

2ei, i = 1, 2, . . . , n; (4.19)

• and the generators of the “conformal group”

2xixs∂xs − xx∂xi + (n + 1)xi − xei, (4.20)

for i = 0, 1, . . . , n. Here x = x0 + x1e1 + · · · + xnen and x are conjugate in thesense of Clifford-valued functions.

Using these basic 1-symmetries we can construct the Clifford-valued opera-tor indeterminates K − A in the space Hol(Cl0,n) as operator action similar tomultiplication on x − a. Namely, let a = a1e1 + · · · + anen and a be conjugatein the sense of Clifford algebra. Define A = A0 + A1e1 + · · · + Anen and Ai fori = 0, 1, . . . , n where

Ai = 2

j =i,j=0

ajJji − 2ai

aj∂xj + 2aiR0 + aa∂xi .

Theorem 4.7. All Cl0,n-analytic polynomial functions f(x) can be represented inthe neighborhood of a given point a in the form

u(x) = U0(K0 −A0, . . . , Kn −An)[1] + · · ·+ Ui(K0 −A0, . . . , Kn −An)[ei],

where Ui(x), i = 0, 1, 2, . . . , 2n, are real homogeneous polynomials being factorizedby the relation x2

0 + x21 + · · ·+ x2

n = 0.

Proof. The proof is analogous [18] to the proof of Theorem 4.6.

Claim 4.8. The Clifford-valued analytic functions have a unique power series ex-pansion in pairwise commutative operator independents K = K0, . . . , Kn.4.4. Axial monogenic polynomials

A Clifford analytic function u(x) is called axial symmetric if u(x) = F (K0)[x].

It was shown in [18] that all axial Clifford-valued analytic functions u(x) maybe represented in the form

Proposition 4.9. Any axial symmetric Clifford analytic function has a structure

u(x) = ϕ(x0, r) + ψ(x0, r)x,

where r is a length of radius vector, r2 = xx and ϕ(p, q), ψ(p, q) are generalizinganalytic functions in sense of [32], namely:

∂pϕ(p, q) + p∂pψ(p, q) = q∂qψ(p, q) + (n− 1)ψ(p, q),∂qϕ(p, q) + p∂qψ(p, q) + q∂pψ(p, q) = 0.

Table 1. Clifford analytic function in the axial symmetric case.

ϕ(p, q) ψ(p, q) Clifford analytic function

(n − 1)p 1 K0[1] = (n − 1)x0 + x

(n − 2)p2 2p K20 [1] = (n − 2)x2

0 − r2 + 2x0x

2p−1(1−2p+q2)n−2

2p−1(1−2p+q2)n−2 e(K0)[1] = 2x0−1−x

(1−2p+q2)n−2

Example. In [18] was defined the homogeneous monogenic polynomials in x

pl(x) =∑

((n− 1)/2)i

((n + 1)/2)j

One can explicitly verify that some axial monomials can be written in theform pl(x) = K l

0[a]. Now one can define the multiplication rule in space of theseaxially symmetric monomials. Namely, by the definition

pl(x) pm(x) = K l0(K

m0 [a]) = K l+m

0 [a] = pl+m(x).

Further, in [18] some commutation relations between the 1-symmetry operatorsare considered.

The knowledge of symmetries gives the key to the study of the structure ofmonogenic functions. It is well known that monogenic Clifford-valued functionscan be represented as convergent series of monogenic homogeneous polynomialfunctions. The problems of the representation of the axial symmetric monogenicfunctions therefore may be solved by the straightforward evaluation of Cliffordanalytic functions in operator indeterminates. (See examples in Table 1.)

5. Polynomial solutions to homogeneous PDE

First we clarify the structure of the exponentially-polynomial solutions to constantcoefficient homogeneous PDE:

Q(∂x)u(x) :=∑

|α|=m

aα∂αx u(x1, . . . , xn) = 0. (5.1)

Here aα are real for all multi-indices α, and ∂x stands for differentiation. Clearly,for m ≥ 2, (5.1) is not an evolution equation and, therefore, technique (1.6) is notstraightforward applicable.

The PDE (5.1) is said to be a homogeneous PDE of order m. Obviously, ahomogeneous PDE admits an infinitesimal dilatation R0 = x1∂1 + · · ·+ xn∂n.

Denote by dk,n the dimension of the space of all homogeneous polynomialsPk of order k in Rn. Thus

dk,n = dimPk =(

n + k − 1k

). (5.2)

Theorem 5.1. The homogeneous PDE (5.1) admit the exponential-polynomial so-lution u = e<λ,x>pk(x), pk(x) ∈ Pk in Cn, n ≥ 2 iff Q(λ) = 0, λ ∈ Cn.

Proof. If λ = 0 and the polynomial pk(x) is of order 0 ≤ k < m the proof istrivial. For k ≥ m and λ = 0, we have no more than dk−m,n linear algebraicconditions on coefficients of pk(x) ∈ Pk in order to fulfill (5.1). So the space ofthe homogeneous polynomial solutions to (5.1) of order k is at least of dimensiondQ,k ≥ dk,n − dk−m,n.

If λ = 0, it is possible to construct the exponential-polynomial solution inthe form u = pk(x)e<λ,x> only if pk(x) and e<λ,x> both are solutions. In fact λ is(necessarily) a root of Q(λ) = 0 with multiplicity l < m and

|β|=ν

|α|=m

). . .( αn

)∂β

xpk(x)λα−β = 0,

for ν = l + 1, . . . , m.

Actually, the symmetries are considered as transformations on the solutionspace of a DE. However, the solution space is not usually known a priori. Thestraightforward computations (cf. [20]) of symmetries turn out to be equivalent tothe solvability of the overdetermined system of PDEs.

Question 5.2. Under what condition is the overdetermined system of PDEs arisingin context of Proposition 0.10 non-trivially solvable?

To proceed with homogeneous PDE, for which we shall build the solutionspace explicitly, consider

5.1. Second-order PDE

Assume Q := Q(∂x) to be the second-order PDO associated with the quadraticform Q(x) = xT Ax in Rn, (n > 2):

Q(∂x)u(x) :=

aij∂xi∂xj u(x) = 0. (5.3)

Suppose the matrix A = aij in (5.3) is not singular. Denote by P (x) the qua-dratic form P (x) := xT A−1x.

Theorem 5.3. Let (p, n − p) denote the signature of the quadratic form Q(x) in(5.3), (i.e., p is the number of positive entries and m is the number of negativeentries in a diagonalization). Then the space of 1-symmetries operators of (5.3)forms N = (n+1)(n+2)/2-dimensional pseudo-orthogonal Lie algebra isomorphicto so(p+1, n− p+1) (cf. Definition 0.13). The basis in SolQ(Rn) (apart from thetrivial identity symmetry) consists of the following N operators:

(i) n generators of translation group in Rn

Di = ∂xi , i = 1, . . . , n; (5.4)

(ii) the generator of dilatation

R0 = x1∂x1 + x2∂x2 + · · ·+ xn∂xn +n− 2

2; (5.5)

(iii) n(n− 1)/2 generators of the rotation (Lorentz) group

Jij =1

4(Pi(x)Qj(∂)− Pj(x)Qi(∂)) i = j = 1, . . . , n; (5.6)

(iv) and the n generators of the special (pseudo-) conformal group

Ki = (n− 2)xi −1

(P (x)Qi(∂)− xi

Pj(x)Qj(∂)). (5.7)

Proof. Without loss of generality, assume Q(x) is already reduced to its canonicaldiagonal form and let (p, n − p) be the signature of Q(x). Hence, we can splitthe coordinates x1, . . . , xn onto two subsets, such that p coordinates with positiveentries come first

Q(∂x) :=

σi∂2xi

, here σi =

1 i ≤ p ,−1 i > p .

Choose the general 1-symmetry operator for Q(∂x) as follows:

L(x, ∂) :=n∑

σibi(x)∂xi + c(x). (5.9)

Following Proposition 0.10, in order to construct all symmetries of Q it isenough to find the function R(x) such that QL− LQ = R(x)Q. Thus

∂xj bi(x) + ∂xibj(x) = 0, σi∂xibi(x) = σj∂xj bj(x) = R(x), (5.10)

for all i = j ∈ 1, . . . , n and

Q[bi(x)] + 2∂xic(x) = 0, Q[c(x)] = 0. (5.11)

for all i.

Applying results of Proposition 2.2 to the system (5.10) yields that thereexists only a finite set of solutions to (5.10). We find the 1-symmetry operators toQ with (2.5). This gives:

Di = ∂xi , i = 1, . . . , n, (5.12)

R0 = x1∂x1 + · · ·+ xn∂xn + (n− 2)/2, (5.13)

Jij = σixi∂xj − σjxj∂xi , i, j = 1, . . . , n, (5.14)

Ki = (n− 2)xi −n∑

σjx2j∂xi + xi

σjxj∂xj , i = 1, . . . , n. (5.15)

There are some additional important relations between operators (5.12)–(5.15) in Sym(Q). For example, it can be easily verified that

Q(K) ≡ Q(x)2Q(∂x). (5.16)

Actually, Q(K) is a trivial symmetry operator (Q(K) ∼ 0).

Using the Definition 0.13, one can verify that the same commutator relationsfor both of the Lie algebras Sym(Q) and so(p+1, n−p+1) holds after the followingcorrespondences:

Di := Γ1,i+1 + Γi,n+2, R0 := Γ1,n+2, Ki := Γ1,i+1 − Γi,n+2, Jij := Γi+1,j+1.

Finally, after back substitution we obtain the required condition on coefficients ofL in the form (5.4)–(5.7).

Theorem 5.4. (cf. [17]) Let u(x) be any locally analytic solution to the homogeneousPDE (5.3). Then there exists a one-to-one correspondence between u(x) and theentire function f(x) such that

u(x) = f(K1, . . . , Kn)[1], (5.17)

where Ki was defined in (5.7) up to choice of basis.

5.2. Laplace and biharmonic equation

x = x1, . . . , xn ∈ Rn and r =√

x22 + · · ·+ x2

Consider axial symmetric harmonics u = u(x1, r) in Rn:

∆u(x1, r) ≡ ∂2x1

u(x1, r) + ∂2ru(x1, r) +

n− 2

r∂ru(x1, r) = 0

By Theorem 5.4 any axial symmetric harmonic function u(x1, r) in Rn may berepresented in the form u(x1, r) = F (K1)[1]. The action of the symmetry operator

K1 from (5.7) on the function u(x1, r) is defined as follows:

K1[u(x1, r)] := (n− 2)x1u(x1, r) + (x21 − r2)∂x1u(x1, r) + 2x1r∂ru(x1, r).

Example. Let f(x) be represented as a formal power series. Then the solution ofthe boundary value problem

∆u(x1, r) = 0, u(x1, 0) = f(x1), f(x) =

∞∑

may be written as u(x1, r) = F (K1)[1] where F (x) is as follows

F (x) =

∞∑

(n− 3)!

(n− 3 + m)!amxm.

Table 2. Some harmonic functions in R3 and their operator form.

u(x, 0) F (x) Harmonic function representation

xm 1m!

xm 16K3

1 [1] := x31 − 3r2/2

11−x

ex eK1 [1] := 1/√

(x1 − 1)2 + r2

ex I0(2√

x) I0(2√

K1)[1] := ex1J0(r)

We finish with the solution of the biharmonic equation ∆2u = 0 as follows:

u(x1, . . . , xn) = f(K1, . . . , Kn)[1] +

xigi(K1, . . . , Kn)[1].

Conclusions

• Every first-order PDO with constant coefficient is the Dirac operator in thecorresponding algebra.• The solution to the Dirac equation in isotopic algebras forms an equivalent

function theory.• The A-analysis in the regular algebras is equivalent to the canonical function

theory on their unital hearts.

Acknowledgments

The authors gratefully acknowledge fruitful conversations with Yuli Eidelman con-cerning the results of this article. The authors are grateful to an anonymous refereefor important remarks and suggestions.

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Samuil D. EidelmanDepartment of MathematicsSolomonov UniversityKiev, Ukrainee-mail: seidelman@amath.kiev.ua

Yakov KrasnovDepartment of MathematicsBar-Ilan UniversityRamat-Gan, 52900, Israele-mail: krasnov@math.biu.ac.il

On the Bessmertnyı Class of HomogeneousPositive Holomorphic Functionson a Product of Matrix Halfplanes

Dmitry S. Kalyuzhnyı-Verbovetzkiı

Abstract. We generalize our earlier results from [9] on the Bessmertnyı classof operator-valued functions holomorphic in the open right poly-halfplanewhich admit representation as a Schur complement of a block of a linear ho-mogeneous operator-valued function with positive semidefinite operator coef-ficients, to the case of a product of open right matrix halfplanes. Several equiv-alent characterizations of this generalized Bessmertnyı class are presented. Inparticular, its intimate connection with the Agler–Schur class of holomorphiccontractive operator-valued functions on the product of matrix unit disks isestablished.

Mathematics Subject Classification (2000). Primary 47A48; Secondary 32A10,47A56, 47A60.

Keywords. Several complex variables, homogeneous, positive, holomorphic,operator-valued functions, product of matrix halfplanes, long resolvent repre-sentations, Agler–Schur class.

1. Introduction

In the PhD Thesis of M.F. Bessmertrnyı [4] (the translation into English of someof its parts can be found in [5, 6, 7]) the class of rational n × n matrix-valuedfunctions of N complex variables z = (z1, . . . , zN) ∈ CN , representable in the form

f(z) = a(z)− b(z)d(z)−1c(z), (1.1)

where a linear (n + p)× (n + p) matrix-valued function

A(z) = z1A1 + · · ·+ zNAN =

[a(z) b(z)c(z) d(z)

](1.2)

The author was supported by the Center for Advanced Studies in Mathematics, Ben-GurionUniversity of the Negev.

140 D.S. Kalyuzhnyı-Verbovetzkiı

has positive semidefinite matrix coefficients Aj , j = 1, . . . , N , with real entries,was considered. Such a representation (1.1)–(1.2), was called in the thesis a longresolvent representation. The motivation of its consideration comes from the elec-trical engineering. Bessmertrnyı has shown that this class is precisely the class ofall characteristic functions of passive 2n-poles, where the impedances of elementsof an electrical circuit are considered as independent variables.

In [9] a more general class BN(U) of holomorphic functions on the open rightpoly-halfplane ΠN := z ∈ CN : Re zk > 0, k = 1, . . . , N, with values in theC∗-algebra L(U) of bounded linear operators on a Hilbert space U , which admita representation (1.1) with a linear pencil A(z) as in (1.2), however consistingof operators from L(U ⊕H) where H is an auxiliary Hilbert space, such thatAj ≥ 0, j = 1, . . . , N , was introduced. Here the Hilbert spaces are supposed tobe complex. This class BN(U) was called the Bessmertnyı class. Any functionf ∈ BN (U) is homogeneous of degree one and takes operator values with positivesemidefinite real parts. Moreover, f can be uniquely extended to a holomorphicand homogeneous of degree one function on the domain

ΩN :=⋃

λ∈T

(λΠ)N ⊂ CN , (1.3)

so that (1.1) holds true for z ∈ ΩN , as well as the homogeneity relation

f(λz) = λf(z), λ ∈ C \ 0, z ∈ ΩN , (1.4)

and the symmetry relation

f(z) = f(z)∗, z ∈ ΩN (1.5)

(here λz = (λz1, . . . , λzN ) and z = (z1, . . . , zN)). In [9] several equivalent char-acterizations of the Bessmertnyı class have been established: in terms of certainpositive semidefinite kernels on ΩN × ΩN , in terms of functional calculus of N -tuples of commuting bounded strictly accretive operators on a common Hilbertspace, and in terms of the double Cayley transform. Let us briefly recall thelast one. The double Cayley transform (over the variables and over the values),F = C(f), of a function f ∈ BN(U) is defined for w in the open unit polydiskDN := w ∈ CN : |wk| < 1, k = 1, . . . , N as

F(w) =

(1 + w1

1− w1, . . . ,

1 + wN

1− wN

)− IU

(1 + w1

1− w1, . . . ,

1 + wN

1− wN

(1.6)For any f ∈ BN(U), its double Cayley transform F = C(f) belongs to the Agler–Schur class ASN (U), i.e., F is holomorphic on DN and ‖F(T)‖ ≤ 1 for ev-ery N -tuple T = (T1, . . . , TN) of commuting strict contractions on a commonHilbert space (see details on this class in [1]). Moreover, there exist Hilbert spaces

X ,X1, . . . ,XN such that X =⊕N

k=1 Xk, and an Agler representation

F(w) = D + CP (w)(IX −AP (w))−1B, w ∈ DN , (1.7)

On the Bessmertnyı Class 141

where P (w) =∑N

k=1 wkPXk, with orthogonal projections PXk

onto Xk, and[

A BC D

]=: U = U−1 = U∗ ∈ L(X ⊕ U).

Conversely, any function F ∈ ASN (U) satisfying the latter condition can be rep-resented as the double Cayley transform, F = C(f), of some function f ∈ BN (U).

Let us recollect that matrices Aj , j = 1, . . . , N , in original Bessmertnyı’sdefinition had real entries, thus functions from his class took matrix values whoseall entries were real at real points z ∈ RN . In [9] we have considered also a “real”version of the (generalized) Bessmertnyı class. Namely, we have defined the realstructure on a Hilbert space U by means of an anti-unitary involution (a counter-part of the complex conjugation), i.e., an operator ι = ιU : U → U such that

ι2 = IU , (1.8)

〈ιu1, ιu2〉 = 〈u2, u1〉 , u1, u2 ∈ U . (1.9)

Such an operator ι is anti-linear, i.e.,

ι(αu1 + βu2) = αu1 + βu2, α, β ∈ C, u1, u2 ∈ U .

An operator A from L(U ,Y), the Banach space of all bounded linear operatorsfrom a Hilbert space U to a Hilbert space Y, is called (ιU , ιY)-real for anti-unitaryinvolutions ιU and ιY if

ιYA = AιU . (1.10)

Such operators A are a counterpart of matrices with real entries. Finally, a functionf on a set Ω ⊂ CN such that z ∈ Ω ⇔ z ∈ Ω, which takes values from L(U ,Y) iscalled (ιU , ιY)-real if

f ♯(z) := ιYf(z)ιU = f(z), z ∈ Ω. (1.11)

If U = Y and ιU = ιY = ι then such a function is called ι-real. We have definedthe “ι-real” Bessmertnyı class ιRBN (U) as the subclass of all ι-real functions fromBN(U). The latter subclass is a counterpart of the original class considered byBessmertnyı. In [9] we have obtained different characterizations for ιRBN (U), too.

In the present paper we introduce and investigate analogous classes of func-tions (either for the “complex” and “real” cases) on more general domains. First,we define a product of matrix halfplanes as

Πn1×n1 × · · · ×ΠnN×nN := Z = (Z1, . . . , ZN) :

Zk ∈ Cnk×nk , Zk + Z∗k > 0, k = 1, . . . , N (1.12)

which serves as a generalization of the open right poly-halfplane ΠN . Then wedefine a counterpart of the domain ΩN as

Ωn1,...,nN :=⋃

λ∈T

(λΠn1×n1 × · · · × λΠnN×nN

), (1.13)

and define the corresponding Bessmertnyı classes of functions on the domainΩn1,...,nN . Consideration of such classes can be also motivated by problems of thetheory of electrical networks since there are situations where “matrix impedances”

are considered as matrix variables (see, e.g., [10]). On the other hand, mathemat-ical tools for such an investigation have recently appeared. Since in [9] the closerelation of the Bessmertnyı classes BN (U) and ιRBN(U) to the Agler–Schur classASN (U) has been established, this has made possible the use of properties of thelatter class as a tool for investigation Bessmertnyı’s classes. In the same mannerwe make use of the recent works of C.-G. Ambrozie and D. Timotin [2], J.A. Balland V. Bolotnikov [3] on the Agler–Schur class of function on so-called polynomi-ally defined domains for the investigation of the Bessmertnyı’s classes of functionson Ωn1,...,nN . A counterpart of the class BN (U) is introduced in Section 2, wherealso a useful decomposition for functions from this class is obtained. In Section 3the relationship between the Bessmertnyı class on Ωn1,...,nN and the correspond-ing Agler–Schur class on a product of matrix disks is established. This allows usto give a characterization of the (generalized) Bessmertnyı class in terms of func-tional calculus for collections of operators. In Section 4 we describe the image ofthis class under the double Cayley transform. Finally, a counterpart of the “real”Bessmertnyı class ιRBN (U) is studied in Section 5.

2. The Bessmertnyı class for a matrix domain

Let us define the class Bn1,...,nN (U) of all L(U)-valued functions f holomorphic onthe domain Ωn1,...,nN defined in (1.13) (see also (1.12)) which are representable as

f(Z) = a(Z)− b(Z)d(Z)−1c(Z) (2.1)

for Z ∈ Ωn1,...,nN , where

A(Z) = G∗1(Z1⊗IM1)G1+· · ·+G∗

N (ZN⊗IMN )GN =

[a(Z) b(Z)c(Z) d(Z)

]∈ L(U ⊕H)

(2.2)for some Hilbert spaces M1, . . . ,MN ,H and operators Gk ∈ L(U ⊕H, Cnk ⊗Mk), k = 1, . . . , N .

Remark 2.1. If a function f is holomorphic on Πn1×n1 × · · · × ΠnN×nN and hasa representation (2.1)–(2.2) there, then f can be extended to Ωn1,...,nN by ho-mogeneity of degree one, and this extension is, clearly, holomorphic and admitsa representation 2.1 in Ωn1,...,nN . That is why we define the class Bn1,...,nN (U)straight away as a class of functions on Ωn1,...,nN . Keeping in mind the possibilityand uniqueness of such extension, we will write sometimes f ∈ Bn1,...,nN (U) forfunctions defined originally on Πn1×n1 × · · · ×ΠnN×nN .

Theorem 2.2. Let f be an L(U)-valued function holomorphic on Πn1×n1 × · · · ×ΠnN×nN . Then f ∈ Bn1,...,nN (U) if and only if there exist holomorphic functionsϕk(Z) on Πn1×n1 × · · · ×ΠnN×nN with values in L(U , Cnk ⊗Mk), k = 1, . . . , N ,such that

f(Z) =

ϕk(Λ)∗(Zk ⊗ IMk)ϕk(Z), Z, Λ ∈ Πn1×n1 × · · · ×ΠnN×nN (2.3)

holds. In this case the functions ϕk(Z) can be uniquely extended to the holomorphicfunctions on Ωn1,...,nN (we use the same notation for the extended functions) whichare homogeneous of degree zero, i.e., for every λ ∈ C\0,

ϕk(λZ) = ϕk(Z), Z ∈ Ωn1,...,nN , (2.4)

and identity (2.3) is extended to all of Z, Λ ∈ Ωn1,...,nN .

Proof. Necessity. Let f ∈ Bn1,...,nN (U). Then (2.1) holds for Z ∈ Ωn1,...,nN , someHilbert spaces H,M1, . . . ,MN and a linear pencil of operators (2.2). Define

ψ(Z) :=

−d(Z)−1c(Z)

]∈ L(U ,U ⊕H), Z ∈ Ωn1,...,nN .

Then for all Z, Λ ∈ Ωn1,...,nN one has

f(Z) = a(Z)− b(Z)d(Z)−1c(Z)

IU −c(Λ)∗d(Λ)−∗ ][

a(Z)− b(Z)d(Z)−1c(Z)0

IU −c(Λ)∗d(Λ)−∗ ][

a(Z) b(Z)c(Z) d(Z)

−d(Z)−1c(Z)

= ψ(Λ)∗A(Z)ψ(Z).

Set ϕk(Z) := Gkψ(Z), k = 1, . . . , N . Clearly, the functions ϕk(Z), k = 1, . . . , N ,are holomorphic on Ωn1,...,nN and satisfy (2.4). Rewriting the equality

f(Z) = ψ(Λ)∗A(Z)ψ(Z), Z ∈ Ωn1,...,nN , (2.5)

in the form

f(Z) =

ϕk(Λ)∗(Zk ⊗ IMk)ϕk(Z), Z, Λ ∈ Ωn1,...,nN , (2.6)

we obtain, in particular, (2.3).

Sufficiency. Let f be an L(U)-valued function holomorphic on Πn1×n1 ×· · ·×ΠnN×nN and representable there in the form (2.3) with some holomorphic functionsϕk(Z) taking values in L(U , Cnk ⊗Mk), k = 1, . . . , N . Set

(Cnk ⊗Mk), Pk := PMk, k = 1, . . . , N,

ϕ(Z) := col(

ϕ1(Z) . . . ϕN (Z))∈ L(U ,N ),

E := (In1 , . . . , InN ) ∈ Πn1×n1 × · · · ×ΠnN×nN ,

where In denotes the identity n× n matrix. From (2.3) we get

f(E) =

ϕk(Λ)∗ϕk(E), Λ ∈ Πn1×n1 × · · · ×ΠnN×nN . (2.7)

In particular,

f(E) =

ϕk(E)∗ϕk(E). (2.8)

By subtracting (2.8) from (2.7) we get

[ϕk(Λ)− ϕk(E)]∗ϕk(E) = 0, Λ ∈ Πn1×n1 × · · · ×ΠnN×nN ,

i.e., the following orthogonality relation holds:

H := clos spanΛ∈Πn1×n1×···×ΠnN×nN [ϕ(Λ)− ϕ(E)]U ⊥ closϕ(E)U =: X .

For any Λ ∈ Πn1×n1 × · · · ×ΠnN×nN and u ∈ U one can represent now ϕ(Λ)u as

ϕ(E) ϕ(Λ)− ϕ(E)]u ∈ X ⊕H.

On the other hand, for any u ∈ U , Λ ∈ Πn1×n1 × · · · ×ΠnN×nN one has

ϕ(E)u ∈ clos spanΛ∈Πn1×n1×···×ΠnN×nN ϕ(Λ)U,(ϕ(Λ)− ϕ(E))u ∈ clos spanΛ∈Πn1×n1×···×ΠnN×nN ϕ(Λ)U.

Thus, clos spanΛ∈Πn1×n1×···×ΠnN×nN ϕ(Λ)U = X ⊕H. Let κ : X ⊕H → N bethe natural embedding defined by

[ϕ(E)u

(ϕ(Λ)− ϕ(E))u

]−→ ϕ(Λ)u =

⎡⎢⎣

ϕ1(Λ)u...

ϕN (Λ)u

⎤⎥⎦ (2.9)

and extended to the whole X ⊕H by linearity and continuity. Set

Gk := (Ink⊗ Pk)κ

[ϕ(E) 0

]∈ L(U ⊕H, Cnk ⊗Mk), k = 1, . . . , N,

ψ(Λ) :=

ϕ(Λ)− ϕ(E)

]∈ L(U ,U ⊕H), Λ ∈ Πn1×n1 × · · · ×ΠnN×nN .

Thenf(Z) = ψ(Λ)∗A(Z)ψ(Z), Z, Λ ∈ Πn1×n1 × · · · ×ΠnN×nN ,

where A(Z) is defined by (2.2). Indeed,

ψ(Λ)∗A(Z)ψ(Z) =

ϕ(Λ)− ϕ(E)

]∗ [ϕ(E) 0

]∗κ∗(

Zk ⊗ Pk

ϕ(E) 00 IH

ϕ(Z)− ϕ(E)

N∑k=1

ϕk(Λ)∗(Zk ⊗ IMk)ϕk(Z) = f(Z).

Now, with respect to the block partitioning of A(Z) we have

A(Z)ψ(Z) =

[a(Z) b(Z)c(Z) d(Z)

ϕ(Z)− ϕ(E)

[a(Z) + b(Z)(ϕ(Z)− ϕ(E))c(Z) + d(Z)(ϕ(Z) − ϕ(E))

[f1(Z)f2(Z)

Since for Z, Λ ∈ Πn1×n1 × · · · ×ΠnN×nN one has

ψ(Λ)∗A(Z)ψ(Z) =[

IU ϕ(Λ)∗ − ϕ(E)∗] [ f1(Z)

]= f(Z),

by setting Λ := E in this equality we get

f1(Z) = f(Z), Z ∈ Πn1×n1 × · · · ×ΠnN×nN .

Therefore, for every Z, Λ ∈ Πn1×n1×· · ·×ΠnN×nN we get [ϕ(Λ)−ϕ(E)]∗f2(Z) = 0.This implies that for every Z ∈ Πn1×n1 × · · · × ΠnN×nN and u ∈ U one hasf2(Z)u ⊥ H. But f2(Z)u ∈ H. Therefore, f2(Z)u = 0, and f2(Z) ≡ 0, i.e.,

c(Z) + d(Z)[ϕ(Z)− ϕ(E)] ≡ 0. (2.10)

Since for every Z ∈ Πn1×n1 × · · · ×ΠnN×nN the operator P (Z) :=∑N

k=1 Zk ⊗ Pk

has positive definite real part, i.e., P (Z) + P (Z)∗ ≥ αZIN > 0 for some scalarαZ > 0, the operator d(Z) = PHκ∗P (Z)κ|H has positive definite real part, too.Therefore, d(Z) is boundedly invertible for all Z ∈ Πn1×n1 × · · · ×ΠnN×nN . From(2.10) we get ϕ(Z)− ϕ(E) = −d(Z)−1c(Z), Z ∈ Πn1×n1 × · · · ×ΠnN×nN , and

f(Z) = f1(Z) = a(Z)− b(Z)d(Z)−1c(Z), Z ∈ Πn1×n1 × · · · ×ΠnN×nN .

Taking into account Remark 2.1, we get f ∈ Bn1,...,nN (U).

Functions ϕ(Z) − ϕ(E) = −d(Z)−1c(Z) and, hence, ψ(Z) are well defined,holomorphic and homogeneous of degree zero on Ωn1,...,nN , thus (2.6) holds. – Theproof is complete.

3. The class Bn1,...,nN(U) and functional calculus

Let us observe now that (2.3) is equivalent to the couple of identities

f(Z) + f(Λ)∗ =

ϕk(Λ)∗((Zk + Λ∗k)⊗ IMk

)ϕk(Z), (3.1)

f(Z)− f(Λ)∗ =

ϕk(Λ)∗((Zk − Λ∗k)⊗ IMk

)ϕk(Z) (3.2)

valid for all Z, Λ ∈ Πn1×n1 × · · · × ΠnN×nN . We will show that the double Cay-ley transform F = C(f) applied to a function f from the Bessmertnyı classBn1,...,nN (U) and defined as

F(W ) =[f((In1 + W1)(In1 −W1)

−1, . . . , (InN + WN )(InN −WN )−1)− IU]

×[f((In1 + W1)(In1 −W1)

−1, . . . , (InN + WN )(InN −WN )−1) + IU]−1

(compare with (1.6)) turns the first of these identities into an Agler-type identitywhich characterizes the Agler–Schur class of holomorphic L(U)-valued functions

on the product of open matrix unit disks

Dn1×n1 × · · · × DnN×nN := W = (W1, . . . , WN ) ∈ Cn1×n1 × · · · × CnN×nN :

WkW ∗k < Ink

, k = 1, . . . , N.

The latter is a special case of the Agler–Schur class of holomorphic L(U)-valuedfunctions on a domain with matrix polynomial defining function, which was studiedin [2] and [3]. This allows us to obtain one more characterization of Bn1,...,nN (U).Let P (w), w ∈ Cn, be a polynomial p× q matrix-valued function, and

DP := w ∈ Cn : ‖P (w)‖ < 1

(here and in the sequel the norm of a p× q matrix means its operator norm withrespect to the standard Euclidean metrics in Cp and Cq). Let CDP denote the set ofcommutative n-tuples T = (T1, . . . , Tn) of bounded linear operators on a commonHilbert space HT subject to the condition ‖P (T)‖ < 1. It was shown in [2] thatthe Taylor joint spectrum σT (T) (see [15, 16] and also [8]) of any T ∈ CDP iscontained in DP . Thus, for any function S holomorphic on DP and any T ∈ CDP

the operator S(T) is well defined by the Taylor functional calculus (see [14, 17]and also [8]). For the domain DP , the Agler–Schur class ASDP (E , E∗) consists ofall holomorphic L(E , E∗)-valued functions F on DP such that

‖F(T)‖ ≤ 1, T ∈ CDP . (3.4)

Recall the following theorem from [3] (the case when E = E∗ = C can be foundin [2]), however in a slightly simplified form which will be sufficient for our pur-pose.

Theorem 3.1. Let F be an L(E , E∗)-valued function holomorphic on DP . Then thefollowing statements are equivalent:

(i) F ∈ ASDP (E , E∗);(ii) there exist an auxiliary Hilbert space M and an L(Cp ⊗M, E∗)-valued func-

tion HL holomorphic on DP such that

IE∗−F(w)F(ω)∗ = HL(w) ((Ip − P (w)P (ω)∗)⊗ IM) HL(ω)∗ (3.5)

holds for all w, ω ∈ DP ;

(iii) there exist an auxiliary Hilbert spaceM and an L(E , Cq⊗M)-valued functionHR holomorphic on DP such that

IE −F(ω)∗F(w) = HR(ω)∗ ((Iq − P (ω)∗P (w)) ⊗ IM)HR(w) (3.6)

holds for all w, ω ∈ DP ;

(iv) there exist an auxiliary Hilbert space M, an L(Cp ⊗M, E∗)-valued functionHL and an L(E , Cq⊗M)-valued function HR, which are holomorphic on DP ,

such that[

IE −F(ω′)∗F(w) F(ω′)∗ −F(ω)∗

F(w′)−F(w) IE∗−F(w′)F(ω)∗

[HR(ω′)∗ 0

0 HL(w′)

](3.7)

Iq − P (ω′)∗P (w) P (ω′)∗ − P (ω)∗

P (w′)− P (w) Ip − P (w′)P (ω)∗

]⊗ IM

)[HR(w) 0

0 HL(ω)∗

holds for all w, w′, ω, ω′ ∈ DP ;

(v) there exists a Hilbert space X and a unitary operator

[A BC D

]∈ L((Cp ⊗X )⊕ E , (Cq ⊗X )⊕ E∗) (3.8)

such that

F (w) = D + C(P (w) ⊗ IX ) (ICq⊗X −A(P (w) ⊗ IX ))−1

B (3.9)

holds for all w ∈ DP .

In [3] it was shown how to obtain from (3.5) a unitary operator (3.8) whichgives the representation (3.9) for an arbitraryF ∈ ASDP (E , E∗). We will show nowhow to get from (3.7) a special unitary operator (3.8) and representation (3.9) foran arbitrary F ∈ ASDP (E , E∗). Let (3.7) hold for such F , where a Hilbert spaceM and functions HL, HR are such as in statement (iv) of Theorem 3.1. Define thelineals

D0 := span

[(P (w) ⊗ IM)HR(w)

[HL(ω)∗

F(ω)∗

]e∗ :

w, ω ∈ DP , e ∈ E , e∗ ∈ E∗ ⊂ (Cp ⊗M)⊕ E ,

R0 := span

[HR(w)F(w)

[(P (ω)∗ ⊗ IM)HL(ω)∗

]e∗ :

w, ω ∈ DP , e ∈ E , e∗ ∈ E∗ ⊂ (Cq ⊗M)⊕ E∗,

and the operator U0 : D0 →R0 which acts on the generating vectors of D0 as[

(P (w) ⊗ IM)HR(w)IE

]e −→

[HR(w)F(w)

]e, w ∈ DP , e ∈ E ,

[HL(ω)∗

F(ω)∗

]e∗ −→

[(P (ω)∗ ⊗ IM)HL(ω)∗

]e∗, ω ∈ DP , e∗ ∈ E∗.

This operator is correctly defined. Moreover, U0 maps D0 isometrically onto R0.Indeed, (3.7) can be rewritten as

[HR(ω′)∗ F(ω′)∗

HL(w′)(P (w′)⊗ IM) IE∗

] [HR(w) (P (ω)∗ ⊗ IM)HL(ω)∗

F(w) IE∗

[HR(ω′)∗(P (ω′)∗ ⊗ IM) IE

HL(w′) F (w′)

] [(P (w) ⊗ IM)HR(w) HL(ω)∗

IE F(ω)∗

which means that for

[(P (w) ⊗ IM)HR(w)

[HL(ω)∗

F(ω)∗

]e∗,

x′ =

[(P (w′)⊗ IM)HR(w′)

]e′ +

[HL(ω′)∗

F(ω′)∗

]e′∗,

one has〈U0x, U0x

′〉 = 〈x, x′〉 .Clearly, U0 can be uniquely extended to the unitary operator U0 : clos(D0) →clos(R0). In the case when

dim((Cp ⊗M)⊕ E)⊖ clos(D0) = dim((Cq ⊗M)⊕ E∗)⊖ clos(R0) (3.10)

there exists a (non-unique!) unitary operator U : (Cp⊗M)⊕E → (Cq ⊗M)⊕E∗such that U |clos(D0) = U0. In the case when (3.10) does not hold one can set

M :=M⊕K, where K is an infinite-dimensional Hilbert space, then (3.10) holds

for M in the place ofM, and there exists a unitary operator U : (Cp⊗M)⊕E →(Cq ⊗ M) ⊕ E∗ such that U |clos(D0) = U0. Thus, without loss of generality wemay consider that (3.10) holds.

Let U have a block partitioning

[A BC D

]: (Cp ⊗M)⊕ E → (Cq ⊗M)⊕ E∗.

Then, in particular,[A BC D

] [(P (w) ⊗ IM)HR(w)

[HR(w)F(w)

], w ∈ DP . (3.11)

Since for w ∈ DP one has ‖P (w)‖ < 1, and since ‖A‖ ≤ 1, we can solve the firstblock row equation of (3.11) for HR(w):

HR(w) = (ICq⊗H −A(P (w) ⊗ IM))−1B, w ∈ DP .

Then from the second block row of (3.11) we get

F(w) = D + C(P (w) ⊗ IM)(ICq⊗M −A(P (w) ⊗ IM))−1B, w ∈ DP ,

i.e., (3.9) with X =M.We are interested here in the case of the Agler–Schur class for the domain

DP where the domain DP is Dn1×n1 × · · · × DnN×nN , and the polynomial whichdefines this domain is

P (W ) = diag(W1, . . . , WN ), W ∈ Dn1×n1 × · · · × DnN×nN .

Here W may be viewed as an (n21 · · ·n2

N )-tuple of scalar variables (Wk)ij , k =1, . . . , N, i, j = 1, . . . , nk. We will write in this case ASn1,...,nN (E , E∗) instead ofASDP (E , E∗), and if E = E∗ we will write ASn1,...,nN (E). The class CDP is identified

for DP = Dn1×n1×· · ·×DnN×nN with the class C(n1,...,nN ) of N -tuples of matricesT = (T1, . . . , TN) ∈ Bn1×n1

T×· · ·×BnN×nN

Tover a common commutative operator

algebra BT ⊂ L(HT), with a Hilbert space HT, such that ‖Tk‖ < 1, k = 1, . . . , N .

Denote by A(n1,...,nN ) the class of N -tuples of matrices R = (R1, . . . , RN ) ∈Bn1×n1

R×· · ·×BnN×nN

Rover a common commutative operator algebraBR ⊂ L(HR),

with a Hilbert space HR, for which there exists a real constant sR > 0 such that

Rk + R∗k ≥ sRICnk

⊗HR, k = 1, . . . , N.

Theorem 3.2. For any R ∈ A(n1,...,nN ),

σT (R) ⊂ Πn1×n1 × · · · ×ΠnN×nN ,

where σT (R) denotes the Taylor joint spectrum of the collection of operators(Rk)ij , k = 1, . . . , N, i, j = 1, . . . , nk.

Proof. It is shown in [13] that the Taylor joint spectrum σT (X) of an n-tuple ofcommuting bounded operators X = (X1, . . . , Xn) on a common Hilbert space HX

is contained in the polynomially convex closure of σπ(X), the approximate pointspectrum of X. The latter is defined as the set of points λ = (λ1, . . . , λn) ∈ Cn forwhich there exists a sequence of vectors hν ∈ HX such that ‖hν‖ = 1, ν ∈ N, and(Xj − λjIHX

)hν → 0 as ν →∞ for all j = 1, . . . , n. Thus it suffices to show that

σπ(R) := σπ ((Rk)ij : k = 1, . . . , N, i, j = 1, . . . , nk) ⊂ Πn1×n1s ×· · ·×ΠnN×nN

whenever R ∈ A(n1,...,nN ) and Rk + R∗k ≥ sICnk⊗HR

> 0, k = 1, . . . , N , where

Πn×ns := M ∈ Cn×n : M + M∗ ≥ sIn,

since Πn1×n1s × · · ·×ΠnN×nN

s is convex, and hence polynomially convex, and sinceΠn1×n1

s × · · · × ΠnN×nNs ⊂ Πn1×n1 × · · · × ΠnN×nN for s > 0. Suppose that Λ =

(Λ1, . . . ,ΛN) ∈ σπ(R). Then there exists a sequence of vectors hν ∈ HR such that‖hν‖ = 1, ν ∈ N, and for k = 1, . . . , N, i, j = 1, . . . , nk one has

((Rk)ij − (Λk)ijIHR)hν → 0 as ν →∞.

Therefore, for every k ∈ 1, . . . , N and uk = col(uk1, . . . , uknk) ∈ Cnk one has

(⟨((Rk)ij + (Rk)∗ji)hν , hν

⟩− ((Λk)ij + (Λk)ji) 〈hν , hν〉

)ukiukj → 0

as ν →∞. Since 〈hν , hν〉 = 1, the subtrahend does not depend on ν. Therefore,

s 〈uk, uk〉 = s limν→∞

〈uk ⊗ hν , uk ⊗ hν〉≤ lim

ν→∞〈(Rk + R∗

k)uk ⊗ hν , uk ⊗ hν〉

= limν→∞

⟨((Rk)ij + (Rk)∗ji)hν , hν

⟩ukiukj

((Λk)ij + (Λk)ji)ukiukj = 〈(Λk + Λ∗k)uk, uk〉 .

Thus, Λk + Λ∗k ≥ sInk

, k = 1, . . . , N , i.e., Λ ∈ Πn1×n1s × · · · ×ΠnN×nN

s , as desired.

Theorem 3.2 implies that for every holomorphic function f on Πn1×n1×· · ·×ΠnN×nN and every R ∈ A(n1,...,nN ) the operator f(R) is well defined by the Taylorfunctional calculus.

The Cayley transform defined by

Rk = (ICnk⊗HT+ Tk)(ICnk⊗HT

− Tk)−1, k = 1, . . . , N, (3.12)

maps the class C(n1,...,nN ) onto the class A(n1,...,nN ), and its inverse is given by

Tk = (Rk − ICnk⊗HR)(Rk + ICnk⊗HR

)−1, k = 1, . . . , N, (3.13)

where HR = HT. Let f be an L(U)-valued function holomorphic on Πn1×n1 ×· · · × ΠnN×nN . Then its double Cayley transform F = C(f) defined by (3.3) isholomorphic on Dn1×n1 × · · · × DnN×nN , and by the spectral mapping theoremand uniqueness of Taylor’s functional calculus (see [11]) one has

F(T) = f(R),

where T ∈ C(n1,...,nN ) and R ∈ A(n1,...,nN ) are related by (3.12) and (3.13).

Theorem 3.3. Let f be an L(U)-valued function holomorphic on Πn1×n1 × · · · ×ΠnN×nN . Then f ∈ Bn1,...,nN (U) if and only if the following conditions are satis-fied:

(i) f(tZ) = tf(Z), t > 0, Z ∈ Πn1×n1 × · · · ×ΠnN×nN ;(ii) f(R) + f(R)∗ ≥ 0, R ∈ A(n1,...,nN );(iii) f(Z∗) := f(Z∗

1 , . . . , Z∗N) = f(Z)∗, Z ∈ Πn1×n1 × · · · ×ΠnN×nN .

Proof. Necessity. Let f ∈ Bn1,...,nN (U). Then (i) and (iii) easily follow from therepresentation (2.1) of f . Condition (ii) on f is equivalent to condition (3.4) on Fwhich is defined by (3.3), i.e., to F ∈ ASn1,...,nN (U). Let us show the latter. Sinceby Theorem 2.2 f satisfies (2.3), and hence (3.1), one can set

Zk = (Ink+ Wk)(Ink

−Wk)−1, Λk = (Ink+ Ξk)(Ink

− Ξk)−1, k = 1, . . . , N,

in (3.1) and get

(IU + F(W ))(IU −F(W ))−1 + (IU −F(Ξ)∗)−1(IU + F(Ξ)∗)

θk(Ξ)∗(

(Ink+ Wk)(Ink

−Wk)−1 + (Ink− Ξ∗

k)−1(Ink+ Ξ∗

k))⊗ IMk

× θk(W ), W, Ξ ∈ Dn1×n1 × · · · × DnN×nN ,

where for k = 1, . . . , N ,

θk(W ) = ϕk

((In1 + W1)(In1 −W1)

−1, . . . , (InN + WN )(InN −WN )−1). (3.14)

We can rewrite this in the form

IU −F(Ξ)∗F(W ) =

θk(Ξ)∗ ((Ink− Ξ∗

kWk)⊗ IMk) θk(W ), (3.15)

where for k = 1, . . . , N ,

θk(W ) =((Ink

−Wk)−1 ⊗ IMk

)θk(W )(IU −F(W )) ∈ L(U , Cnk ⊗Mk). (3.16)

The identity (3.15) coincides with (3.6) for our case, with

HR(W ) = col(θ1(W ), . . . , θN (W )) ∈ L

(Cnk ⊗Mk)

P (W ) = diag(W1, . . . , WN ).

Note, that without loss of generality we may consider all of Mk’s equal to some

spaceM, say,M =⊕N

k=1Mk. Then HR(W ) ∈ L (U , Cn1+···+nN ⊗M). By The-orem 3.1, this means that F ∈ ASn1,...,nN (U).

Sufficiency. Let f satisfy conditions (i)–(iii). Since (ii) is equivalent to F ∈ASn1,...,nN (U), where F is defined by (3.3), the identity (3.15) holds with someL (U , Cnk ⊗M)-valued functions θk holomorphic on Dn1×n1 × · · · ×DnN×nN , k =1, . . . , N , with an auxiliary Hilbert space M (spaces Mk can be chosen equal in(3.15)). Set

Wk = (Zk − Ink)(Zk + Ink

)−1, Ξk = (Λk − Ink)(Λk + Ink

)−1, k = 1, . . . , N,

in (3.15), and by virtue of (3.3) get (3.1) with

ϕk(Z) = ((Ink+ Zk)−1)⊗ IM)

× θk

((Z1 − In1)(Z1 + In1)

−1, . . . , (ZN − InN )(ZN + InN )−1)

× (IU + f(Z)) ∈ L(U , Cnk ⊗M), k = 1, . . . , N (3.17)

(in fact, passing from (3.1) to (3.15) is invertible, and (3.17) is obtained from(3.14) and (3.16), and vice versa). The property (iii) implies f(X) = f(X)∗ forevery N -tuple X = (X1, . . . , XN) ∈ Πn1×n1 × · · · × ΠnN×nN of positive definitematrices (we will denote this set by P(n1,...,nN )), and for any such X and t > 0 by(3.1) one has:

f(X) + f(tX) = (1 + t)N∑

ϕk(tX)∗(Xk ⊗ IM)ϕk(X),

f(tX) + f(X) = (1 + t)N∑

ϕk(X)∗(Xk ⊗ IM)ϕk(tX),

2[f(X) + f(X)] =

ϕk(X)∗(2Xk ⊗ IM)ϕk(X),

2t[f(tX) + f(tX)] =

2ϕk(tX)∗(2tXk ⊗ IM)ϕk(tX).

By (i), the left-hand sides of these equalities coincide and equal (1+ t)f(X), hence

f(X) =

ϕk(tX)∗(Xk ⊗ IM)ϕk(X) =

ϕk(X)∗(Xk ⊗ IM)ϕk(tX)

ϕk(X)∗(Xk ⊗ IM)ϕk(X) =N∑

ϕk(tX)∗(Xk ⊗ IM)ϕk(tX).

It follows from the latter equalities that

0 ≤N∑

[ϕk(tX)− ϕk(X)]∗(Xk ⊗ IM)[ϕk(tX)− ϕk(X)]

ϕk(tX)∗(Xk ⊗ IM)ϕk(tX)−N∑

ϕk(tX)∗(Xk ⊗ IM)ϕk(X)

−N∑

ϕk(X)∗(Xk ⊗ IM)ϕk(tX) +

ϕk(X)∗(Xk ⊗ IM)ϕk(X) = 0.

Thus ϕk(tX) − ϕk(X) = 0 for every X ∈ P(n1,...,nN ), t > 0 and k = 1, . . . , N .For fixed k ∈ 1, . . . , N and t > 0 the function hk,t(Z) := ϕk(tZ) − ϕk(Z) isholomorphic on Πn1×n1 × · · ·×ΠnN×nN and takes values in L(U , Cnk ⊗M). Thenfor any fixed k ∈ 1, . . . , N, t > 0, u ∈ U and m ∈ Cnk ⊗M the scalar functionhk,t,u,m(Z) := 〈hk,t(Z)u, m〉Cnk⊗M is holomorphic on Πn1×n1×· · ·×ΠnN×nN and

vanishes on P(n1,...,nN ). The latter set is the uniqueness subset in Πn1×n1 × · · · ×ΠnN×nN , thus by the uniqueness theorem for holomorphic functions of severalvariables (see, e.g., [12]), hk,t,u,m(Z) ≡ 0, hence hk,t(Z) ≡ 0, which means:

ϕk(tZ) = ϕk(Z), t > 0, Z ∈ Πn1×n1 × · · · ×ΠnN×nN .

It follows from the latter equality that for every Z, Λ ∈ Πn1×n1 × · · · × ΠnN×nN

and t > 0 one has

f(Z) + tf(Λ)∗ = f(Z) + f(tΛ)∗ =N∑

ϕk(tΛ)∗ ((Zk + tΛ∗k)⊗ IM)ϕk(Z)

ϕk(Λ)∗ ((Zk + tΛ∗k)⊗ IM) ϕk(Z)

ϕk(Λ)∗ (Zk ⊗ IM)ϕk(Z) + tϕk(Λ)∗ (Λ∗k ⊗ IM)ϕk(Z),

and the comparison of the coefficients of the two linear functions in t, at thebeginning and at the end of this chain of equalities, gives:

f(Z) =

ϕk(Λ)∗ (Zk ⊗ IM)ϕk(Z), Z, Λ ∈ Πn1×n1 × · · · ×ΠnN×nN ,

i.e., (2.3) with Mk = M, k = 1, . . . , N . By Theorem 2.2, f ∈ Bn1,...,nN (U). Theproof is complete.

Corollary 3.4. Let f be an L(U)-valued function holomorphic on Ωn1,...,nN . Thenf ∈ Bn1,...,nN (U) if and only if the following conditions are satisfied:

(i) f(λZ) = λf(Z), λ ∈ C \ 0, Z ∈ Ωn1,...,nN ;

(ii) f(R) + f(R)∗ ≥ 0, R ∈ A(n1,...,nN );(iii) f(Z∗) = f(Z)∗, Z ∈ Ωn1,...,nN .

Proof. If f ∈ Bn1,...,nN (U) then (i) and (iii) follow from the representation (2.1)–(2.2) of f , and (ii) follows from Theorem 3.3.

Conversely, statements (i)–(iii) of the corollary imply statements (i)–(iii) ofTheorem 3.3, which in turn imply that f ∈ Bn1,...,nN (U).

Remark 3.5. By Corollary 3.4, its conditions (i)–(iii) on holomorphic L(U)-valuedfunctions on Ωn1,...,nN give an equivalent definition of the class Bn1,...,nN (U), whichseems to be more natural than the original definition given above in “existence”terms.

4. The image of the class Bn1,...,nN(U) under

the double Cayley transform

It was shown in the proof of Theorem 3.3 that if f ∈ Bn1,...,nN (U) then the doubleCayley transform of f , F = C(f), defined by (3.3), belongs to the Agler–Schurclass ASn1,...,nN (U). In fact, we are able to proof a stronger statement.

Theorem 4.1. A holomorphic L(U)-valued function F on Dn1×n1 × · · · ×DnN×nN

can be represented as F = C(f) for some f ∈ Bn1,...,nN (U) if and only if thefollowing conditions are fulfilled:

(i) There exist a Hilbert space X and an operator

[A BC D

]∈ L((Cn1+···+nN ⊗X )⊕ U) (4.1)

such that for W = (W1, . . . , WN ) ∈ Dn1×n1 × · · · × DnN×nN one has

F(W ) = D + C(P (W ) ⊗ IX )(ICn1+···+nN ⊗X −A(P (W ) ⊗ IX ))−1B, (4.2)

where P (W ) = diag(W1, . . . ,WN) and U = U∗ = U−1.(ii) 1 /∈ σ(F(0)).

Proof. Necessity. Let f ∈ Bn1,...,nN (U). Then (3.1) and (3.2) hold. As we haveshown in Theorem 3.3, the identity (3.1) implies the identity (3.15) for F = C(f),with holomorphic L(U , Cnk⊗Mk)-valued functions θk, k = 1, . . . , N , on Dn1×n1×· · ·×DnN×nN defined by (3.14) and (3.16). Analogously, the identity (3.2) implies

F(W )−F(Ξ)∗ =

θk(Ξ)∗ ((Wk − Ξ∗k)⊗ IMk

) θk(W ), (4.3)

W, Ξ ∈ Dn1×n1 × · · · × DnN×nN .

Let us rewrite (3.15) and (4.3) in a somewhat different way. Since by Theorem 3.3f ∈ Bn1,...,nN (U) satisfies f(Z∗) = f(Z)∗, Z ∈ Πn1×n1 × · · · × ΠnN×nN , one hasalso

F(W ∗) = F(W )∗, W ∈ Dn1×n1 × · · · × DnN×nN .

Therefore, (3.15) and (4.3) are equivalent to the following two identities, respec-tively:

IU −F(W )F(Ξ)∗ =

θk(W ) ((Ink−WkΞ∗

k)⊗ IMk) θk(Ξ)∗, (4.4)

F(W )−F(Ξ)∗ =N∑

θk(W ) ((Wk − Ξ∗k)⊗ IMk

) θk(Ξ)∗, (4.5)

W, Ξ ∈ Dn1×n1 × · · · × DnN×nN ,

where θk(W ) = θk(W ∗)∗ are holomorphic L(Cnk ⊗Mk,U)-valued functions onDn1×n1 × · · · × DnN×nN . We will show that the identities (4.4) and (4.5) allowus to construct a Hilbert space X and an operator U satisfying condition (i) ofthis theorem. To this end, we will apply the construction from Section 3 (next toTheorem 3.1) to F = C(f). In this case E = E∗ = U . Without loss of generality we

may consider all ofMk’s equal. Say, set M :=⊕N

k=1Mk and regard

HRk = θk ∈ L(U , Cnk ⊗M), HL

k = θk ∈ L(Cnk ⊗M,U), k = 1, . . . , N.

Then (3.15), (4.3), (4.4) and (4.5) imply (3.7), and

HL(W ∗) = HR(W )∗,

where HL(W ) = col(HL1 (W), . . . ,HL

N(W)), HR = col(HR1 (W), . . . ,HR

N(W)), W ∈Dn1×n1×· · ·×DnN×nN . Thus, D0 = R0, and the operator U0 acts on the generatingvectors of D0 as follows:

[(P (W ) ⊗ IM)HR(W )

]u −→

[HR(W )F(W )

]u −→

[(P (W )⊗ IM)HR(W )

W ∈ Dn1×n1 × · · · × DnN×nN .

We used here the relations F(W ∗) = F(W )∗, HL(W ∗) = HR(W )∗, P (W ∗) =

diag(W∗1 , . . . ,W

∗N) = P(W)∗. Thus U0 = U−1

0 . Therefore, U0 = U0

−1. Since p =

q = n1 + · · ·+ nN , E = E∗ = U , (3.10) holds. Then the operator

U = U0 ⊕ I((Cn1+···+nN ⊗M)⊕U)⊖clos(D0) ∈ L((Cn1+···+nN ⊗M)⊕ U)

satisfies U = U−1. Since we have also U∗ = U−1, (i) is satisfied with X =M.

Statement (ii) follows in the same way as in [9, Theorem 4.2], with E =(In1 , . . . , InN ) in the place of e = (1, . . . , 1).

Sufficiency. Let the conditions (i) and (ii) on F be satisfied. Then in thesame way as in [9, Theorem 4.2] one can see that 1 /∈ σ(F(W )) for all W ∈Dn1×n1 × · · · × DnN×nN . Thus, the function

F (W ) := (IU + F(W ))(IU −F(W ))−1

is correctly defined and holomorphic on Dn1×n1 × · · · × DnN×nN . It is easy to seethat

F (W ) + F (Ξ)∗ = 2(IU −F(Ξ)∗)−1(IU −F(Ξ)∗F(W ))(IU −F(W ))−1, (4.6)

F (W )− F (Ξ)∗ = 2(IU −F(Ξ)∗)−1(F(W )−F(Ξ)∗)(IU −F(W ))−1. (4.7)

As shown in [3], it follows from (4.2) that

IU −F(Ξ)∗F(W ) = B∗(ICn1+···+nN ⊗X − (P (Ξ)∗ ⊗ IX )A∗)−1

× ((ICn1+···+nN − P (Ξ)∗P (W )) ⊗ IX )

× (ICn1+···+nN ⊗X −A(P (W ) ⊗ IX ))−1B.

Since U = U∗, we get

IU −F(Ξ)∗F(W ) =N∑

B∗(ICn1+···+nN ⊗X − (P (Ξ)∗ ⊗ IX )A)−1

× (PCnk ⊗ IX )((Ink− Ξ∗

kWk)⊗ IX )(PCnk ⊗ IX )

× (ICn1+···+nN ⊗X −A(P (W ) ⊗ IX ))−1B.

Analogously,

F(W )−F(Ξ)∗ =

B∗(ICn1+···+nN ⊗X − (P (Ξ)∗ ⊗ IX )A)−1

× (PCnk ⊗ IX )((Wk − Ξ∗k)⊗ IX )(PCnk ⊗ IX )

× (ICn1+···+nN ⊗X −A(P (W )⊗ IX ))−1B.

Thus, from (4.6) and (4.7) we get

F (W ) + F (Ξ)∗ =

ξk(Ξ)∗((Ink− Ξ∗

kWk)⊗ IX )ξk(W ), (4.8)

F (W )− F (Ξ)∗ =

ξk(Ξ)∗((Wk − Ξ∗k)⊗ IX )ξk(W ), (4.9)

ξk(W ) =√

2(PCnk⊗IX )(ICn1+···+nN ⊗X −A(P (W )⊗ IX ))−1B(IU −F(W ))−1,

for all W ∈ Dn1×n1 × · · · × DnN×nN and k = 1, . . . , N .

Since for Zk, Λk ∈ Πnk×nk we have

Ink− (Λ∗

k + Ink)−1(Λ∗

k − Ink)(Zk − Ink

)(Zk + Ink)−1

= 2(Λ∗k + Ink

)−1(Zk + Λ∗k)(Zk + Ink

)−1,

(Zk − Ink)(Zk + Ink

)−1 − (Λ∗k + Ink

)−1(Λ∗k − Ink

= 2(Λ∗k + Ink

)−1(Zk − Λ∗k)(Zk + Ink

)−1,

by setting Wk := (Zk − Ink)(Zk + Ink

)−1 and Ξk := (Λk − Ink)(Λk + Ink

)−1 in(4.8) and (4.9) we get the identities (3.1) and (3.2) for

f(Z) = F ((Z1 − In1)(Z1 + In1)−1, . . . , (ZN − InN )(ZN − InN )−1),

ϕk(Z) =√

2((Zk + Ink)−1 ⊗ IX )

× ξk((Z1 − In1)(Z1 + In1)−1, . . . , (ZN − InN )(ZN − InN )−1)

for k = 1, . . . , N . Thus, by Theorem 2.2 we finally get F = C(f) where f ∈Bn1,...,nN (U). The proof is complete.

5. The “real” case

In Section 1 we have mentioned the notions of an anti-unitary involution (AUI) ι =ιU on a Hilbert space U (a counterpart of the operator ιn of complex conjugationon Cn), a (ιU , ιY)-real operator A ∈ L(U ,Y) (a counterpart of matrix with realentries), and a (ιU , ιY)-real operator-valued function f (a counterpart of functionwhich takes real scalar or matrix values at real points). Some basic properties ofAUI were described in [9, Proposition 6.1]. We will need also the following property.

Proposition 5.1. Let ιU and ιH be AUIs on Hilbert spaces U and H, respectively.Then the operator ιU⊗H = ιU ⊗ ιH on U ⊗H which is defined on elementarytensors u⊗ h as

(ιU ⊗ ιH)(u⊗ h) = ιUu⊗ ιHh (5.1)

and then extended to all of U ⊗H by linearity and continuity, is defined correctlyand is an AUI on U ⊗H.

Proof. First, let us observe that ιU⊗H is correctly defined. To this end, note that

for arbitrary x′ =∑l

α=1 u′α ⊗ h′

α and x′′ =∑m

β=1 u′′β ⊗ h′′

β from U ⊗H we have

〈ιU⊗Hx′, ιU⊗Hx′′〉U⊗H =

⟨ιUu′

α ⊗ ιHh′α, ιUu′′

β ⊗ ιHh′′β

⟩U⊗H

⟨ιUu′

α, ιUu′′β

⟩U⟨ιHh′

α, ιHh′′β

⟩H =

⟨u′′

β , u′α

⟩U⟨h′′

β , h′α

⟨u′′

β ⊗ h′′β, u′

α ⊗ h′α

⟩U⊗H = 〈x′′, x′〉U⊗H ,

i.e., ιU⊗H is an anti-isometry on linear combinations of elementary tensors. Thus,it is uniquely extended to an operator on all of U ⊗H, and the property (1.9) of

the extended operator follows by continuity. Since for arbitrary x′ =∑l

α=1 u′α⊗h′

and x′′ =∑m

β=1 u′′β ⊗ h′′

β from U ⊗H we have

⟨ι2U⊗Hx′, x′′⟩

U⊗H =

⟨ι2Uu′

α ⊗ ι2Hh′α, u′′

β ⊗ h′′β

⟩U⊗H

⟨u′

α ⊗ h′α, u′′

β ⊗ h′′β

⟩U⊗H

= 〈x′, x′′〉U⊗H ,

by continuity the property (1.8) of ιU⊗H follows as well. Thus, ιU⊗H is an AUI onU ⊗H.

Let U be a Hilbert space, and let ι = ιU be an AUI on U . Denote byιRBn1,...,nN (U) the subclass of Bn1,...,nN (U) consisting of ι-real functions. Thefollowing theorem gives several equivalent characterizations of the “ι-real-valuedBessmertnyı class” ιRBn1,...,nN (U) which specify for this case the characterizationsobtained above for the “complex-valued Bessmertnyı class” Bn1,...,nN (U).

Theorem 5.2. Let f be a holomorphic L(U)-valued function on Ωn1,...,nN , andι = ιU be an AUI on a Hilbert space U . The following statements are equivalent:

(i) f ∈ ιRBn1,...,nN (U);

(ii) there exist a representation (2.1) of f and AUIs ιMkon Mk, k = 1, . . . , N ,

such that the operators Gk in (2.2) are (ιU ⊕ ιH, ιnk⊗ ιMk

)-real;

(iii) there exist a representation (2.3) of f and AUIs ιMkon Mk such that the

holomorphic functions ϕk(Z) are (ιU , ιnk⊗ ιMk

)-real, k = 1, . . . , N ;

(iv) there exist a Hilbert space X and an operator U as in (4.1) such that F = C(f)satisfies (4.2) and U = U∗ = U−1; moreover, there exists an AUI ιX on Xsuch that the operator U is ((ιn1+···+nN ⊗ ιX )⊕ ιU )-real.

Proof. (i)⇒ (iii) Let (i) hold. By Theorem 2.2 there exists a representation (2.3) off with holomorphic L(U , Cnk⊗Mk)-valued functions ϕk on Πn1×n1×· · ·×ΠnN×nN .Let ιMk

be an AUI onMk, and let ιnkbe a standard AUI on Cnk , i.e., a complex

conjugation. Set

Mk :=Mk ⊕Mk and ιMk:=

[0 ιMk

], k = 1, . . . , N.

Clearly, ιMkis an AUI on Mk.

Define the rearrangement isomorphisms Vk : Cnk ⊗ (Mk ⊕Mk) −→ (Cnk ⊗Mk)⊕ (Cnk ⊗Mk) by

⎡⎢⎢⎢⎢⎢⎣

...m1nk

⎤⎥⎥⎥⎥⎥⎦−→

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

...m1nk

...m2nk

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

ιnk⊗ ιMk

= V −1k

[0 ιnk

⊗ ιMk

ιnk⊗ ιMk

]Vk. (5.2)

ϕk(Z) :=1√2V −1

[ϕk(Z)

(ιnk⊗ ιMk

)ϕk(Z)ιU

where Z = (Z1, . . . , ZN ), and (Zk)ij = (Zk)ij , k = 1, . . . , N, i, j = 1, . . . , nk. Byproperties of AUIs, ϕk(Z) is holomorphic on Πn1×n1 × · · · × ΠnN×nN . Moreover,ϕk is (ιU , ιnk

⊗ ιMk)-real. Indeed, due to (5.2) we have

ϕk♯(Z) = (ιnk

⊗ ιMk)ϕk(Z)ιU

=1√2V −1

[0 ιnk

⊗ ιMk

ιnk⊗ ιMk

ϕk(Z)(ιnk⊗ ιMk

)ϕk(Z)ιU

=1√2V −1

[(ιnk⊗ ιMk

)2ϕk(Z)ι2U(ιnk⊗ ιMk

)ϕk(Z)ιU

=1√2V −1

[ϕk(Z)

(ιnk⊗ ιMk

)ϕk(Z)ιU

= ϕk(Z).

Let us show that

f(Z) =

ϕk(Λ)∗(Zk ⊗ IMk)ϕk(Z), Z, Λ ∈ Πn1×n1 × · · · ×ΠnN×nN .

To this end, let us show first that for k = 1, . . . , N :((ιnk⊗ ιMk

)ϕk(Λ)ιU)∗

= ιU ϕk(Λ)∗(ιnk⊗ ιMk

), Λ ∈ Πn1×n1 × · · · ×ΠnN×nN .

Indeed, for any m ∈ Cnk ⊗ Mk, u ∈ U , Λ ∈ Πn1×n1 × · · · ×ΠnN×nN one has⟨(

(ιnk⊗ ιMk

)ϕk(Λ)ιU)∗

m, u⟩U

=⟨m, (ιnk

⊗ ιMk)ϕk(Λ)ιUu

⟩Cnk⊗Mk⟨

ϕk(Λ)ιUu, (ιnk⊗ ιMk

Cnk⊗Mk

=⟨ιUu, ϕk(Λ)∗(ιnk

⊗ ιMk)m⟩U

=⟨ιU ϕk(Λ)∗(ιnk

⊗ ιMk)m, u

Now, for any Z, Λ ∈ Πn1×n1 × · · · ×ΠnN×nN :

ϕk(Λ)∗(Zk ⊗ IMk)ϕk(Z) =

[ϕk(Λ)

(ιnk⊗ ιMk

)ϕk(Λ)ιU

× Vk(Zk ⊗ IMk)V −1

[ϕk(Z)

(ιnk⊗ ιMk

)ϕk(Z)ιU

[ϕk(Λ)

(ιnk⊗ ιMk

)ϕk(Λ)ιU

]∗ [Zk ⊗ IMk

00 Zk ⊗ IMk

ϕk(Z)(ιnk⊗ ιMk

)ϕk(Z)ιU

ϕk(Λ)∗(Zk ⊗ IMk)ϕk(Z)

ιUϕk(Λ)∗(ιnk⊗ ιMk

)(Zk ⊗ IMk)(ιnk

⊗ ιMk)ϕk(Z)ιU

ϕk(Λ)∗(Zk ⊗ IMk)ϕk(Z) +

ιUϕk(Λ)∗(Zk ⊗ IMk)ϕk(Z)ιU

2(f(Z) + ιUf(Z)ιU ) = f(Z),

where we used (5.3), unitarity of Vk, and identity ιnkZkιnk

= Zk. Thus, (iii) followsfrom (i).

(iii)⇒ (ii) Let (iii) hold. As in the sufficiency part of the proof of Theorem 2.3we set

(Cnk ⊗Mk), ϕ(Z) := col(ϕ1(Z), . . . , ϕN (Z)) ∈ L(U ,N ),

Pk := PMk, E = (In1 , . . . , InN ) ∈ Πn1×n1 × · · · ×ΠnN×nN ,

H := clos spanΛ∈Πn1×n1×···×ΠnN×nN (ϕ(Λ)− ϕ(E))U ⊂ N ,

Gk := (Ink⊗ Pk)κ

[ϕ(E) 0

]∈ L(U ⊕H, Cnk ⊗Mk),

where κ : X ⊕H → N is defined by (2.9). For ψ(Z) =

ϕ(Z)− ϕ(E)

]one has

ϕ(E)U = U ⊕ 0, therefore the linear span of vectors of the form ψ(Z)u, Z ∈Πn1×n1 × · · · × ΠnN×nN , u ∈ U , is dense in U ⊕H. Set ιN :=

⊕Nk=1(ιnk

⊗ ιMk).

By the assumption, we have for k = 1, . . . , N :

(ιnk⊗ ιMk

)ϕk(Z) = ϕk(Z)ιU , Z ∈ Πn1×n1 × · · · ×ΠnN×nN .

Therefore,

ιN (ϕ(Z) − ϕ(E))u = (ϕ(Z)− ϕ(E))ιUu ∈ H, u ∈ U .

Thus ιNH ⊂ H. Moreover,H = ι2NH ⊂ ιNH, therefore ιNH = H. Set ιH := ιN |H.Clearly, ιH is an AUI on H, and

(ιU ⊕ ιH)ψ(Z) = ψ(Z)ιU , Z ∈ Πn1×n1 × · · · ×ΠnN×nN .

Let us verify that Gk is (ιU ⊕ ιH, ιnk⊗ ιMk

)-real, k = 1, . . . , N .

(ιnk⊗ ιMk

)Gkψ(Z)u

= (ιnk⊗ ιMk

)(Ink⊗ Pk)κ

[ϕ(E) 0

ϕ(Z)− ϕ(E)

= (ιnk⊗ ιMk

)(Ink⊗ Pk)κ

[ϕ(E)

ϕ(Z)− ϕ(E)

]u = (ιnk

⊗ ιMk)ϕk(Z)u

= ϕk(Z)ιUu = (Ink⊗ Pk)κ

[ϕ(E) 0

ϕ(Z)− ϕ(E)

= Gkψ(Z)ιUu = Gk(ιU ⊕ ιH)ψ(Z)u.

Since the linear span of vectors of the form ψ(Z)u, Z ∈ Πn1×n1×· · ·×ΠnN×nN , u ∈U , is dense in U ⊕H, the operator Gk is (ιU ⊕ ιH, ιnk

⊗ ιMk)-real, as desired.

(ii) ⇒ (i) Let f satisfy (ii). Then by Theorem 2.2 f ∈ Bn1,...,nN (U). Let usshow that the operator-valued linear function A(Z) from (2.2) is ιU ⊕ ιH-real.Since Gk is (ιU ⊕ ιH, ιnk

⊗ ιMk)-real, one has Gk(ιU ⊕ ιH) = (ιnk

⊗ ιMk)Gk and

(ιU ⊕ ιH)G∗k = G∗

k(ιnk⊗ ιMk

), k = 1, . . . , N . The latter equality follows from thefact that for every h ∈ Cnk ⊗Mk, x ∈ U ⊕H:

〈(ιU ⊕ ιH)G∗kh, x〉U⊕H = 〈(ιU ⊕ ιH)x, G∗

kh〉U⊕H= 〈Gk(ιU ⊕ ιH)x, h〉

Cnk⊗Mk= 〈(ιnk

⊗ ιMk)Gkx, h〉

Cnk⊗Mk

= 〈(ιnk⊗ ιMk

)h, Gkx〉Cnk⊗Mk

= 〈G∗k(ιnk

⊗ ιMk)h, x〉U⊕H .

Therefore,

(ιU ⊕ ιH)A(Z)(ιU ⊕ ιH) =N∑

(ιU ⊕ ιH)G∗k(Zk ⊗ IMk

)Gk(ιU ⊕ ιH)

G∗k(ιnk

⊗ ιMk)(Zk ⊗ IMk

)(ιnk⊗ ιMk

G∗k(ιnk

Zkιnk⊗ IMk

G∗k(Zk ⊗ IMk

)Gk = A(Z).

The latter is equivalent to the identities

ιUa(Z)ιU = a(Z), ιUb(Z)ιH = b(Z),

ιHc(Z)ιU = c(Z), ιHd(Z)ιH = d(Z).

Since ι2H = IH and

(ιHd(Z)−1ιH) · (ιHd(Z)ιH) = (ιHd(Z)ιH) · (ιHd(Z)−1ιH) = IH,

one hasιHd(Z)−1ιH = (ιHd(Z)ιH)−1 = d(Z)−1.

Therefore,

f ♯(Z) = ιUf(Z)ιU = ιU (a(Z)− b(Z)d(Z)−1c(Z))ιU

= ιUa(Z)ιU − (ιU b(Z)ιH) · (ιHd(Z)−1ιH) · (ιHc(Z)ιU )

= a(Z)− b(Z)d(Z)−1c(Z) = f(Z),

and f is ιU -real. Thus, (i) follows from (ii).(iv) ⇒ (i) Let (iv) hold. Then the operator U = U∗ = U−1 from (4.1) is

((ιn1+···+nN ⊗ ιX )⊕ ιU )-real, i.e.,[

ιn1+···+nN ⊗ ιX 00 ιU

A BC D

ιn1+···+nN ⊗ ιX 00 ιU

[A BC D

This is equivalent to the following identities:

(ιn1+···+nN ⊗ ιX )A(ιn1+···+nN ⊗ ιX ) = A, (ιn1+···+nN ⊗ ιX )BιU = B,

ιUC(ιn1+···+nN ⊗ ιX ) = C, ιUDιU = D.

Moreover, for W ∈ Dn1×n1 × · · · × DnN×nN one has

(ιn1+···+nN ⊗ ιX )(ICn1+···+nN ⊗X −A(P (W ) ⊗ IX ))(ιn1+···+nN ⊗ ιX )

= (ιn1+···+nN ⊗ ιX )2 − (ιn1+···+nN ⊗ ιX )A(ιn1+···+nN ⊗ ιX )

× (ιn1+···+nN ⊗ ιX )(P (W )⊗ IX )(ιn1+···+nN ⊗ ιX )

= ICn1+···+nN ⊗X − A(P (W )⊗ IX ).

Therefore,

(ιn1+···+nN ⊗ ιX )(ICn1+···+nN ⊗X −A(P (W ) ⊗ IX ))−1(ιn1+···+nN ⊗ ιX )

= (ICn1+···+nN ⊗X −A(P (W )⊗ IX ))−1

(we already used an analogous argument above). Thus,

F ♯(W ) = ιUF(W )ιU

= ιU[D + C(P (W )⊗ IX )(ICn1+···+nN ⊗X −A(P (W )⊗ IX ))−1B

= D + C(P (W )⊗ IX )(ICn1+···+nN ⊗X −A(P (W ) ⊗ IX ))−1B

= F(W ), W ∈ Dn1×n1 × · · · × DnN×nN ,

i.e., F is ιU -real. Applying the inverse double Cayley transform to F , one can seethat f is ιU -real on Πn1×n1 × · · · × ΠnN×nN , and hence on Ωn1,...,nN . Thus, (i)follows from (iv).

(iii) ⇒ (iv) Let f satisfy (2.3) with holomorphic (ιU , ιnk⊗ ιMk

)-realL(U , Cnk ⊗Mk)-valued functions ϕk on Πn1×n1 × · · · ×ΠnN×nN . As in the proofof Theorem 4.1, we get for F = C(f) consecutively: identities (3.15) and (4.3)with holomorphic L(U , Cnk ⊗Mk)-valued functions θk on Πn1×n1 ×· · ·×ΠnN×nN

given by (3.14), (3.16) which are, moreover, (ιU , ιnk⊗ ιMk

)-real; then identi-

ties (4.4) and (4.5) with holomorphic L(Cnk ⊗ Mk,U)-valued functions θk onΠn1×n1 × · · · ×ΠnN×nN which are, moreover, (ιnk

⊗ ιMk, ιU )-real. Without loss of

generality, we consider all ofMk’s equal, i.e., setM :=⊕N

k=1Mk and regard

HRk = θk ∈ L(U , Cnk ⊗M), HL

k = θk ∈ L(Cnk ⊗M,U), ιM :=

Then HRk is (ιU , ιnk

⊗ ιM)-real, and HLk is (ιnk

⊗ ιM, ιU )-real, k = 1, . . . , N .

Set X := M. Let us observe that the subspace D0 (and hence, clos(D0)) is((ιn1+···+nN ⊗ ιX )⊕ ιU )-invariant. Indeed, for

[(P (W )⊗ IX )HR(W )

[HR(W ′)F(W ′)

]u′ ∈ D0,

with some u, u′ ∈ U and W, W ′ ∈ Dn1×n1 × · · · × DnN×nN , we have

((ιn1+···+nN ⊗ ιX )⊕ ιU )x

[(ιn1+···+nN P (W )⊗ ιX )HR(W )

[(ιn1+···+nN ⊗ ιX )HR(W ′)

ιUF(W ′)

[(P (W )ιn1+···+nN ⊗ ιX )HR(W )

[HR(W ′)ιUF(W

′)ιU

[(P (W )⊗ IX )HR(W )

]ιUu +

[HR(W ′)

F(W′)

]ιUu ∈ D0.

Therefore, the subspace

D⊥0 := ((Cn1+···+nN ⊗X ) ⊕ U)⊖ clos(D0)

is also ((ιn1+···+nN ⊗ ιX )⊕ ιU )-invariant. Indeed, for any h1 ∈ D0, h2 ∈ D⊥0 :

〈((ιn1+···+nN ⊗ ιX )⊕ ιU )h2, h1〉 = 〈((ιn1+···+nN ⊗ ιX )⊕ ιU )h1, h2〉 = 0,

thus h2 ∈ D⊥0 implies ((ιn1+···+nN ⊗ ιX )⊕ ιU )h2 ∈ D⊥

Now, it is easy to check that U0 and therefore U0 are ((ιn1+···+nN ⊗ ιX )⊕ ιU )-

real. Since U = U0⊕ID⊥0, and D⊥

0 is ((ιn1+···+nN⊗ιX )⊕ιU )-invariant, the operator

U = U∗ = U−1 is ((ιn1+···+nN ⊗ ιX )⊕ ιU )-real, as required.

The proof is complete.

References

[1] J. Agler. On the representation of certain holomorphic functions defined on a poly-disc. In: Topics in operator theory: Ernst D. Hellinger memorial volume, volume 48of Oper. Theory Adv. Appl., pages 47–66. Birkhauser, Basel, 1990.

[2] C.-G. Ambrozie and D. Timotin. A von Neumann type inequality for certain domainsin Cn. Proc. Amer. Math. Soc., 131(3) (2003), 859–869 (electronic).

[3] J.A. Ball and V. Bolotnikov. Realization and interpolation for Schur-Agler-class func-tions on domains with matrix polynomial defining function in Cn. J. FunctionalAnalysis, to appear.

[4] M.F. Bessmertnyı. Functions of several complex variables in the theory of finite linearstructures. PhD thesis, Kharkov University, Kharkov, 1982. (Russian).

[5] M.F. Bessmertnyı. On realizations of rational matrix functions of several com-plex variables. In: Interpolation theory, systems theory and related topics (TelAviv/Rehovot, 1999), volume 134 of Oper. Theory Adv. Appl., pages 157–185.Birkhauser, Basel, 2002. Translated from the Russian by D. Alpay and V. Kat-snelson.

[6] M.F. Bessmertnyı. On realizations of rational matrix functions of several complexvariables. II. In: Reproducing kernel spaces and applications, volume 143 of Oper.Theory Adv. Appl., pages 135–146. Birkhauser, Basel, 2003. Translated from theRussian by V. Katsnelson.

[7] M.F. Bessmertnyı. On realizations of rational matrix functions of several complexvariables. III. In: Current trends in operator theory and its applications (BlacksburgVA, 2002), volume 149 of Oper. Theory Adv. Appl., pp. 133–138. Birkhauser, Basel,2004. Translated from the Russian by D.S. Kalyuzhnyı-Verbovetzkiı.

[8] R.E. Curto. Applications of several complex variables to multiparameter spectral the-ory. In: Surveys of some recent results in operator theory, Vol. II, volume 192 ofPitman Res. Notes Math. Ser., pages 25–90. Longman Sci. Tech., Harlow, 1988.

[9] D.S. Kalyuzhnyı-Verbovetzkiı. On the Bessmertnyı class of homogeneous positiveholomorphic functions of several variables. In: Current trends in operator theory andits applications (Blacksburg VA, 2002), volume 149 of Oper. Theory Adv. Appl., pp.255–289. Birkhauser, Basel, 2004.

[10] G. Kron. Tensor analysis of networks. Wiley, New York, 1939.

[11] M. Putinar. Uniqueness of Taylor’s functional calculus. Proc. Amer. Math. Soc.,89(4) (1983), 647–650.

[12] B.V. Shabat. Introduction to complex analysis. Part II, volume 110 of Translations ofMathematical Monographs. American Mathematical Society, Providence, RI, 1992.Functions of several variables, Translated from the third (1985) Russian edition byJ. S. Joel.

[13] Z. Slodkowski and W. Zelazko. On joint spectra of commuting families of operators.Studia Math., 50 (1974), 127–148.

[14] J.L. Taylor. The analytic-functional calculus for several commuting operators. ActaMath., 125 (1970), 1–38.

[15] J.L. Taylor. A joint spectrum for several commuting operators. J. Functional Anal-ysis, 6 (1970), 172–191.

[16] F.-H. Vasilescu. A characterization of the joint spectrum in Hilbert spaces. Rev.Roumaine Math. Pures Appl., 22(7) (1977), 1003–1009.

[17] F.-H. Vasilescu. A Martinelli type formula for the analytic functional calculus. Rev.Roumaine Math. Pures Appl., 23(10) (1978), 1587–1605.

Dmitry S. Kalyuzhnyı-VerbovetzkiıDepartment of MathematicsBen-Gurion University of the NegevP.O. Box 653Beer-Sheva 84105, Israele-mail: dmitryk@math.bgu.ac.il

Rational Solutions of the Schlesinger Systemand Isoprincipal Deformations ofRational Matrix Functions II

Victor Katsnelson and Dan Volok

Abstract. In this second article of the series we study holomorphic familiesof generic rational matrix functions parameterized by the pole and zero loci.In particular, the isoprincipal deformations of generic rational matrix func-tions are proved to be isosemiresidual. The corresponding rational solutionsof the Schlesinger system are constructed and the explicit expression for therelated tau function is given. The main tool is the theory of joint systemrepresentations for rational matrix functions with prescribed pole and zerostructures.

Mathematics Subject Classification (2000). Primary: 47A56;Secondary: 34M55.

Keywords. Isoprincipal, isosemiresidual, joint system representation, Fuchsiansystem, Schlesinger system.

Notation

• C stands for the complex plane.• C∗ stands for the punctured complex plane:

C∗ = C \ 0.• C stands for the extended complex plane (= the Riemann sphere):

C = C ∪∞.

• z stands for the complex variable.• Cn stands for the n-dimensional complex space.• In the coordinate notation, a point t ∈ Cn will be written as t = (t1, . . . , tn).

Victor Katsnelson thanks Ruth and Silvia Shoham for endowing the chair that supports hisrespective research. Victor Katsnelson was also supported by the Minerva foundation.

166 V. Katsnelson and D. Volok

• Cn∗ is the set of points t ∈ Cn, whose coordinates t1, . . . , tn are pairwise

different:

Cn∗ = Cn \

1≤i,j≤ni=j

t : ti = tj.

• Cm×n stands for the set of all m× n matrices with complex entries.• For A ∈ Cm×n, A∗ ∈ Cn×m is the adjoint matrix, Im(A) is the image subspace

of A in Cm (= the linear span of the columns of A) and Nul(A) is the nullsubspace of A in Cn.• [·, ·] denotes the commutator: for A, B ∈ Cm×m, [A, B] = AB −BA.• I stands for the identity matrix of an appropriate dimension.

10. Simple singularity of a meromorphic matrix function

(For Sections 1–9, see Operator Theory: Advances and Applications, Vol. 149,pp. 291–348.)

Definition 10.1. Let R(z) be a Cm×m-valued function, holomorphic in a puncturedneighborhood of a point t ∈ C. The point t is said to be a simple pole of the matrixfunction R if

R(z) =Rt

z − t+ H(z),

where Rt ∈ Cm×m is a constant matrix and the function H is holomorphic at thepoint t. The matrix Rt is said to be the residue of the function R at the pointt. Furthermore, if r = rank(Rt) and ft ∈ Cm×r and gt ∈ Cr×m are matricesproviding the factorization Rt = ftgt, we shall say that ft is the left semiresidueof R at t and gt is the right semiresidue of R at t.

Remark 10.2. The left and right semiresidues ft, gt are defined up to the transfor-mation

ft → ftc, gt → c−1gt,

where c ∈ Cr×r is an invertible matrix.

Definition 10.3. Let R(z) be a Cm×m-valued function, holomorphic and invertiblein a punctured neighborhood of a point t ∈ C.

1. The point t is said to be regular for the function R if both the function Rand the inverse function R−1 are holomorphic functions in the entire (non-punctured) neighborhood of the point t, i.e., if R and R−1 are holomorphicat the point t.

2. The point t is said to be singular for the function R if at least one of thefunctions R and R−1 is not holomorphic at the point t.

In particular, the point t is singular for the function R if R is holomorphic at thepoint t, but its value R(t) is a degenerate matrix. In this case, the point t is saidto be a zero of the function R.

Isoprincipal Deformations of Rational Matrix Functions II 167

Definition 10.4. Let R(z) be a Cm×m-valued function, holomorphic and invertiblein a punctured neighborhood of a point t ∈ C, and let t be a singular point of R.The singular point t is said to be simple if one of the following holds:

1. The point t is a simple pole of the function R and a holomorphy point of theinverse function R−1.

2. The point t is a simple pole of the inverse function R−1 and a holomorphypoint of the function R itself.

Remark 10.5. Note that, according to Definition 10.4, if t is a simple singularpoint of the function R then R is a single-valued meromorphic function in theentire (non-punctured) neighborhood of t.

Our main goal is to study a matrix function in a neighborhood of its sim-ple singular point from the point of view of linear differential systems. Thus weconsider the left logarithmic derivative of the function R:

QlR(z)

def= R′(z)R(z)−1.

Remark 10.6. One can also consider the right logarithmic derivative of R:

QrR(z) = R(z)−1R′(z).

But then −QrR is the left logarithmic derivative of the inverse function R−1:

QlR−1(z) = (R−1(z))′R(z) = −R(z)−1R′(z)R(z)−1R(z) = −Qr

Thus in this work we shall deal mainly with left logarithmic derivatives. Therefore,we shall use the notation QR instead of Ql

QR(z)def= R′(z)R(z)−1,

and omit the word “left” when referring to the left logarithmic derivative.

Proposition 10.7. Let R(z) be a Cm×m-valued function, holomorphic and invertiblein a punctured neighborhood of a point t ∈ C, and let t be a simple singular pointof R. Then the point t is a simple pole for the logarithmic derivative1 QR of R.Moreover, for the residue and the constant term of the Laurent expansion

QR(z) =Qt

z − t+ C + o(1) as z → t (10.1)

the following relations hold.

1. If t is a pole of R then

Q2t = −Qt, (10.2a)

QtCQt = −CQt (10.2b)

andIm(Qt) = Im(Rt), (10.3)

where Rt is the residue of R at t.

1See Remark 10.6.

2. If t is a zero of R then

Q2t = Qt, (10.4a)

QtCQt = QtC (10.4b)

Nul(Qt) = Nul(Rt), (10.5)

where Rt is the residue of R−1 at t.

Proof. First, let us assume that t is a pole of R and let

R(z) =Rt

z − t+ A0 + A1(z − t)) + A2(z − t)2 + · · · , (10.6)

R−1(z) = B0 + B1(z − t) + B2(z − t)2 + · · · , (10.7)

be the Laurent expansions of the functions R and R−1 at t. Then

R′(z) = − Rt

(z − t)2+ A1 + 2A2(z − t) + · · · (10.8)

Multiplying the Laurent expansions term by term, we obtain from (10.7) and (10.8)

QR(z) = − RtB0

(z − t)2− RtB1

z − t−RtB2 + A1B0 + o(1). (10.9)

Substituting the expansions (10.6), (10.7) into the identity

R−1(z)R(z) = R(z)R−1(z) = I,

we observe that

RtB0 = B0Rt = 0 and RtB1 + A0B0 = I. (10.10)

Hence the first term of the expansion (10.9) vanishes and we obtain the expansion(10.1) with

Qt = −RtB1 = A0B0 − I, (10.11)

C = −RtB2 + A1B0. (10.12)

(I + Qt)Qt = (A0B0)(−RtB1) = −A0(B0Rt)B1 = 0,

i.e., (10.2a) holds. Furthermore,

(I + Qt)CQt = (A0B0)(−RtB2 + A1B0)(−RtB1) =

= A0(B0Rt)B2RtB1 −A0B0A1(B0Rt)B1 = 0,

i.e., (10.2b) holds as well. Finally,

QtRt = (A0B0 − I)Rt = A0(B0Rt)−Rt = −Rt,

which, together with (10.11), implies (10.3). This completes the proof in the casewhen t is a pole of R. The case when t is a zero of R can be treated analogously.

Remark 10.8. Since for any p× q matrix A the subspace Nul(A) is the orthogonalcomplement of the subspace Im(A∗) in Cq, the relation (10.5) can be rewritten as

Im(Q∗t ) = Im(R∗

The latter relation, together with (10.4a), means that Q∗t is a (non-orthogonal, in

general) projector onto the subspace Im(R∗t ) ⊂ Cm. Hence the right semiresidue

of R−1 at its pole t is also the right semiresidue of QR at t.

Analogously, the relations (10.2a) and (10.3) mean that −Qt is a (non-orthogonal, in general) projector onto the subspace Im(Rt) ⊂ Cm. Hence theleft semiresidue of R at its pole t is also the left semiresidue of QR at t.

Proposition 10.7 implies that a Cm×m-valued function R(z) in a puncturedneighborhood of its simple singular point t may be viewed as a fundamental solu-tion of a linear differential system

R′(z) = Q(z)R(z), (10.13)

for which t is a Fuchsian singularity (see the first part of this work [KaVo] fordetails and references) and whose coefficients satisfy the relations (10.2) or (10.4).The next proposition shows that (10.2) or (10.4) are the only requirements adifferential system (10.13) with a Fuchsian singularity t has to satisfy in order forits fundamental solution in a punctured neighborhood of t to be single-valued andhave a simple singular point at t:

Proposition 10.9. Let Q(z) be a Cm×m-valued function, holomorphic and single-valued in a punctured neighborhood Ω of a point t. Let the point t be a simple polefor Q(z), let

Q(z) =Qt

z − t+ C + o(1) as z → t (10.14)

be the Laurent expansion of the function Q at the point t and let R be a fundamentalsolution of the linear differential system

R′(z) = Q(z)R(z), z ∈ Ω. (10.15)

Assume that one of the following two cases takes place.

1. The coefficients Qt, C of the expansion (10.14) satisfy the relations

Q2t = −Qt, (10.16a)

QtCQt = −CQt. (10.16b)

2. The coefficients Qt, C of the expansion (10.14) satisfy the relations

Q2t = Qt, (10.17a)

QtCQt = QtC. (10.17b)

Then R is a single-valued function in Ω and t is a simple singular point of R; inthe first case t is a pole of R, in the second case t is a zero of R.

Proof. Once again, we shall prove only the first statement. Thus we assume thatthe relations (10.2a), (10.2b) hold and consider the transformation

U(z) = (I + Qt + (z − t)Qt)R(z).

Then, because of (10.16a), the inverse transformation is given by

R(z) = (I + Qt + (z − t)−1Qt)U(z).

Substituting these formulae and the Laurent expansion of M into the linear system(10.15), we obtain the following linear system for U :

U ′(z) =

((I + Qt)CQt

z − t+ V (z)

)U(z),

where the function V (z) is holomorphic in the entire (non-punctured) neighbor-hood of the point t. In view of (10.16b), the coefficients of this system are holo-morphic at the point t, hence U is holomorphic and invertible in the entire neigh-borhood of t and R has a simple pole at t. Since

R−1(z) = U−1(z)(I + Qt + (z − t)Qt),

R−1 is holomorphic at t and hence has a zero at t.

An important role in the theory of Fuchsian differential systems is played bymultiplicative decompositions of fundamental solutions (see Section 5 of [KaVo]).In the present setting we are interested in decompositions of the following form:

Definition 10.10. Let R(z) be a Cm×m-valued function, holomorphic and invertiblein a punctured neighborhood Ω of a point t. Let R admit in Ω the factorization

R(z) = Ht(z)Et(ζ), ζ = z − t, z ∈ Ω, (10.18)

where the factors Ht(z) and Et(ζ) possess the following properties:

1. Ht(z) is a Cm×m-valued function, holomorphic and invertible in the entireneighborhood Ω ∪ t ;

2. Et(ζ) is a Cm×m-valued function, holomorphic and invertible in the punc-tured plane C∗ = C \ 0.

Then the functions Et(ζ) and Ht(z) are said to be, respectively, the principal andregular factors of R at t.

Remark 10.11. The multiplicative decomposition (10.18), which appears in Defi-nition 10.10, is always possible. This follows, for example, from the results due toG.D.Birkhoff (see [Birk1]). The principal factor Et(ζ) is, in a sense, the multiplica-tive counterpart of the principal part of the additive (Laurent) decomposition: itcontains the information about the nature of the singularity t of R. Of course,the principal and regular factors at the point t are determined only up to thetransformation

Et(ζ)→M(ζ)Et(ζ), Ht(z)→ Ht(z)M−1(z − t), (10.19)

where M(z) is an invertible entire Cm×m-valued function. However, once the choiceof the principal factor Et is fixed, the regular factor Ht is uniquely determined andvice-versa.

A possible choice of the principal factor of the function R at its simple singularpoint t is described in the following

Lemma 10.12. Let R(z) be a Cm×m-valued function, holomorphic and invertiblein a punctured neighborhood of a point t and let t be a simple singular point of R.Then a principal factor Et(ζ) of R at t can be chosen as follows.

1. If t is a pole of R, choose any matrix L ∈ Cm×m, satisfying the conditions

L2 = −L, Nul(L) = Nul(Rt), (10.20)

where Rt is the residue of R at t, and set for ζ ∈ C∗

Et(ζ) = I + L− ζ−1L. (10.21)

2. If t is a zero of R, choose any matrix L ∈ Cm×m, satisfying the conditions

L2 = L, Im(L) = Im(Rt), (10.22)

where Rt is the residue of R−1 at t, and set for ζ ∈ C∗

Et(ζ) = I − L + ζL. (10.23)

Proof. Let us assume that t is a pole of R and that the function Et is givenby (10.21), where the matrix L satisfies the conditions (10.20). Then Et(ζ) isholomorphic in C∗; its inverse E−1

t (ζ) is given by

E−1t (ζ) = I + L− ζL

and is holomorphic in C∗, as well. Let us now show that the function

H(z)def= R(z)E−1

t (z − t)

is holomorphic and invertible at t.Indeed, in a neighborhood of t the principal part of the Laurent expansion

of H equals toRt(I + L)

z − t. But by (10.20) Im(L∗) = Im(R∗

t ) and hence

Im((I + L∗)R∗t ) = Im((I + L∗)L∗) = Im((L2 + L)∗) = 0.

Therefore, Rt(I + L) = 0 and H is holomorphic at t.In the same way, the principal part of the Laurent expansion of H−1 equals

to − LB0

z − t, where B0 = R−1(t) is the constant term of the Laurent expansion of

R−1 at t. But RtB0 = 0 (see (10.10) in the proof of Proposition 10.7), hence

Im(B∗0L∗) = Im(B∗

0R∗t ) = 0,

LB0 = 0 and H−1 is holomorphic at t, as well.The proof in the case when t is a zero of R is completely analogous.

Remark 10.13. Let us note that the formulae (10.21) and (10.23) can be rewrittenin the unified form

Et(ζ) = ζL(= eL log ζ).

This is precisely the form of the principal factor (with Q = 0) which appears inProposition 5.6 of [KaVo].

Remark 10.14. The relations (10.20) mean that −L∗ is a projector onto Im(R∗t ).

This is equivalent to L being of the form L = pgt, where gt is the right semiresidueof the function R at its pole t and p ∈ Cm×rank(Rt) is such that gtp = −I. Anal-ogously, the relations (10.22) mean that L is a projector onto Im(Rt). This isequivalent to L being of the form L = ftq, where ft is the left semiresidue of thefunction R−1 at its pole t and q ∈ Crank(Rt)×m is such that qft = I. For example,one can choose the matrix L mentioned in Lemma 10.12 as follows:

−g∗t (gtg

∗t )−1gt if t is a pole of R,

ft(f∗t ft)

−1f∗t if t is a zero of R.

11. Rational matrix functions of simple structure

In this section we apply the local results obtained in Section 10 to the study ofrational matrix functions.

Definition 11.1. A Cm×m-valued rational function R(z) is said to be a rationalmatrix function of simple structure if it meets the following conditions:

1. detR(z) ≡ 0;2. all singular points of R are simple;3. z =∞ is a regular point of R.

The set of all poles of the function R is said to be the pole set of the function Rand is denoted by PR. The set of all zeros of the function R is said to be the zeroset of the function R and is denoted by ZR.

Remark 11.2. Note that if R is a rational matrix function of simple structure thenthe inverse function R−1 is a rational matrix function of simple structure, as well,and ZR = PR−1 .

Below we formulate the “global” counterparts of Propositions 10.7 and 10.9in order to characterize Fuchsian differential systems whose fundamental solutionsare rational matrix functions of simple structure.

Theorem 11.3. Let R(z) be a rational matrix function of simple structure withthe pole set PR and the zero set ZR. Then its logarithmic derivative2 QR(z) is a

2See Remark 10.6.

rational function with the set of poles PR ∪ ZR; all the poles of QR are simple.Furthermore, the function QR admits the additive decomposition

QR(z) =∑

t∈PR∪ZR

z − t, (11.1)

and its residues Qt ∈ Cm×m satisfy the following relations:∑

t∈PR∪ZR

Qt = 0, (11.2)

−Qt if t ∈ PR,

Qt if t ∈ ZR,(11.3)

QtCtQt =

−CtQt if t ∈ PR,

QtCt if t ∈ ZR,(11.4)

Ct =∑

t′∈PR∪ZR

t′ =t

t− t′. (11.5)

Proof. Since both functions R and R−1 are holomorphic in C \ (PR ∪ ZR), thelogarithmic derivative QR is holomorphic there, as well. According to Proposition10.7, each point of the set PR ∪ ZR is a simple pole of QR, hence we can writefor QR the additive decomposition

QR(z) = QR(∞) +∑

t∈PR∪ZR

z − t,

where Qt are the residues of QR. Since R is holomorphic at ∞, the entries of itsderivative R′ decay as o(|z|−1) when z →∞. The rate of decay for the logarithmicderivative QR is the same, because R−1, too, is holomorphic at∞. Thus we obtainthe additive decomposition (11.1) for QR and the relation (11.2) for the residuesQt. Now the relations (11.3), (11.4) follow immediately from Proposition 10.7,once we observe that the matrix Ct given by (11.5) is but the constant term of theLaurent expansion of QR at its pole t.

Theorem 11.4. Let P and Z be two finite disjoint subsets of the complex plane Cand let Q(z) be a Cm×m-valued rational function of the form

Q(z) =∑

t∈P∪Z

z − t, (11.6)

where Qt ∈ Cm×m. Let the matrices Qt satisfy the relations∑

t∈P∪ZQt = 0, (11.7)

−Qt if t ∈ P ,

Qt if t ∈ Z,(11.8)

QtCtQt =

−CtQt if t ∈ P ,

QtCt if t ∈ Z,(11.9)

Ct =∑

t′∈P∪Zt′ =t

t− t′. (11.10)

Let R(z) be a fundamental solution of the Fuchsian differential system

R′(z) = Q(z)R(z). (11.11)

Then R is a rational matrix function of simple structure such that

PR = P , ZR = Z.

Proof. Since the condition (11.7) implies that the point ∞ is a regular point forthe Fuchsian system (11.11), we may, without loss of generality, consider the fun-damental solution R satisfying the initial condition R(∞) = I. Then R(z) is amatrix function, holomorphic (a priori, multi-valued) and invertible in the (multi-connected) set C \ (P ∪ N ). However, for t ∈ P ∪ N the function Q admits in aneighborhood of t the Laurent expansion

Q(z) =Qt

z − t+ Ct + o(1)

with the constant term Ct given by (11.10). The coefficients Qt and Ct satisfy therelations (11.8), (11.9), hence by Proposition 10.9 the function R is meromorphicat t. Since this is true for every t ∈ P∪N , the function R is rational (in particular,single-valued). Proposition 10.9 also implies that every t ∈ P (respectively, t ∈ Z)is a simple pole (respectively, a zero) of the function R and a zero (respectively, asimple pole) of the inverse function R−1. Therefore, R is a rational matrix functionof simple structure with the pole set P and the zero set Z.

We close this section with the following useful

Lemma 11.5. Let R be a rational matrix function of simple structure. For t ∈PR∪ZR let Rt denote the residue of the function R at t if t ∈ PR, and the residueof the inverse function R−1 at t if t ∈ ZR. Then

t∈PR

rank(Rt) =∑

t∈ZR

rank(Rt). (11.12)

Proof. Let us consider the logarithmic derivative QR of R. Its residues Qt satisfyby Theorem 11.3 the relations (11.2) and (11.3). From (11.3) it follows that

rank(Qt) =

− trace(Qt) if t ∈ PR,

trace(Qt) if t ∈ ZR.

But (11.2) implies∑

t∈PR

trace(Qt) +∑

t∈ZR

trace(Qt) = 0,

hence ∑

t∈PR

rank(Qt) =∑

t∈ZR

rank(Qt).

Finally, by Proposition 10.7 (see (10.3), (10.5) there),

rank(Rt) = rank(Qt), ∀t ∈ PR ∪ ZR.

Thus (11.12) holds.

12. Generic rational matrix functions

Definition 12.1. A Cm×m-valued rational function R(z) is said to be a generic3

rational matrix function if R is a rational matrix function of simple structure andall the residues of the functions R and R−1 have rank one.

Lemma 12.2. Let R be a generic rational matrix function. Then the cardinalitiesof its pole and zero sets4 coincide:

#PR = #ZR. (12.1)

Proof. Since all the residues of R and R−1 are of rank one, the statement followsimmediately from Lemma 11.5.

Let R be a Cm×m-valued generic rational function. In what follows, we assumethat R is normalized by

R(∞) = I. (12.2)

Let us order somehow the pole and zero sets of R:

PR = t1, . . . , tn, ZR = tn+1, . . . , t2n, (12.3)

where n = #PR = #ZR. Then we can write for R and R−1 the additive decom-positions

R(z) = I +

z − tk, (12.4a)

R−1(z) = I +

z − tk, (12.4b)

3In [Kats2] such functions are called “rational matrix functions in general position”.4See Definition 11.1.

where for 1 ≤ k ≤ n (respectively, n + 1 ≤ k ≤ 2n) we denote by Rk the residueof R (respectively, R−1) at its pole tk. Since each matrix Rk is of rank one, therepresentations (12.4) can be rewritten as

R(z) = I +

z − tkgk, (12.5a)

R−1(z) = I +

z − tkgk, (12.5b)

where for 1 ≤ k ≤ n (respectively, n + 1 ≤ k ≤ 2n) fk ∈ Cm×1 and gk ∈ C1×m arethe left and right semiresidues 5 of R (respectively, R−1) at tk. Furthermore, weintroduce two n× n diagonal matrices:

AP = diag(t1, . . . , tn), AZ = diag(tn+1, . . . , t2n), (12.6)

two m× n matrices :

FP =(f1 . . . fn

), FZ =

(fn+1 . . . f2n

), (12.7)

and two n×m matrices :

⎛⎜⎝

⎞⎟⎠ , GZ =

⎛⎜⎝

...g2n

⎞⎟⎠ . (12.8)

The matrices AP and AZ are said to be, respectively, the pole and zero matricesof R. The matrices FP and GP are said to be, respectively, the left and right polesemiresidual matrices of R. Analogously, the matrices FZ and GZ are said to bethe left and right zero semiresidual matrices of R.

Remark 12.3. It should be mentioned that for a fixed ordering (12.3) of the poleand zero sets the pole and the zero matrices AP and AN are defined uniquely, andthe semiresidual matrices FP , GP , FZ , GZ are defined essentially uniquely, up tothe transformation

FP → FPDP , GP → D−1P GP , (12.9a)

FZ → FZDZ , GZ → D−1Z GZ , (12.9b)

where DP , DZ ∈ Cn×n are arbitrary invertible diagonal matrices. Once the choiceof the left pole semiresidual matrix FP is fixed, the right pole semiresidual matrixGP is determined uniquely, etc.

In terms of the matrices AP , AZ , FP , GZ , FZ , GZ , the representations (12.5)take the following form:

R(z) = I + FP (zI −AP)−1

GP , (12.10a)

R−1(z) = I + FZ (zI −AZ)−1

GZ . (12.10b)

5See Definition 10.1.

The representations (12.10) are not quite satisfactory for the following rea-sons. Firstly, in view of the identity RR−1 = R−1R = I, the information containedin the pair of representations (12.10) is redundant: each of these representationsdetermines the function R (and R−1) uniquely. Secondly, if, for example, the di-agonal matrix AP and the matrices FP , GP of appropriate dimensions are chosenarbitrarily then the rational function (12.10a) need not be generic. In our investi-gation we shall mainly use another version of the system representation of rationalmatrix functions, more suitable for application to linear differential equations. Thisis the so-called joint system representation (see [Kats2] for details and references)of the function R(z)R−1(ω) of two independent variables z and w. A key role inthe theory of the joint system representation is played by the Lyapunov equations.These are matricial equations of the form

UX −XV = Y, (12.11)

where the matrices U, V, Y ∈ Cn×n are given, and the matrix X ∈ Cn×n is un-known. If the spectra of the matrices U and V are disjoint, then the Lyapunovequation (12.11) is uniquely solvable with respect to X for arbitrary right-handside Y . The solution X can be expressed, for example, as the contour integral

(zI − U)−1Y (zI − V )−1dz, (12.12)

where Γ is an arbitrary contour, such that the spectrum of U is inside Γ and thespectrum of V is outside Γ (see, for instance, Chapter I, Section 3 of the book[DaKr]).

With the generic rational function R we associate the following pair of Lya-punov equations (with unknown SZP , SPZ ∈ Cn×n):

AZSZP − SZPAP = GZFP , (12.13a)

APSPZ − SPZAZ = GPFZ . (12.13b)

Since the spectra of the pole and zero matrices AP and AZ do not intersect (theseare the pole and zero sets of R), the Lyapunov equations (12.13a) and (12.13b)are uniquely solvable. In fact, since the matrices AP and AZ are diagonal, thesolutions can be given explicitly (using the notation (12.6)–(12.8)):

(gn+ifj

tn+i − tj

, SPZ =

(gifn+j

ti − tn+j

. (12.14)

The matrices SZP and SPZ are said to be, respectively, the zero-pole and pole-zerocoupling matrices of R.

Proposition 12.4. Let R(z) be a generic rational matrix function normalized byR(∞) = I. Then

1. the coupling matrices SZP and SPZ of R are mutually inverse:

SZPSPZ = SPZSZP = I; (12.15)

2. for the semiresidual matrices of R the following relations hold:

GZ = −SZPGP , FP = FZSZP ; (12.16)

3. the function R admits the joint representation

R(z)R−1(ω) = I + (z − ω)FP(zI −AP)−1S−1ZP(ωI −AZ)−1GZ , (12.17)

where AP , AZ are the pole and zero matrices of R.

Proof. The proof of Proposition 12.4 can be found in [Kats2].

Remark 12.5. Note that, since R(∞) = I, one can recover from the joint repre-sentation (12.17) when z →∞ or ω →∞ the separate representations

R(z) = I − FP (zI −AP)−1S−1ZPGZ , (12.18a)

R−1(ω) = I + FPS−1ZP(ωI −AZ)−1GZ , (12.18b)

which, in view of (12.16), coincide with the representations (12.10).

Remark 12.6. In view of (12.15), (12.16), one can also write the joint representa-tion for R in terms of the matrices FZ , GP and the solution SPZ of the Lyapunovequation (12.13b):

R(z)R−1(ω) = I − (z − ω)FZS−1PZ(zI −AP)−1SPZ(ωI −AZ)−1S−1

PZGP . (12.19)

Thus we may conclude that the pole and zero sets together with a pair of thesemiresidual matrices (either right pole and left zero or left pole and right zero)determine the normalized generic rational function R uniquely.

Remark 12.7. The theory of system representations for rational matrix functionswith prescribed zero and pole structures first appeared in [GKLR], and was furtherdeveloped in [BGR1], [BGR2], and [BGRa].

The joint representations (12.17), (12.19) suggest that this theory can beapplied to the investigation of families of rational functions parameterized by thezeros’ and poles’ loci and the corresponding6 deformations of linear differentialsystems. The version of this theory adapted for such applications was presentedin [Kats1] and [Kats2]. Also in [Kats2] one can find some historical remarks and alist of references.

Proposition 12.8. Let R(z) be a generic rational matrix function normalized byR(∞) = I. Then its logarithmic derivative7 admits the representation:

R′(z)R−1(z) = FP(zI −AP )−1S−1ZP(zI −AZ)−1GZ (12.20)

where AP and AZ are the pole and zero matrices of R; FP and GZ are the left poleand right zero semiresidual matrices of R; SZP is the zero-pole coupling matrixof R.

6See Propositions 11.3, 11.4.7See Remark 10.6.

Proof. Differentiating (12.17) with respect to z, we obtain

R′(z)R−1(ω) = FP(I − (z − ω)(zI −AP )−1

)(zI −AP)−1S−1

ZP(ωI −AZ)−1GZ .

Now set ω = z to obtain (12.20).

Remark 12.9. The representation (12.20) for the logarithmic derivative QR of thenormalized generic rational matrix function R can also be rewritten in terms ofthe matrices FZ , GP and the solution SPZ of the Lyapunov equation (12.13b) (seeRemark 12.6):

R′(z)R−1(z) = −FZS−1PZ(zI −AP)−1SPZ(zI −AZ)−1S−1

PZGP . (12.21)

13. Generic rational matrix functions with prescribed local data

In the previous section we discussed the question, how to represent a genericrational function R in terms of its local data (the pole and zero sets and theresidues). The main goal of this section is to construct a (normalized) genericrational matrix R(z) function with prescribed local data. In view of Proposition12.4, Remark 12.6 and Remark 12.3, such data should be given in the form of twodiagonal matrices of the same dimension (the pole and zero matrices) and twosemiresidual matrices of appropriate dimensions (either right pole and left zero orright zero and left pole)8.

Thus we consider the following

Problem 13.1. Let two diagonal matrices

AP = diag(t1, . . . , tn), AZ = diag(tn+1, . . . , t2n), ti = tj unless i = j,

and two matrices: F ∈ Cm×n, G ∈ Cn×m, be given.ZP-version: Find a generic Cm×m-valued rational function R(z) such that

1. R(∞) = I;2. the matrices AP and AZ are, respectively, the pole and zero matrices of R;3. the matrix F is the left pole semiresidual matrix FP of R: F = FP ;4. the matrix G is the right zero semiresidual matrix GZ of R: G = GZ .

PZ-version: Find a generic Cm×m-valued rational function R(z) such that

1. R(∞) = I;2. the matrices AP and AZ are, respectively, the pole and zero matrices of R;3. the matrix F is the left zero semiresidual matrix FZ of R: F = FZ ;4. the matrix G is the right pole semiresidual matrix GP of R: G = GP .

Proposition 13.2.

1. The ZP-version of Problem 13.1 is solvable if and only if the solution S ofthe Lyapunov equation

AZS − SAP = GF (13.1)

is an invertible matrix.

8Here we use the terminology introduced in Section 12.

2. The PZ-version of Problem 13.1 is solvable if and only if the solution S ofthe Lyapunov equation

APS − SAZ = GF (13.2)

is an invertible matrix.

Proof. The proof of Proposition 13.2 can be found in [Kats2]. Here we would liketo note that the solutions of the Lyapunov equations (13.1) and (13.2) can bewritten explicitly as, respectively,

tn+i − tj

and S =

ti − tn+j

, (13.3)

where gi is the ith row of the matrix G and fj is the jth column of the matrixF . Note also that the necessity of S being invertible in both cases follows fromProposition 12.4.

In view of Proposition 13.2, we propose the following terminology:

Definition 13.3. Let AP , AZ , F, G be the given data of Problem 13.1. Then:

1. the solution S of the Lyapunov equation

AZS − SAP = GF (13.4)

is said to be the ZP-coupling matrix related to the data AP , AZ , F, G;2. the solution S of the Lyapunov equation

APS − SAZ = GF (13.5)

is said to be the PZ-coupling matrix related to the data AP , AZ , F, G;3. the data AP , AZ , F, G are said to be ZP-admissible if the ZP-coupling matrix

related to this data is invertible;4. the data AP , AZ , F, G are said to be PZ-admissible if the PZ-coupling matrix

related to this data is invertible.

Proposition 13.4. Let AP , AZ , F, G be the given data of Problem 13.1.

1. If the data AP , AZ , F, G are ZP-admissible then the ZP-version of Problem13.1 has the unique solution R(z) given by

R(z) = I − F (I −AP)−1S−1G, (13.6)

where S is the ZP-coupling matrix related to the data AP , AZ , F, G. Thelogarithmic derivative of R is given by

R′(z)R−1(z) = F (zI −AP )−1S−1(zI −AZ)−1G. (13.7)

2. If the data AP , AZ , F, G are PZ-admissible then the PZ-version of Problem13.1 has the unique solution R(z) given by

R(z) = I + FS−1(zI −AP)−1G, (13.8)

where S is the PZ-coupling matrix related to the data AP , AZ , F, G. Thelogarithmic derivative of R is given by

R′(z)R−1(z) = −FS−1(zI −AP)−1S(zI −AZ)−1S−1G. (13.9)

Proof. If the data AP , AZ , F, G are ZP-admissible then, according to Proposition13.2, the ZP-version of Problem 13.1 has a solution R. Then the ZP-couplingmatrix S related to the data AP , AZ , F, G is the zero-pole coupling matrix of R.According to Proposition 12.4, the function R admits the representation (13.6) andhence9 is determined uniquely. Now the representation (13.7) for the logarithmicderivative follows from Proposition 12.8.

Analogous considerations hold also in the case when the data AP , AZ , F, Gare PZ-admissible (see Remarks 12.6, 12.9).

It was already mentioned (see Remark 12.3) that in the definition of thesemiresidual matrices there is a certain freedom. Accordingly, certain equivalencyclasses rather than individual matrices F, G should serve as data for Problem 13.1.The appropriate definitions are similar to the definition of the complex projectivespace Pk−1 as the space of equivalency classes of the set Ck \ 0 (two vectorsh′, h′′ ∈ Ck \ 0 are declared to be equivalent if h′, h′′ are proportional, i.e.,h′′ = dh′ for some d ∈ C∗).

Definition 13.5. 1. Let Cm×n∗,c denote the set of m × n matrices which have no

zero columns. Two matrices F ′, F ′′ ∈ Cm×n∗,c are declared to be equivalent:

F ′ c∼ F ′′, if

F ′′ = F ′Dc, (13.10)

where Dc is a diagonal invertible matrix.

The space P(m−1)×nc is a factor-set of the set Cm×n

∗,c modulo the equivalency

relationc∼.

2. Let Cn×m∗,r denote the set of n ×m matrices which have no zero rows. Two

matrices G′, G′′ ∈ Cn×m∗,r are declared to be equivalent: G′ r∼ G′′, if

G′′ = DrG′, (13.11)

where Dr is a diagonal invertible matrix.

The space Pn×(m−1)r is a factor-set of the set Cn×m

∗,r modulo the equivalency

relationr∼.

The factor spaces P(m−1)×nc and P

n×(m−1)r inherit topology from the spaces Cm×n

and Cn×m∗,r , respectively. They can be provided naturally with the structure of

complex manifolds.

If F ′ and F ′′ are twoc∼ - equivalent m× n matrices, and G′ and G′′ are two

r∼ - equivalent n×m matrices, then the solutions S′, S′′ of the Lyapunov equation

9See Remark 12.5.

(13.1) with F ′, G′ F ′′, G′′, substituted instead of F, G, and the same AP , AZ arerelated by

S′′ = DrS′Dc, (13.12)

where Dc, Dr are the invertible diagonal matrices, which appear in (13.10), (13.11).Similar result holds also for the Lyapunov equations (13.2).

However, since diagonal matrices commute, the expressions on the right-hand side of (13.6) will not be changed if we replace the matrices F, G, S with thematrices FDc, DrG, DrSDc, respectively.

Thus, the following result holds:

Proposition 13.6. Given AP and AZ , solution of Problem 13.1 depends not on the

matrices F, G themselves but on their equivalency classes in P(m−1)×nc , P

n×(m−1)r .

Remark 13.7. In view of Remark 12.3, if R is a generic rational matrix functionthen its left and right pole semiresidual matrix FP and GP can be considered

separately as elements of the sets P(m−1)×nc and P

n×(m−1)r , respectively. However,

simultaneously the matrices FP and GP can not be considered so. The same holdsfor the pair of the zero semiresidual matrices, as well.

14. Holomorphic families of generic rational matrix functions

Definition 14.1. Let D be a domain 10 in the space C2n∗ and for every t =

(t1, . . . , t2n) ∈ D let R(z, t) be a generic Cm×m-valued rational function of zwith the pole and zero matrices

AP(t) = diag(t1, . . . , tn), AZ(t) = diag(tn+1, . . . , t2n). (14.1)

Assume that for every t0 ∈ D and for every fixed z ∈ C\t01, . . . , t02n the functionR(z, t) is holomorphic with respect to t in a neighborhood of t0. Assume also that

R(∞, t) ≡ I. (14.2)

Then the family R(z, t)t∈D is said to be a normalized holomorphic family ofgeneric rational functions parameterized by the pole and zero loci.

Given a normalized holomorphic family R(z, t)t∈D of generic rational func-tions parameterized by the pole and zero loci, we can write for each fixed t ∈ D thefollowing representations for the functions R(z, t), R−1(z, t) and the logarithmicderivative

QR(z, t)def=

∂R(z, t)

∂zR−1(z, t) (14.3)

10One can also consider a Riemann domain over C2n∗

(see Definition 5.4.4 in [Hor]).

(see (11.1), (12.4)):

R(z, t) = I +

z − tk, (14.4a)

R−1(z, t) = I +

z − tk, (14.4b)

QR(z, t) =2n∑

z − tk. (14.4c)

The residues Rk(t), Qk(t), considered as functions of t, are defined in the wholedomain D. It is not hard to see that these functions are holomorphic in D:

Lemma 14.2. Let D be a domain in C2n∗ and let R(z, t)t∈D be a normalized

holomorphic family of generic rational functions, parameterized by the pole andzero loci. For each fixed t ∈ D and 1 ≤ k ≤ n (respectively, n + 1 ≤ k ≤ 2n) letRk(t) be the residue of the rational function R(·, t) (respectively, R−1(·, t)) at itspole tk. Likewise, for each fixed t ∈ D and 1 ≤ k ≤ 2n let Qk(t) be the residue ofthe logarithmic derivative QR(·, t) at its pole tk. Then Rk(t), Qk(t) considered asfunctions of t are holomorphic in D.

Proof. Let us choose an arbitrary t0 ∈ D and n pairwise distinct points z1, . . . , zn

in C \ t01, . . . , t0n. From the expansion (14.4a) we derive the following system oflinear equations with respect to the residue matrices Rk(t):

zℓ − tk= R(zℓ, t)− I, ℓ = 1, . . . , n. (14.5)

The matrices R(zℓ, t) − I on the right-hand side of the system (14.5) are holo-morphic with respect to t in a neighborhood of t0. The determinant of this linearsystem

∆(t) = det

zℓ − tk

1≤ℓ,k≤n

is holomorphic in a neighborhood of t0, as well. In fact, the determinant ∆(t)(known as the Cauchy determinant) can be calculated explicitly (see, for example,[[PS], part VII, Section 1, No. 3]):

∆(t) = ±

∏1≤p<q≤n

(zp − zq)(tp − tq)

n∏ℓ,k=1

(zℓ − tk).

In particular, ∆(t0) = 0. Hence, for k = 1, . . . , n, the functions Rk(t) are holomor-phic in a neighborhood of t0. Since this is true for any t0 ∈ D, these functions areholomorphic in the whole domain D. The proof for Rk(t), when n + 1 ≤ k ≤ 2n,and for Qk(t) is completely analogous.

Remark 14.3. Note that, on the one hand, the functions R(z, t), R−1(z, t) arerational with respect to z, and hence are holomorphic with respect to z inC \ t1, . . . , t2n. On the other hand, for every fixed z ∈ C, these functions are

holomorphic with respect to t in D \ ⋃2nk=1t : tk = z. Thus, by Hartogs the-

orem (see for example [Shab], Chapter I, sections 2.3, 2.6), the matrix functionsR(z, t), R−1(z, t) are jointly holomorphic in the variables z, t outside the singular

set⋃2n

k=1x, t : tk = x. In view of Lemma 14.2, the same conclusion follows fromthe representations (14.4a), (14.4b).

In order to employ the joint system representation techniques described inSections 12 and 13 in the present setting, we have to establish the holomorphynot only of the residues but also of the semiresidues of R(·, t). In view of Remarks10.2 and 12.3, we have a certain freedom in definition of the semiresidues andthe semiresidual matrices. Thus one should take care in choosing the semiresidualmatrices of R(·, t) for each fixed t in order to obtain holomorphic functions of t.In general, it is possible to define the holomorphic semiresidues only locally (werefer the reader to Appendix B of the present paper where the global holomorphicfactorization of a matrix function of rank one is discussed).

Lemma 14.4. Let M(t) be a Cm×m-valued function, holomorphic in a domain Dof CN , and let

rankM(t) = 1 ∀t ∈ D. (14.6)

Then there exist a finite open covering Upmp=1 of D, a collection fp(t)mp=1

of Cm×1-valued functions and a collection gp(t)mp=1 of C1×m-valued functionssatisfying the following conditions.

1. For p = 1, . . . , m the functions fp(t) and gp(t) are holomorphic in Up.2. Whenever Up′ ∩ Up′′ = ∅, there exists a (scalar) function ϕp′,p′′(t), holomor-

phic and invertible in Up′ ∩ Up′′ , such that for every t ∈ Up′ ∩ Up′′

fp′′(t) = fp′(t)ϕp′,p′′(t), gp′′(t) = ϕ−1p′,p′′(t)gp′(t). (14.7)

3. For p = 1, . . . , m the function M(t) admits the factorization

M(t) = fp(t)gp(t), t ∈ Up. (14.8)

Proof. Let fp(t) be the pth column of the matrix M(t) and let

Up = t ∈ D : fp(t) = 0, p = 1, . . . , m. (14.9)

Then, in view of (14.6) and (14.9), Upmp=1 is an open covering of D. Furthermore,from (14.6) and (14.9) it follows that for 1 ≤ p, q ≤ m there exists a unique (scalar)function ϕp,q(t), holomorphic in Up, such that

fq(t) = fp(t)ϕp,q(t), t ∈ Up. (14.10)

Now define gp(t) as

gp(t) =(ϕp,1(t) . . . ϕp,m(t)

), t ∈ Up. (14.11)

Then the function gp(t) is holomorphic in Up and, according to (14.10), the factor-ization (14.8) holds for these fp(t) and gp(t). (14.10) also implies that wheneverUp′ ∩ Up′′ = ∅ we have

ϕp′′,p′(t)ϕp′,k(t) = ϕp′′,k(t), t ∈ Up′ ∩ Up′′ , 1 ≤ k ≤ m.

In particular,

ϕp′′,p′(t)ϕp′,p′′(t) = 1, t ∈ Up′ ∩ Up′′ ,

and (14.7) follows.

Theorem 14.5. Let D be a domain in C2n∗ and let R(z, t)t∈D be a normalized

holomorphic family of Cm×m-valued generic rational functions parameterized bythe pole and zero loci. Then there exist a finite open covering11 Dαα∈A of D,two collections FP,α(t)α∈A, FZ,α(t)α∈A of Cm×n-valued functions and twocollections GP,α(t)α∈A, GZ,α(t)α∈A of Cn×m-valued functions satisfying thefollowing conditions.

1. For each α ∈ A the functions FP,α(t), FZ,α(t), GP,α(t), GZ,α(t) are holo-morphic in Dα.

2. Whenever Dα′ ∩Dα′′ = ∅, there exist diagonal matrix functions DP,α′,α′′(t),DZ,α′,α′′(t), holomorphic and invertible in Dα′ ∩ Dα′′ , such that for everyt ∈ Dα′ ∩ Dα′′

FP,α′′(t) = FP,α′(t)DP,α′,α′′(t), GP,α′′(t) = D−1P,α′,α′′(t)GP,α′(t), (14.12a)

FZ,α′′(t) = FZ,α′(t)DZ,α′,α′′(t), GZ,α′′(t) = D−1Z,α′,α′′(t)GZ,α′(t). (14.12b)

3. For each α ∈ A and t ∈ Dα the matrices FP,α(t), FZ,α(t), GP,α(t), GZ,α(t)are, respectively, the left pole, left zero, right pole, right zero semiresidualmatrices of the generic rational function R(·, t), i.e., the representations

R(z, t) = I + FP,α(t) (zI −AP(t))−1

GP,α(t), (14.13a)

R−1(z, t) = I + FZ,α(t) (zI −AZ(t))−1

GZ,α(t) (14.13b)

hold true for all t ∈ Dα.

Proof. Let Rk(t), k = 1, . . . , 2n, be the holomorphic residue functions as in Lemma14.2. Since for every fixed t the rational function R(·, t) is generic, each matrixRk(t) is of rank one. Hence there exists a finite open covering Uk,pmp=1 of D, suchthat in each open set Uk,p the function Rk(t) admits the factorization Rk(t) =fk,p(t)gk,p(t) as in Lemma 14.4. Now it suffices to define A as the set of 2n-tuples(p1, . . . , p2n) such that ∩2n

k=1Uk,pk= ∅, the open covering Dαα∈A of D by

D(p1,...,p2n) =

Uk,pk,

11The index α runs over a finite indexing set A.

and the collections FP,α(t)α∈A, FZ,α(t)α∈A, GP,α(t)α∈A, GZ,α(t)α∈A by

FP,(p1,...,p2n)(t) =(f1,p1(t) . . . fn,pn(t)

FZ,(p1,...,p2n)(t) =(fn+1,pn+1(t) . . . f2n,p2n(t)

GP,(p1,...,p2n)(t) =

⎛⎜⎝

g1,p1(t)...

gn,pn(t)

⎞⎟⎠ , GZ,(p1,...,p2n)(t) =

⎛⎜⎝

gn+1,pn+1(t)...

g2n,p2n(t)

⎞⎟⎠ .

Definition 14.6. Let D be a domain in C2n∗ and let R(z, t)t∈D be a nor-

malized holomorphic family of Cm×m-valued generic rational functions param-eterized by the pole and zero loci. Let a finite open covering Dαα∈A of D,collections FP,α(t)α∈A, FZ,α(t)α∈A of Cm×n-valued functions and collec-tions GP,α(t)α∈A, GZ,α(t)α∈A of Cn×m-valued functions satisfy the condi-tions 1.–3. of Theorem 14.5. Then the collections FP,α(t)α∈A, FZ,α(t)α∈A,GP,α(t)α∈A, GZ,α(t)α∈A are said to be the collections of, respectively, the leftpole, left zero, right pole, right zero semiresidual functions related to the familyR(z, t)t∈D.

Now we can tackle the problem of recovery of a holomorphic family of genericmatrix functions from the semiresidual data. Once again, let D be a domain inC2n

∗ and let R(z, t)t∈D be a normalized holomorphic family of Cm×m-valuedgeneric rational functions, parameterized by the pole and zero loci. Let collectionsFP,α(t)α∈A, FZ,α(t)α∈A, GP,α(t)α∈A, GZ,α(t)α∈A be the collections ofthe semiresidual matrices related to the family R(z, t)t∈D. Then for each α ∈ A

and a fixed t ∈ Dα the matrices AP(t), AZ(t) are the pole and zero matrices ofthe generic rational function R(z, t), and FP,α(t), FZ,α(t), GP,α(t), GZ,α(t) are,respectively, the left pole, left zero, right pole, right zero semiresidual matricesof R(·, t). The appropriate coupling matrices SZP,α(t), SPZ,α(t) satisfying theLyapunov equations

AZ(t)SZP,α(t)− SZP,α(t)AP(t) = GZ,α(t)FP,α(t), (14.14a)

AP(t)SPZ,α(t)− SPZ,α(t)AZ(t) = GP,α(t)FZ,α(t) (14.14b)

are given by

SZP,α(t) =

(gn+i,α(t)fj,α(t)

tn+i − tj

, SPZ,α(t) =

(gi,α(t)fn+j,α(t)

ti − tn+j

(14.15)where for 1 ≤ k ≤ n gk,α(t) is the kth row of GP,α(t), gn+k,α(t) is the kth rowof GZ,α(t), fk,α(t) is the kth column of FP,α(t), fn+k,α(t) is the kth columnof FZ,α(t) (compare with similar expressions (12.7), (12.8), (12.14)). From theexplicit expressions (14.15) it is evident that SZP,α(t), SPZ,α(t), considered asfunctions of t, are holomorphic in Dα. According to Proposition 12.4, the functions

SZP,α(t), SPZ,α(t) are mutually inverse:

SZP,α(t)SPZ,α(t) = SPZ,α(t)SZP,α(t) = I, t ∈ Dα, (14.16)

and the following relations hold:

GZ,α(t) = −SZP,α(t)GP,α(t), FP,α(t) = FZ,α(t)SZP,α(t). (14.17)

Furthermore, for t ∈ Dα the function R(z, t) admits the representation

R(z, t)R−1(ω, t) = I+

+ (z − ω)FP,α(t)(zI −AP(t))−1S−1ZP,α(t)(ωI −AZ(t))−1GZ,α(t), (14.18)

and, in view of Proposition 12.8, its logarithmic derivative with respect to z admitsthe representation

∂R(z, t)

∂zR−1(z, t) = FP,α(t)(zI −AP(t))−1S−1

ZP,α(t)(zI −AZ(t))−1GZ,α(t).

(14.19)

The representations (14.18), (14.19) above are local: each of them holds in theappropriate individual subset Dα. Note, however, that whenever Dα′ ∩ Dα′′ = ∅,by Theorem 14.5 we have

SZP,α′′(t) = D−1Z,α′,α′′(t)SZP,α′(t)DP,α′,α′′(t), t ∈ Dα′ ∩ Dα′′ ,

where the functions DZ,α′,α′′(t), DP,α′,α′′(t) are as in (14.12). Hence the expres-sions (14.18), (14.19) coincide in the intersections of the subsets Dα (although theindividual functions FP,α(t), SZP,α(t), GZ,α(t) do not). Here it is a self-evidentfact, because these expressions represent globally defined objects. In the next sec-tion, where we shall use such local representations to construct globally definedobjects, it will become a requirement.

15. Holomorphic families of generic rational matrix functionswith prescribed local data

This section can be considered as a t - dependent version of Section 13. Here weconsider the problem, how to construct a normalized holomorphic family of genericrational functions12 with prescribed local data. The nature of such data is sug-gested by the considerations of the previous section (see, in particular, Theorem14.5).

Let D be a domain in C2n∗ and let Dαα∈A be a finite open covering of

D. We assume that Fα(t)α∈A and Gα(t)α∈A are collections of, respectively,Cm×n-valued and Cn×m-valued functions, satisfying the following conditions:

1. For each α ∈ A the functions Fα(t), Gα(t) are holomorphic in Dα.2. Whenever Dα′ ∩ Dα′′ = ∅, there exist diagonal matrix functions Dr,α′,α′′(t),

Dc,α′,α′′(t), holomorphic and invertible in Dα′ ∩ Dα′′ , such that for everyt ∈ Dα′ ∩ Dα′′

Fα′′ (t) = Fα′(t)Dc,α′,α′′(t), Gα′′ (t) = Dr,α′,α′′(t)Gα′(t). (15.1)

Remark 15.1. Conditions 1. and 2. imply that the collections Fα(t)α∈A and

Gα(t)α∈A represent holomorphic mappings from D into the spaces13 P(m−1)×nc

and Pn×(m−1)r , respectively.

For each α ∈ A and t ∈ Dα we consider the Lyapunov equation

AP(t)Sα(t)− Sα(t)AZ(t) = Gα(t)Fα(t), (15.2)

where AP(t) and AZ(t) are as in (14.1). Its solution Sα(t) is the PZ-coupling14

matrix, related to the data AP(t), AZ(t), Fα(t), Gα(t). Considered as a function oft, Sα(t) is holomorphic in Dα, because the right-hand side Gα(t)Fα(t) is holomor-phic in Dα. The collection of functions Sα(t)α∈A is said to be the collection ofPZ-coupling functions related to the pair of collections Fα(t)α∈A, Gα(t)α∈A.In view of (15.1), whenever Dα′ ∩ Dα′′ = ∅, we have

Sα′′(t) = Dr,α′,α′′(t)Sα′′(t)Dc,α′,α′′(t), t ∈ Dα′ ∩Dα′′ , (15.3)

where Dr,α′,α′′(t), Dc,α′,α′′(t) are diagonal, holomorphic and invertible matrixfunctions. Hence either ∀α ∈ A detSα(t) ≡ 0 or ∀α ∈ A detSα(t) ≡ 0. In thelatter case the pair of collections Fα(t)α∈A and Gα(t)α∈A is said to be PZ-admissible, and the set

ΓPZ =⋃

α∈A

t ∈ Dα : detSα(t) = 0 (15.4)

is said to be the PZ-singular set related to the pair of collections Fα(t)α∈A,Gα(t)α∈A .

In the same way, we can consider the collection of functions Sα(t)α∈A,where for each α ∈ A and t ∈ Dα the matrix Sα(t) is the solution of the Lyapunovequation

AZ(t)Sα(t)− Sα(t)AP(t) = Gα(t)Fα(t). (15.5)

This collection is said to be the collection of ZP-coupling functions related to thepair of collections Fα(t)α∈A, Gα(t)α∈A. If for every α ∈ A detSα(t) ≡ 0 in Dα

then the pair of collections Fα(t)α∈A, Gα(t)α∈A is said to be ZP-admissible,and the set

ΓZP =⋃

α∈A

t ∈ Dα : detSα(t) = 0 (15.6)

is said to be the ZP-singular set related to the pair of collections Fα(t)α∈A,Gα(t)α∈A.

13See Definition 13.5.14See Definition 13.3.

Remark 15.2. Note that, in view of (15.1), if for each α ∈ A Sα(t) satisfies theLyapunov equation (15.2) (or (15.5)) then the subsets

Γα = t ∈ Dα : detSα(t) = 0of the appropriate singular set agree in the intersections of the sets Dα:

Γα′ ∩ (Dα′ ∩ Dα′′) = Γα′′ ∩ (Dα′ ∩ Dα′′) ∀α′, α′′.

Theorem 15.3. Let D be a domain in C2n∗ and let Dαα∈A be a finite open cover-

ing of D. Let Fα(t)α∈A and Gα(t)α∈A are collections of, respectively, Cm×n-valued and Cn×m-valued functions, satisfying the following conditions:

Fα′′ (t) = Fα′(t)Dc,α′,α′′(t), Gα′′ (t) = Dr,α′,α′′(t)Gα′(t).

3. The pair of collections Fα(t)α∈A, Gα(t)α∈A is ZP-admissible.

Let ΓZP denote the ZP-singular set related to the pair of collectionsFα(t)α∈A, Gα(t)α∈A. Then there exists a unique normalized holomorphic fam-ily R(z, t)t∈D\ΓZP

of rational generic functions parameterized by the pole andzero loci such that for every α ∈ A and t ∈ Dα \ΓZP the matrices Fα(t) and Gα(t)are, respectively, the left pole and right zero semiresidual matrices of R(·, t):

Fα(t) = FP,α(t), Gα(t) = GZ,α(t), ∀t ∈ Dα \ ΓZP . (15.7)

It is locally given by

R(z, t) = I − Fα(t)(zI −AP(t))−1S−1α (t)Gα(t), t ∈ Dα \ ΓZP , (15.8)

where Sα(t)α∈A is the collection of ZP-coupling functions related to the pair ofcollections Fα(t)α∈A, Gα(t)α∈A. Furthermore, the logarithmic derivative ofR(z, t) with respect to z admits the local representation

∂R(z, t)

∂zR−1(z, t) =

= Fα(t)(zI −AP (t))−1S−1α (t)(zI −AZ(t))−1Gα(t),

t ∈ Dα \ ΓZP . (15.9)

Proof. In view of Proposition 13.4 and condition 3, for each α ∈ A and t ∈ Dα\ΓZPthere exists a unique generic rational function Rα(·, t), normalized by Rα(∞, t) =I, with the pole and zero matrices AP(t), AZ(t) and the prescribed left zero andright pole semiresidual matrices (15.7). The function Rα(·, t) and its logarithmicderivative admit the representations

Rα(z, t) = I − Fα(t)(zI −AP (t))−1S−1α (t)Gα(t), (15.10)

R′α(z, t)R−1

α (z, t) = Fα(t)(zI −AP(t))−1S−1α (t)(zI −AZ(t))−1Gα(t).

From the representation (15.10) and condition 1 it follows that the familyRα(z, t)t∈Dα\ΓZP

is holomorphic. In view of condition 2 (see also (15.3)), wehave

Rα′(z, t) = Rα′′(z, t), ∀t ∈ (Dα′ ∩ Dα′′) \ ΓZP .

Hence we can define the holomorphic family R(z, t)t∈D\ΓZPby

R(z, t) = Rα(z, t), t ∈ Dα \ ΓZP

to obtain the local representations (15.8), (15.9). The uniqueness of such a familyfollows from the uniqueness of each function Rα(·, t).

Theorem 15.4. Let D be a domain in C2n∗ and let Dαα∈A be a finite open cover-

ing of D. Let Fα(t)α∈A and Gα(t)α∈A are collections of, respectively, Cm×n-valued and Cn×m-valued functions, satisfying the following conditions:

Fα′′ (t) = Fα′(t)Dc,α′,α′′(t), Gα′′ (t) = Dr,α′,α′′(t)Gα′(t).

3. The pair of collections Fα(t)α∈A, Gα(t)α∈A is PZ-admissible.

Let ΓPZ denote the PZ-singular set related to the pair of collectionsFα(t)α∈A, Gα(t)α∈A. Then there exists a unique normalized holomorphic fam-ily R(z, t)t∈D\ΓPZ

of rational generic functions parameterized by the pole andzero loci such that for every α ∈ A and t ∈ Dα \ΓPZ the matrices Fα(t) and Gα(t)are, respectively, the left zero and right pole semiresidual matrices of R(·, t):

Fα(t) = FZ,α(t), Gα(t) = GP,α(t), ∀t ∈ Dα \ ΓPZ . (15.11)

It is locally given by

R(z, t) = I + Fα(t)S−1α (t)(zI −AP(t))−1Gα(t), t ∈ Dα \ ΓPZ , (15.12)

where Sα(t)α∈A is the collection of PZ-coupling functions related to the pair ofcollections Fα(t)α∈A, Gα(t)α∈A. Furthermore, the logarithmic derivative ofR(·, t) admits the local representation

∂R(z, t)

∂zR−1(z, t)

= −Fα(t)S−1α (t)(zI −AP(t))−1Sα(t)(zI −AZ(t))−1S−1

α (t)Gα(t),

t ∈ Dα \ ΓPZ . (15.13)

Proof. The proof is analogous to that of Theorem 15.3.

16. Isosemiresidual families of generic rational matrix functions

In the present section we shall consider an important special case of holomorphicfamilies of generic rational functions, parameterized by the pole and zero loci t.Namely, we are interested in the case when (either left pole and right zero or rightpole and left zero) semiresidual functions of t, determining the family as explainedin Section 14, are constant.

Definition 16.1. Let D be a domain in C2n∗ and let R(z, t)t∈D be a normalized

holomorphic family of Cm×m-valued generic rational functions, parameterized bythe pole and zero loci.

1. The family R(z, t)t∈D is said to be ZP-isosemiresidual15 if there exists apair of matrices F ∈ Cm×n and G ∈ Cn×m such that for every t ∈ D thematrices F and G are, respectively, the left pole and right zero semiresidualmatrices of the generic rational function R(·, t):

F = FP(t), G = GZ(t), ∀t ∈ D.

2. The family R(z, t)t∈D is said to be PZ-isosemiresidual if there exists apair of matrices F ∈ Cm×n and G ∈ Cn×m such that for every t ∈ D thematrices F and G are, respectively, the left zero and right pole semiresidualmatrices of the generic rational function R(·, t):

F = FZ(t), G = GP(t), ∀t ∈ D.

Let us assume that a pair of matrices F ∈ Cm×n and G ∈ Cn×m is given.How to construct a (PZ- or ZP-) isosemiresidual normalized holomorphic familyof Cm×m-valued generic rational functions, parameterized by the pole and zeroloci, for which the constant functions

F (t) ≡ F, G(t) ≡ G (16.1)

would be the appropriate semiresidual functions? This is a special case of theproblem considered in Section 15 (see Theorem 15.3). Note, however, that in thiscase the prescribed semiresidual functions (16.1) are holomorphic in the domainC2n

∗ . Therefore, we may consider its open covering consisting of the single set –the domain itself.

Following the approach described in Section 15, we consider the solutionsSPZ(t), SZP(t) of the Lyapunov equations

AP(t)SPZ(t)− SPZ(t)AZ(t) = GF, (16.2a)

AZ(t)SZP(t)− SZP(t)AP (t) = GF, (16.2b)

AP(t) = diag(t1, . . . , tn), AZ(t) = diag(tn+1, . . . , t2n). (16.3)

15Iso- (from..ισoς - equal - in Old Greek) is a combining form.

Then the functions SPZ(t), SZP(t) are given explicitly by

SPZ(t) =

ti − tn+j

1≤i,j≤n

, (16.4a)

SZP(t) =

tn+i − tj

1≤i,j≤n

, (16.4b)

where gi and fj denote, respectively, the ith row of G and the jth column of F .In particular, the functions SPZ(t), SZP(t) are rational with respect to t andholomorphic in C2n

∗ . The next step is to verify that the constant functions (16.1)are PZ- or ZP-admissible (that is, suitable for the construction of a holomorphicfamily of generic rational functions – see Theorem 15.3). This means to check thatdetSPZ(t) ≡ 0 or detSZP(t) ≡ 0. Note that, since the functions SPZ(t), SZP(t)are identical up to the permutation of variables tk ↔ tn+k, 1 ≤ k ≤ n, theseconditions are equivalent.

Definition 16.2. A pair of matrices F ∈ Cm×n and G ∈ Cn×m is said to beadmissible if the Cn×n-valued rational functions SPZ(t), SZP(t) given by (16.4)satisfy the (equivalent) conditions

detSPZ(t) ≡ 0, detSZP(t) ≡ 0.

It turns out that the admissibility of a given pair of matrices F ∈ Cm×n,G ∈ Cn×m can be checked by means of a simple criterion, described below.

Recall that for a matrix M =(mi,j

)ni,j=1

detM =∑

(−1)σm1,σ(1) · · ·mn,σ(n), (16.5)

where σ runs over all n! permutations of the set 1, . . . , n, and (−1)σ is equal toeither 1 or −1 depending on the parity of the permutation σ.

Definition 16.3. A matrix M ∈ Cn×n is said to be Frobenius-singular if for some ℓ,1 ≤ ℓ ≤ n, there exist indices 1 ≤ α1 < · · · < αℓ ≤ n; 1 ≤ β1 < · · · < βn−ℓ+1 ≤ n,such that mαi,βj = 0 for all 1 ≤ i ≤ ℓ, 1 ≤ j ≤ n− ℓ + 1.

Theorem 16.4. A matrix M ∈ Cn×n is Frobenius-singular if and only if all n!summands (−1)σm1,σ(1) · · ·mn,σ(n) of the sum (16.5) representing the determinantdetM are equal to zero.

Theorem 16.4 is due to G. Frobenius, [Fro1]. The proof of this theorem canbe also found in [Ber], Chapter 10, Theorem 9. The book [LoPl] contains somehistorical remarks concerning this theorem. See the Preface of [LoPl], especiallypp. xiii–xvii of the English edition (to which pp. 14–18 of the Russian translationcorrespond).

Proposition 16.5. A pair of matrices F ∈ Cm×n, G ∈ Cn×m is admissible if andonly if their product GF is not a Frobenius-singular matrix.

Proof. Assume first that the matrix GF is Frobenius-singular. Then, according to(16.4a),

detSPZ(t) =∑

(−1)σ m1,σ(1) · · ·mn,σ(n)

(t1 − t1+σ(1)) · · · ((tn − tn+σ(n)), (16.6)

where σ runs over all n! permutations of the set 1, . . . , n and mi,j = gifn+j . Ifthe matrix GF is Frobenius-singular then, according to Theorem 16.4, all thenumerators m1,σ(1) · · ·mn,σ(n) of the summands in (16.6) are equal to zero, andhence detSPZ(t) ≡ 0.

Conversely, if the matrix GF is not Frobenius-singular then, according to thesame Theorem 16.4, there exists a permutation σ0 such that

m1,σ0(1) · · ·mn,σ0(n) = 0.

Let us choose and fix n pairwise different numbers t01, . . . t0n and set t0n+σ0(1) =

t01 − ε, . . . , tn+σ0(n) = t0n − ε, where ε = 0. Then as ε→ 0 the summand

(−1)σ0m1,σ0(1) · · ·mn,σ0(n)

(t1 − tn+σ0(1)(ε)) · · · (tn − tn+σ0(n)(ε))= (−1)σ0(m1,σ0(1) · · ·mn,σ0(n))ε

is the leading term of the sum on the right-hand side of (16.6): all other summandsgrow at most as O(ǫ−(n−1)).

Definition 16.6. Let a pair of matrices F ∈ Cm×n, G ∈ Cn×m be such that theirproduct GF is not a Frobenius-singular matrix and let the Cn×n-valued rationalfunctions SPZ(t), SZP(t) be given by (16.4).

1. The set

ΓPZ = t ∈ C2n∗ : detSPZ(t) = 0 (16.7)

is said to be the PZ-singular set related to the pair F, G.2. The set

ΓZP = t ∈ C2n∗ : detSZP(t) = 0 (16.8)

is said to be the ZP-singular set related to the pair F, G.

Remark 16.7. Note that, since det SPZ(t), detSZP(t) are polynomials in (ti −tj)

−1, the singular sets ΓPZ , ΓZP related to the pair F, G are complex algebraicvarieties of codimension one in C2n

Combining Proposition 16.5 and Theorems 15.3, 15.4, we obtain

Theorem 16.8. Let matrices F ∈ Cm×n and G ∈ Cn×m be such that their productGF is not a Frobenius-singular matrix, and let ΓPZ , ΓZP be the related singularsets. Then the following statements hold true.

1. There exists a unique PZ-isosemiresidual family R(z, t)t∈C2n∗ \ΓPZ

Cm×m-valued generic rational functions such that for every t ∈ C2n∗ \ΓPZ the

matrices F and G are, respectively, the left zero and right pole semiresidualmatrices of R(·, t):

F = FZ(t), G = GP (t), ∀t ∈ C2n∗ \ ΓPZ . (16.9)

It is given by

R(z, t) = I + FS−1PZ(t)(zI −AP(t))−1G, t ∈ C2n

∗ \ ΓPZ (16.10)

where the function SPZ(t) satisfying the equation (16.2a) is given by (16.4a).Furthermore, the logarithmic derivative of R(z, t) with respect to z admits therepresentation

∂R(z, t)

∂zR−1(z, t)

= −FS−1PZ(t)(zI −AP(t))−1SPZ(t)(zI −AZ(t))−1S−1

PZ(t)G,

t ∈ C2n∗ \ ΓPZ . (16.11)

2. There exists a unique ZP-isosemiresidual family R(z, t)t∈C2n∗ \ΓZP

Cm×m-valued generic rational functions such that for every t ∈ C2n∗ \ΓZP the

matrices F and G are, respectively, the left pole and right zero semiresidualmatrices of R(·, t):

F = FP(t), G = GZ(t), ∀t ∈ C2n∗ \ ΓZP . (16.12)

It is given by

R(z, t) = I − F (zI −AP(t))−1S−1ZP(t)G, t ∈ C2n

∗ \ ΓZP , (16.13)

where the function SZP(t) satisfying the equation (16.2b) is given by (16.4b).Furthermore, the logarithmic derivative of R(z, t) with respect to z admits therepresentation

∂R(z, t)

∂zR−1(z, t)

= F (zI −AP(t))−1S−1ZP(t)(zI −AZ(t))−1G,

t ∈ C2n∗ \ ΓZP . (16.14)

17. Isoprincipal families of generic rational matrix functions

Our interest in holomorphic families of generic rational functions is motivated byour intent to construct rational solutions of the Schlesinger system (see Section 18below). Indeed, given a holomorphic family R(z, t) : t ∈ D of generic rationalfunctions, we can consider the linear differential system

∂R(z, t)

∂z= QR(z, t)R(z, t), (17.1)

where QR(z, t) is the logarithmic derivative of R(z, t) with respect to z. Accord-ing to Lemma 14.2 (and in view of (14.3), (14.4c)), the system (17.1) can berewritten as

∂R(z, t)

(2n∑

z − tk

)R(z, t), (17.2)

where the functions Qk(t) are holomorphic in D. The system (17.2) can be viewedas a holomorphic family (=deformation) of Fuchsian systems parameterized by thesingularities’ loci. It was proved in [KaVo] (Theorem 8.2) that in the case whenthe deformation (17.2) is isoprincipal the functions Qk(t) satisfy the Schlesingersystem.

Definition 17.1. Let D be a domain in C2n∗ and let R(z, t)t∈D be a normalized

holomorphic family of Cm×m-valued generic rational functions, parameterized bythe pole and zero loci. Assume that for 1 ≤ k ≤ 2n there exist Cm×m-valuedfunctions Ek(·), holomorphic and invertible in C∗, such that for every t ∈ D thefunction Ek is the principal factor16 of the function R(·, t) at tk: there exists aCm×m-valued function Hk(·, t), holomorphic and invertible in a neighborhood oftk, such that

R(z, t) = Hk(z, t)Ek(z − tk). (17.3)

Then the family R(z, t)t∈D is said to be isoprincipal.

Theorem 17.2. Let D be a domain in C2n∗ and let R(z, t)t∈D be a normalized

holomorphic family of Cm×m-valued generic rational functions, parameterized bythe pole and zero loci. The family R(z, t) : t ∈ D is isoprincipal if and only if itis PZ-isosemiresidual.

Proof. First, assume that the family R(z, t)t∈D is PZ-isosemiresidual. Then,according to Definition 16.1, there exist F ∈ Cm×n and G ∈ Cn×m such that forevery t ∈ D the matrices F and G are, respectively, the left zero and right polesemiresidual matrices of the generic rational function R(·, t). Then, by Lemma10.12 (see also Remark 10.14), for k = 1, . . . , 2n and independently of t the prin-cipal factor Ek(ζ) of R(·, t) at tk can be chosen in the form

Ek(ζ) =

I + Lk − ζ−1Lk, if 1 ≤ k ≤ n,

I − Lk + ζLk, if 1 + n ≤ k ≤ 2n,(17.4)

−g∗k(gkg∗k)−1gk, if 1 ≤ k ≤ n,

fk(f∗kfk)−1f∗

k , if 1 + n ≤ k ≤ 2n,(17.5)

gk is the kth row of G and fk is the (k − n)th column of F . Hence, by Definition17.1 the family R(z, t) : t ∈ D is isoprincipal.

Conversely, assume that the family R(z, t)t∈D is isoprincipal. Then, ac-

cording to Definition 17.1, for 1 ≤ k ≤ 2n there exist functions Ek(·), holomorphic

and invertible in C∗, such that for every t ∈ D the function Ek is the principalfactor of the function R(·, t) at tk. Let t0, t ∈ D be fixed and let F ∈ Cm×n andG ∈ Cn×m be, respectively, the left zero and right pole semiresidual matrices ofthe generic rational function R(·, t0). Let us denote by gk the kth row of G and

by fk the (k− n)th column of F . Then, in view of Remark 10.11, the function Ek

is of the form

Ek(ζ) = Mk(ζ)Ek(ζ),

where Mk(·) is a Cm×m-valued function, holomorphic and invertible in C and Ek(·)is given by (17.4), (17.5). Hence for z in a neighborhood of tk R(z, t) admits therepresentation

R(z, t) = Hk(z, t)Mk(z − tk)Ek(z − tk),

where Hk(·, t) is a Cm×m-valued function, holomorphic and invertible at tk. Then,for k = 1, . . . , n, the residue Rk(t) of R(z, t) at tk is given by

Rk(t) =(Hk(tk, t)Mk(0)g∗k(gkg∗k)−1

Therefore, G is the right pole semiresidual matrix of the function R(z, t), as well.Analogously, for k = n + 1, . . . , 2n

E−1(ζ) = I − Lk + ζ−1Lk,

Lk = fk(f∗kfk)−1f∗

hence the residue Rk(t) of R−1(z, t) at tk is given by

Rk(t) = fk

((f∗

kfk)−1f∗kM−1

k (0)H−1k (tk, t)

Therefore, F is the left zero semiresidual matrix of the function R(z, t), as well.This completes the proof.

Theorem 17.2 reduces the construction of an isoprincipal family to the con-struction of an isosemiresidual family. The latter problem has already been con-sidered in Section 16. According to Theorems 16.8 and 17.2, from any pair ofmatrices F ∈ Cm×n and G ∈ Cn×m, such that the product GF is not a Frobenius-singular matrix, we can construct an isoprincipal family of generic rational func-tions R(z, t)t∈C2n

∗ \ΓPZ, where ΓPZ denotes the PZ-singular set related to the

pair of matrices F , G. This family is given by

R(z, t) = I + FS−1PZ(t)(zI −AP(t))−1G, (17.6)

where the function SPZ(t), satisfying (16.2a), is given by (16.4a). The logarithmicderivative of R(z, t) with respect to z is given by

∂R(z, t)

∂zR−1(z, t)

= −FS−1PZ(t)(zI −AP(t))−1SPZ(t)(zI −AZ(t))−1S−1

PZ(t)G, (17.7)

and we obtain the following expressions for its residues Qk(t)

Qk(t) = −FS−1PZ(t)I[k]SPZ(t)(tkI −AZ(t))−1S−1

PZ(t)G,

1 ≤ k ≤ n, (17.8a)

Qk(t) = −FS−1PZ(t)(tkI −AP (t))−1SPZ(t)I[k−n]S

−1PZ(t)G,

n + 1 ≤ k ≤ 2n. (17.8b)

Here we use the notation

I[k]def= diag(δ1,k, . . . , δn,k),

where δi,j is the Kronecker delta.

Remark 17.3. Note that, according to (16.4a), the function SPZ(t) is a rationalfunction of t. Hence also the functions Qk(t) are rational functions of t.

18. Rational solutions of the Schlesinger system

It can be checked that the rational functions Qk(t) given by (17.8a) satisfy theSchlesinger system ⎧

⎪⎪⎪⎨⎪⎪⎪⎩

∂tℓ=

[Qℓ, Qk]

tℓ − tk, k = ℓ,

∂tk=∑

ℓ =k

[Qℓ, Qk]

tk − tℓ.

(18.1)

It is also not very difficult to check that

V (t)def= −FS−1

PZ(t)G (18.2)

is the potential function for this solution:

Qk(t) =∂V (t)

∂tk, k = 1, . . . , 2n. (18.3)

Furthermore, one can show that the rational function detSPZ(t) admits thefollowing integral representation

detSPZ(t) = det SPZ(t0) · exp

1≤j≤2n,j =i

trace(

∂V (t)∂ti· ∂V (t)

ti − tjdti

. (18.4)

where t0 and t are two arbitrary points the domain C2n∗ \ΓPZ , and γ is an arbitrary

path which begins at t0, ends at t and is contained in C2n∗ \ ΓPZ .

However, the explanation of these facts lies in the considerations of Sections2 and 3 of the first part [KaVo] of this work. The matrix functions Qk(t) satisfy

the Schlesinger system, and the function −V (t) is a Laurent coefficient at z =∞of the normalized solution (17.6) of the Fuchsian system

dR(z, t)

⎛⎝ ∑

1≤k≤2n

z − tk

⎞⎠R(z, t), (18.5)

R(z, t) = I − V (t)

z+ o(|z|−1) as z →∞, (18.6)

while the function

τ(t)def= det SPZ(t), (18.7)

is the tau-function related to the solution Q1(t), . . . , Q2n(t) of the Schlesingersystem.

More detailed explanation of these and other related facts will be given inthe third part of this work.

Appendix

B. The global factorization of a holomorphicmatrix function of rank one

Let M(t) = ‖mp,q(t)‖1≤p,q≤m be a Cm×m-valued function of the variable t ∈ D,where D is a domain in CN . (We can even assume that D is a Riemann domain17

of dimension N over CN .) In our considerations N = 2n and D ⊆ C2n∗ . Let the

matrix function M be holomorphic in D and let

rankM(t) = 1 ∀t ∈ D. (B.1)

We will try to represent M in the form

M(t) = f(t)g(t), (B.2)

wheref(t) and g(t) are, respectively, a Cm×1-valued function and a C1×m-valuedfunction, both of them holomorphic18 in D.

Let us recall that, according to Lemma 14.4, there exist a finite open coveringUpmp=1 of D, a collection fp(t)mp=1 of Cm×1-valued functions and a collection

gp(t)mp=1 of C1×m-valued functions satisfying the following conditions.

1. For p = 1, . . . , m the functions fp(t) and gp(t) are holomorphic in Up.2. For p = 1, . . . , m the function M(t) admits the factorization

M(t) = fp(t)gp(t), t ∈ Up. (B.3)

17See Definition 5.4.4 in [Hor].18In general, such a global factorization is impossible even if the factors f(t) and g(t) are onlyrequired to be continuous rather than holomorphic: one of the obstacles is of topological nature.

3. Whenever Up′∩Up′′ = ∅, there exists a (scalar) function ϕp′,p′′(t), holomorphicand invertible in Up′ ∩ Up′′ , such that

fp′′(t) = fp′(t)ϕp′,p′′(t), gp′′(t) = ϕ−1p′,p′′(t)gp′(t) ∀t ∈ Up′ ∩ Up′′ . (B.4)

In particular,

ϕp,p(t) = 1 ∀t ∈ Up, (B.5a)

ϕp,p′(t) = ϕp,p′′(t)ϕp′′,p′(t) ∀t ∈ Up ∩ Up′ ∩ Up′′ . (B.5b)

The equalities (B.3), p = 1, . . . , k, are nothing more that the factorizationsof the form (B.2), with holomorphic factors fp(t) and gp(t). However, the factor-ization (B.3) is only local: for each p the equality (B.3) holds in the open subsetUp of the set D. For different p′ and p′′, the factors fp′ , gp′ and fp′′ , gp′′ may notagree in the intersections Up′ ∩ Up′′ . To glue the factorizations (B.3) for differentp together, we seek scalar functions ϕp(t) which are holomorphic in Up, do notvanish there and satisfy the condition

fp′(t)ϕp′ (t) = fp′′(t)ϕp′′ (t) ∀t ∈ Up′ ∩ Up′′ . (B.6)

Then, in view of (B.3),

ϕ−1p′ (t)gp′(t) = ϕ−1

p′′ (t)gp′′(t) ∀t ∈ Up′ ∩ Up′′ . (B.7)

Assuming that such functions ϕ p , 1 ≤ p ≤ k, are found, we set

f(t)def= fp(t)ϕp(t) if t ∈ Up, (B.8a)

g(t)def= ϕ−1

p (t)gp(t) if t ∈ Up. (B.8b)

The relations (B.6), (B.7) ensure that these definitions are not contradictory. Thusthe functions f(t) and g(t) are defined for every t ∈ D. Moreover, these functionsare holomorphic in D and provide the factorization (B.2).

From (B.4) it follows that the condition (B.6) is equivalent to the condition

ϕp′(t) = ϕp′,p′′(t)ϕp′′(t) ∀t ∈ Up′ ∩ Up′′ , (B.9)

where ϕp′,p′′(t) are the functions appearing in (B.4). Thus, to ensure that theconditions (B.6), (B.7) are in force, we have to solve the so-called second Cousinproblem (see [Shab], [Oni], [Leit] and [Hor]):

Problem B.1. Let D be a complex manifold and let Uαα of D be an opencovering of D. For each α, β such that Uα ∩ Uβ = ∅ let a C-valued function ϕα,β ,holomorphic and non-vanishing in Uα ∩ Uβ, be given.

Find a collection of C-valued functions ϕαα with the following properties:

1. For every α the function ϕα is holomorphic in Uα and does not vanish there.2. Whenever Uα ∩ Uβ = ∅, the relation

ϕα = ϕα,βϕβ (B.10)

holds in Uα ∩ Uβ.

A necessary condition for the solvability of the second Cousin problem in D

with the given data Uαα, ϕα,βα,β is the so-called “cocycle condition”:

ϕα,γ = ϕα,βϕβ,γ in every non-empty triple intersection Uα ∩ Uβ ∩ Uγ ,ϕα,α = 1 in every Uα.

(B.11)In our case this condition is fulfilled – see (B.5). However, the cocycle conditionalone is not sufficient to guarantee the existence of a solution to the second Cousinproblem – it depends on D itself, as well.

Proposition B.2. (J.-P.Serre, [Ser1]; see also [Shab], Section 16; [Hor], sections5.5 and 7.4; [Oni], Section 4.4.) If D is a Stein manifold19 which satisfies thecondition 20

H2(D, Z) = 0, (B.12)

then the second Cousin problem in D with arbitrary given data Uα, ϕβ,α satis-fying the cocycle condition (B.11) is solvable.

As we have seen, the factorization problem (B.2) can be reduced to solvingthe second Cousin with a certain data. Thus, the following result holds:

Theorem B.3. Let M(t) be a Cm×m-valued function, holomorphic for t ∈ D, whereD is a Riemann domain over CN . Assume that M satisfies the condition

rankM(t) = 1 ∀t ∈ D.

If D possesses the property: the second Cousin problem in D with arbitrary givendata Uαα, ϕα,βα,β satisfying the cocycle condition (B.11) is solvable, then thematrix function M(t) admits the factorization of the form

M(t) = f(t) · g(t),

where the factors f(t) and g(t) are, respectively, a Cm×1-valued function and aC1×m-valued function, holomorphic and non-vanishing for t ∈ D. In particular,such is the case if D is a Stein manifold satisfying the condition (B.12).

References

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19A domain D in CN is a Stein manifold if and only if D is pseudoconvex, or, what is equivalent,D is a holomorphy domain.20 H2(D, Z) is the second cohomology group of D with integer coefficients.

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Henkin – red. (Itogi Nauki i Tehniki. Sovremennye problemy matem-atiki. Fundamentalьnye napravleni. Tom 10.) VINITI, Moskva,1986, ss. 1–284. English transl.: [SCV-En]

[SCV-En] Several Complex Variables. IV. Gindikin, S.G. and G.M. Henkin – eds.(Encyclopaedia of Mathematical Sciences, Vol. 10.) Springer-Verlag, Berlin ·

Heidelberg · NewYork, 1990, pp. 1–252. Translated from Russian original:[SCV-Ru]

[Ser1] Serre, J.-P. Quelques problemes globaux relatifs aux varietes de Stein.(French). Pp. 57–68 in: Colloque sur les fonctions de plusieurs variables, tenua Bruxelles, mars 1953. Georges Thone, Liege, 1953; Masson&Cie, Paris, 1953.Reprinted in: [Ser2], pp. 259–270. Russian transl.:Serr, Ж. -P. Nekotorye globalьnye zadaqi, svzannye s mnogoobrazimiXteina. Ss. 363–371 v [Ras], a takжe ss. 344–354 v [Ser3].

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[Ser3] Serr, Ж.P. Sobranie soqinenii. Tom 1. Moskva, Nezavisimyi Moskov-skii Universitet, Moskovskii Centr Nepreryvnogo MatematiqeskogoObrazovani, 2002. 464 ss.

[Shab] Xabat, B.V. Vvedenie v Kompleksnyi Analiz. Qastь II. FunkciiNeskolьkih Peremennyh. Tretьe izdanie. Nauka, Moskva, 1985. 464 ss.English transl.:Shabat, B.V. Introduction to complex analysis. Part II. Functions of severalvariables. (Translations of Mathematical Monographs, 110.) American Mathe-matical Society, Providence, RI, 1992. x+371 pp.French transl.:Chabat, B. Introduction a l’analyse complexe. Tome 2. Fonctions de plusieursvariables. (Traduit du Russe: Mathematiques.) [Translations of Russian Works:Mathematics], Mir, Moscow, 1990. 420 pp.

Victor KatsnelsonDepartment of MathematicsWeizmann Institute of ScienceRehovot 76100, Israele-mail: victor.katsnelson@weizmann.ac.il

Dan VolokDepartment of MathematicsBen–Gurion University of the NegevBeer-Sheva 84105, Israele-mail: volok@math.bgu.ac.il

Preservation of the Norms of Linear OperatorsActing on Some Quaternionic Function Spaces

M. Elena Luna-Elizarraras∗ and Michael Shapiro∗∗

Abstract. There are considered real and quaternionic versions of some classi-cal linear spaces, such as Lebesgue spaces, the spaces of continuous functions,etc., as well as linear operators acting on them. We prove that, given a reallinear bounded operator on one of those spaces its quaternionic extensionkeeps being bounded and its norm does not change.

Mathematics Subject Classification (2000). Primary 47A30, 47B38; Secondary46B25, 46E15.

Keywords. Quaternionic function spaces, quaternionic linear operators, normsof quaternionic extensions of linear operators.

1. Introduction

1.1. Functional analysis and operator theory almost always use the fields R, ofreal numbers, and C, of complex numbers, as the sets of scalars. At the sametime, the last decades have shown an explosive growth of research in numerousareas of hypercomplex analysis, in particular, in quaternionic analysis which hasled to considering linear spaces and linear operators where the linearity involvesthe scalars from the skew-field H of (real) quaternions.

Certain basic as well as more advanced facts can be found, for instance, in thebooks [Ad], [BDS], [DSS], [GuSp1], [GuSp2] and in the articles [AgKu], [ShVa1],[ShVa2], [Sh], [AS], [ASV], [Shar], [SharAl1], [SharAl2] but in many other sourcesas well.

Real and complex linearities has been considered mostly “in parallel”, i.e., thetheorems are proved for both cases simultaneously although there are some pe-culiarities distinguishing between them. In particular, it proved to be interesting

∗ The research was partially supported by CONACYT project and by COFAA-IPN.∗∗ The research was partially supported by CONACYT projects as well as by Instituto PolitecnicoNacional in the framework of COFAA and CGPI programs.

206 M.E. Luna-Elizarraras and M. Shapiro

to look at what is occurring while imbedding real spaces and operators into thewider complex spaces and acting there operators since the choice of the latter isnot unique, at least, starting from a normed real space one can construct manycomplex normed spaces extending the initial real one. The study of the phenomenaarising in this setting, seems to have started in [Ri] and afterwards the problemhas been subject to periodic interest of the mathematicians who have found manyintrinsic features as well as intersections with other areas, see, e.g., [VeSe], [Ve],[So], [Kr1], [Kr2], [Zy], [De], [FIP].

1.2. Quite similar questions arise, equally naturally, for real spaces and operatorsbut now compared with their quaternionic extensions, that is, the H-linear spacesand acting between them H-linear operators generated by the original R-linearobjects. Some of them are treated in the paper, more specifically, we consider thequaternionic analogs of some classical function spaces (Lp-functions; continuous;of bounded variation; etc.) endowed with the structure of a left, or right, H-linearspace, and linear operators acting on or between them. As in complex case, thosequaternionic linear operators can have many norms extending the classical ones intheir real antecedents but we choose to introduce the norms by the usual formulaswith the quaternionic modulus instead of the real or complex one, thus ensuringthat the resulting spaces remain to be Banach spaces. We prove that, under suchconditions, in all the situations we are concerned in the norm of the quaternionicextension of a real linear operator, does not increase.

1.3. We are not aware of any directly preceding work in quaternionic situation.For Lp-spaces, perhaps the nearest one is [GaMa] which is concerned with n-tuples of R-linear operators, so for n = 4 the corresponding 4-tuple can be seenas an R-linear operator being a kind of “dequaternionization” of the quaternionicextension of an initial real linear operator. Besides, we use strongly and efficientlythe multiplicative structure of quaternions and a very specific character of theskew-field H in all the reasonings. Lemma 3.1 is a good, but not unique, exampleof the “quaternionic flavor”, especially if one would like to compare it with itspredecessors.

1.4. To fix the notation, we note that the quaternionic imaginary units in thepaper are e1, e2, e3 : e1e2 = −e2e1 = e3; e2e3 = −e3e2 = e1; e3e1 = −e1e3 =e2; e

21 = e

22 = e

33 = −1. The real unit 1 is written, sometimes, as e0, so that given

α ∈ H we write

αℓeℓ with α0, α1, α2, α3 ⊂ R.

The quaternionic conjugate to α is

α := α0 −3∑

αℓeℓ.

Preservation of the Norms of Linear Operators 207

2. Quaternionic extension of a real space

2.1. We begin this section with stating definitions and notations.

A left-H-linear space is an abelian group (M, +) together with a function · : H ×M −→ M such that for all α, β ∈ H and for all a, b ∈ M the next conditions aresatisfied:

(i) α · (a + b) = α · a + α · b.(ii) (α + β) · a = α · a + β · a.(iii) (αβ) · a = α(β · a).(iv) 1 · a = a.

As always we write αa instead α · a. The above function H×M −→M is usuallycalled the action of H over M , or the multiplication by quaternionic scalars on theleft. Analogously can be defined a right-H-linear space.

2.2. Given an R-linear space X , we want to get a left-H-linear space M such thatX is injected into M and the action of H over M restricted to R ⊂ H, coincideswith the one defined on the R-linear space X . There are at least two equivalentways for doing this. The first of them, commented here, makes use of the ideasthat help in the construction of the field H from the field R.

2.2.1. Take the set M := X ×X ×X ×X over which we define the action of H as

follows: given α =

αiei ∈ H, and x = (x0, x1, x2, x3) ∈M , then

αx = (α0 + α1e1 + α2e2 + α3e3)(x0, x1, x2, x3):= (α0x0 − α1x1 − α2x2 − α3x3, α0x1 + α1x0 + α2x3 − α3x2,

α0x2 − α1x3 + α2x0 + α3x1, α0x3 + α1x2 − α2x1 + α3x0) .

It is easy to verify that the above definition satisfies the requirements to make Ma left H-space.

It is clear that the set M is an R-linear space, and the subset (r, 0, 0, 0) | r ∈ Xis an R-subspace of M which is identified with X and so every element of the form(r, 0, 0, 0) ∈M is identified with the element r ∈ X .

Let us note thate1(r, 0, 0, 0) = (0, r, 0, 0),e2(r, 0, 0, 0) = (0, 0, r, 0),e3(r, 0, 0, 0) = (0, 0, 0, r),

hence every element that belongs to M , can be expressed as

(x0, x1, x2, x3) = x0 + e1(x1, 0, 0, 0) + e2(x2, 0, 0, 0) + e3(x3, 0, 0, 0)= x0 + e1x1 + e2x2 + e3x3 ,

and M has the decomposition:

M = X + e1X + e2X + e3X .

Let us denote M with the above multiplication by quaternionic scalars by XH andit will be called the H-extension of X .

2.2.2. The language of tensor products is quite useful to give a second way ofdefining the H-extension of X . Indeed, consider

X := H⊗R X = (α, z) | α ∈ H, z ∈ X ,

where H is seen just as R4. By definition, X is an R-linear space and X =X ⊕ X ⊕ X ⊕ X but it can be converted into a left-H-linear space introducinga multiplication by quaternionic scalars by the formula: given an arbitrary λ ∈ H

and (α, z) ∈ X, one sets

λ · (α, z) := (λα, z) .

One can check up then, that we have arrived at XH.

2.2.3. Since the multiplication in H is non-commutative, the two above reasoningshave their “right-hand side copy”.

2.3. Consider now an R-linear space X , which has an additional structure beingalso a normed space. Let as above XH be its H-extension. We wonder if there existsa “norm” ‖ · ‖H on XH which extends the norm ‖ · ‖ on X . Mostly the concept of anorm is introduced for real or complex spaces, but the properties of the skew-fieldof quaternions allows us to generalize it directly.

Definition 2.1. If M is a left-H-linear space, a function

‖ · ‖H : M −→ R

is called a norm on M (or simply a norm) if it satisfies the usual conditions:

1. ‖x‖H ≥ 0 ∀ x ∈M and ‖x‖H = 0 if and only if x = 0.2. ‖αx‖H =| α | ·‖x‖H ∀ x ∈M and ∀ α ∈ H.3. ‖x + y‖H ≤ ‖x‖H + ‖y‖H ∀ x, y ∈M .

Now we are ready to define an H-extension of a norm.

Definition 2.2. Let (X, ‖ · ‖) be an R-linear normed space. We say that an H-normin XH is an extension of the norm on X if ‖ · ‖H restricted to X ⊂ XH, coincideswith the norm ‖ · ‖.

2.4. Another notions that can be “H-extended” are the notions of an operatorand its norm. Let X and Y be R-linear spaces and let XH and Y H be theircorresponding H-extensions. Let T : X −→ Y be an R-linear operator. The H-extension of T , denoted by T H, is the operator T H : XH −→ Y H defined by

T H[x0 + e1x1 + e2x2 + e3x3] := T [x0] + e1T [x1] + e2T [x2] + e3T [x3] .

It is direct to check that T H is H-linear, that is

T H[λx] = λT [x] , ∀ λ ∈ H and ∀ x ∈ X .

2.4.1. Let X and Y be R-linear normed spaces . Their corresponding H-extensionsXH and Y H can be endowed with many norms extending the given norms. Let usfix some norms on XH and Y H. Then it is easy to show that if T : X −→ Yis additionally a bounded R-linear operator, then T H is bounded too. Moreover,obviously ‖T ‖ ≤ ‖T H‖. There does not exist a general answer for the questionwhen the reciprocal inequality is valid. In this paper we study some cases in whichthe norm of an operator and the norm of its H-extension are the same.

3. Norms of operators between the quaternionic Lp-spaces

3.1. For K = R, C or H, and for a real number p such that 1 < p < +∞ ,consider the K-linear normed space LK

p := Lp(Ω, Σ, µ; K) where Ω is a space witha σ-algebra Σ, a σ-finite measure µ, and where the norm is

‖f‖p :=

| f(x) |p dx

, ∀ f ∈ LK

Following a tradition, LKp is considered as a left-K-linear space although for K = H

it can be seen as a bi-linear space.

It is easily seen that LCp and LH

p are, respectively, the complex and the quaternionic

extensions of the R-linear space LRp with the additional property that their norms

extend the norm on LRp .

The next lemma provides an important tool in the proofs of the theorems.

Lemma 3.1. Given A =

aiei ∈ H, it follows that

|〈A, B〉|p dSB = |A|pCp , (3.1)

where S3 is the unit sphere in R4, Cp :=

|〈1, x〉|p dSx and 〈·, ·〉 is the inner

(scalar) product in R4 : 〈A, B〉 :=

Ak Bk =1

2(AB + B A) .

Proof. Define A :=A

‖A‖ . It is known that all rotations of H = R4 are of the form

ϕ(y) = αyβ−1, with α and β points of S3. So, taking α = A and β = 1, we define

a rotation ϕ : S3 −→ S3 by ϕ(x) := A x. Then we make the change of variable

B := ϕ(x) = A x and since the Lebesgue measure on the unit sphere S3 is invariant

under orthogonal transformations, we have that∫

|〈A, B〉|p dSB =

∣∣∣〈| A | A, B〉∣∣∣p

dSB =| A |p∫

∣∣∣〈 A, B〉∣∣∣p

=| A |p∫

|〈ϕ(1), ϕ(x)〉|p dSx =| A |p∫

|〈 1, x〉|p dSx .

3.1.1. The above lemma has its origin in [Zy, p.181] and [VeSe] where the com-plexification of real spaces and operators has been considered. Lemma 3.1 exploitsefficiently the rich multiplicative structure of the skew-field of quaternions; onecan compare it with Lemma 1 in [GaMa]; see also [St].

Theorem 3.2. Let T : LRp := Lp(Ω1, Σ1, µ1; R) −→ LR

q := Lq(Ω2, Σ2, µ2; R) be an

R-linear operator and let T H : LHp −→ LH

q be its H-extension. Then for 1 ≤ p ≤q <∞, it follows that

‖T H‖LHp→LH

q= ‖T ‖LR

p→LRq.

Proof. It is enough to show that ‖T H‖LHp→LH

q≤ ‖T ‖LR

p→LRq.

Let f0, f1, f2, f3 ∈ Lp, and for ℓ ∈ 0, 1, 2, 3write T [fℓ] =: gℓ. Then F :=

eℓ fℓ

and G :=

eℓ gℓ are elements of LHp and LH

q respectively. Let B :=

Bℓ eℓ be

a point of S3, then h :=

Bℓ fℓ ∈ LRp , and

T [h] =

Bℓ T [fℓ] =

Bℓ gℓ ∈ LR

Using the definition of norm of an operator, it is obtained that

‖T ‖LRp→LR

q≥‖T [h]‖LR

‖h‖LRp

∥∥∥∥∥3∑

Bℓ gℓ

∥∥∥∥∥LR

q∥∥∥∥∥3∑

Bℓ fℓ

∥∥∥∥∥LR

that is,∥∥∥∥∥

Bℓ gℓ

∥∥∥∥∥LR

≤ ‖T ‖LRp→LR

∥∥∥∥∥3∑

Bℓ fℓ

∥∥∥∥∥LR

From the definitions of the norms on the spaces LRp and LR

q , the above inequalitycan be written as:

∣∣∣∣∣3∑

Bℓ gℓ(α)

∣∣∣∣∣

dµ2(α)

∣∣∣∣∣3∑

Bℓ fℓ(β)

∣∣∣∣∣

dµ1(β)

Raising to the power q both sides of the above inequality and integrating with

respect to the variable B :=

Bℓ eℓ over the sphere S3, we get:

∣∣∣∣∣3∑

Bℓ gℓ(α)

∣∣∣∣∣

dµ2(α) dSB

≤ ‖T ‖qLR

p→LRq

∣∣∣∣∣3∑

Bℓ fℓ(β)

∣∣∣∣∣

dµ1(β)

Now Fubini-Toneli’s Theorem applies to the left side, and raising to the power1

qboth sides we get:

∣∣∣∣∣3∑

Bℓ gℓ(α)

∣∣∣∣∣

dSB dµ2(α)

⎡⎣∫

∣∣∣∣∣3∑

Bℓ fℓ(β)

∣∣∣∣∣

dµ1(β)

⎤⎦

Observe that for α ∈ Ω2 fixed, g :=3∑

gℓ(α) eℓ is a quaternion. Then Lemma 3.1

applies to the left side of the above inequality, with A = g, which gives:

∣∣∣∣∣3∑

Bℓ gℓ(α)

∣∣∣∣∣

dSB dµ2(α)

|〈B, g〉|q dSB dµ2(α)

| g |q Cq dµ2(α)

= C1/qq

⎛⎝∫

| gℓ(α) |2)q/2

dµ2(α)

⎞⎠

where Cq :=

| 〈1, x〉 |q dSx .

Now we apply Jessen’s inequality (see [Du-Sch, p. 530]) to the second member of

inequality (3.2) and use again Lemma 3.1, with A :=

fℓ(β) eℓ :

‖T ‖LRp→LR

⎡⎣∫

∣∣∣∣∣3∑

Bℓ fℓ(β)

∣∣∣∣∣

dµ1(β)

⎤⎦

⎧⎨⎩

∣∣∣∣∣3∑

Bℓ fℓ(β)

∣∣∣∣∣

dµ1(β)

⎫⎬⎭

= ‖T ‖LRp→LR

| 〈B, A〉 |q dSB

dµ1(β)

= ‖T ‖LRp→LR

(| A |q Cq)p/q

dµ1(β)

= ‖T ‖LRp→LR

| A |p dµ1(β)

= ‖T ‖LRp→LR

⎧⎨⎩

| fℓ(β) |2)p/2

dµ1(β)

⎫⎬⎭

From (3.2), (3.3) and (3.4),

⎛⎝∫

| gℓ(α) |2)q/2

dµ2(α)

⎞⎠

⎧⎨⎩

|fℓ(β)|2)p/2

dµ1(β)

⎫⎬⎭

which implies

‖T H[F ]‖LHq≤ ‖T ‖LR

p→LRq· ‖F‖LH

Since F is arbitrary, this means that

‖T H‖LHp→LH

q≤ ‖T ‖LR

p→LRq

Remark 3.3. This theorem can be obtained from Lemma 1 in [GaMa]. We preferto have given a proof which uses strongly a specific character of our quaternionicsituation, in particular, we base the proof on Lemma 3.1.

4. Norms of operators on spaces of additive functions

4.1. Let us denote by ba(S, Σ; K) =: ba(K), the set of K-valued bounded additivefunctions defined on the field Σ of subsets of a set S; that is, Σ is a family of subsetsof S which is closed under the finite umber of operations of union, intersection,and complement. Obviously ba(S, Σ; K) is a K-linear space (although for K = Hit can be seen as a bi-linear space) which is made a normed space with the normgiven by the formula

‖µ‖ba(K) := var(µ) :=

| µ(Ei) | : Eini=1, n ∈ N, is a finite collection of disjoint sets of Σ

Here we make a mention about the notation in the case K = H. When multipli-cation by quaternions is made on the right, we write (µ α)(E) = µ(E)α for anyα ∈ H, µ ∈ ba(S, Σ; H) and E ∈ Σ.It is proved in [DuSch] that for the case K = R or C, it is a Banach space. Forthe case K = H, this can be proved analogously.It is clear that

ba(S, Σ; C) = (ba(S, Σ; R))C

where XC is the complexification of a real space X , and

ba(S, Σ; H) = (ba(S, Σ; R))H .

Theorem 4.1. Given T ∈ L(ba(S, Σ; R)) then ‖T ‖ = ‖T H‖.

Proof. Let λ :=3∑

eℓ λℓ ∈ ba(S, Σ; H) and µ := T H[λ] =3∑

eℓ T [λℓ] =3∑

eℓ µℓ,

with µℓ := T [λℓ], and let y :=3∑

yℓ eℓ be a point of the unit sphere S3. It follows

that ∥∥∥∥∥3∑

yℓ µℓ

∥∥∥∥∥ba(R)

≤ ‖T ‖∥∥∥∥∥

yℓ λℓ

∥∥∥∥∥ba(R)

Integrating over the sphere S3 we get:∫

∥∥∥∥∥3∑

yℓ µℓ

∥∥∥∥∥ba(R)

dSy ≤ ‖T ‖ba∫

∥∥∥∥∥3∑

yℓ λℓ

∥∥∥∥∥ba(R)

dSy . (4.1)

In general, given ν ∈ ba(S, Σ; K), it will be useful to consider var(ν) as the limitof a generalized sequence, and this concept allows one to generalize the notion ofthe limit. For doing this, there is considered the following reasoning.A partially ordered set (D,≤) is said to be directed if every finite subset of D hasan upper bound. A map f : D −→ X of a directed set D into a set X is calleda generalized sequence of elements in X , or simply a generalized sequence in X .

If f : D −→ X is a generalized sequence in the topological space X , it is said toconverge to the point p ∈ X , if to every neighborhood N of p there corresponds ad0 ∈ D, such that d ≥ d0 implies f(d) ∈ N . In this case, it is also said that thelimit of f exists and is equal to p, or symbolically, lim

Df(d) = p.

With additional hypothesis, this notion of convergence gives rise to a related notionof uniform convergence: as before, let D be a directed set, A an arbitrary set, andX a metric space with metric . Suppose that f maps (d, a) ∈ D×A into X . Thenthe statement lim

Df(d, a) = g(a) uniformly on A, or uniformly for a ∈ A, means

that for every ǫ > 0 there exists a d0 ∈ D such that (f(d, a), g(a)) < ǫ for d > d0

and for every a ∈ A.Now we can see var(ν) as the limit of a generalized sequence in the followingmanner. Let F be the family of all finite collections Ei of disjoint sets in Σ,which are ordered by Ei ⊆ Fj to mean that each Ei is the union of some ofthe sets Fj . Then by construction, F is a directed set. Observe that if Ei ⊆ Fj,then ∑

|ν(Ei)| ≤∑|ν(Fj)| .

Define the function f : F −→ K by

f (Eini=1) :=

|ν(Ei)| .

var(ν) = limF

f (Eini=1) = limF

|ν(Ei)| . (4.2)

For any y =

yℓeℓ ∈ S3 and

eℓ νℓ ∈ ba(S, Σ; H), we have that

yℓ νℓ ∈

ba(S, Σ; R) and∥∥∥∥∥

yℓ νℓ

∥∥∥∥∥ba(R)

yℓ νℓ

)= lim

Ej∈F

∣∣∣∣∣3∑

yℓ νℓ(Ej)

∣∣∣∣∣ . (4.3)

Since the norm on a Banach space is continuous, the function var

yℓ νℓ

continuous over the sphere S3. From definition of the order relation on F, it followsthat the real sequence ⎧

⎨⎩

∣∣∣∣∣3∑

yℓ νℓ(Ej)

∣∣∣∣∣

⎫⎬⎭

is not decreasing.Recall that Dini’s Lemma assures that if a monotone sequence of continuous func-tions defined on a compact converges point-wise to a continuous function, then italso converges uniformly. This lemma extends easily onto the case of generalized

sequences. Since S3 is compact, and (4.2) says that the generalized sequence (4.4)

converges point-wise to the continuous function var

yℓ νℓ

)over S3, applying

Dini’s Lemma we conclude that it also converges uniformly. Then the integral andthe limit can be interchanged in the next reasoning:

∥∥∥∥∥3∑

yℓ νℓ

∥∥∥∥∥ba(R)

∣∣∣∣∣3∑

yℓ νℓ(Ej)

∣∣∣∣∣ dSy

= limEj

∣∣∣∣∣3∑

yℓ νℓ(Ej)

∣∣∣∣∣ dSy

= limEj

∣∣∣∣∣〈3∑

νℓ(Ej)eℓ,3∑

yℓ eℓ〉∣∣∣∣∣ dSy .

Applying Lemma 3.1 for p = 1 we have:

∥∥∥∥∥3∑

yℓ νℓ

∥∥∥∥∥ dSy = limEj

∣∣∣∣∣3∑

νℓ(Ej)eℓ

∣∣∣∣∣ C1

∥∥∥∥∥3∑

eℓ νℓ

∥∥∥∥∥ba(H)

Applying this result to the left and right sides of inequality (4.1), we obtain that

∥∥∥∥∥3∑

yℓ µℓ

∥∥∥∥∥ba(R)

dSy = C1

∥∥∥∥∥3∑

eℓ µℓ

∥∥∥∥∥ba(H)

≤ ‖T ‖∫

∥∥∥∥∥3∑

yℓ λℓ

∥∥∥∥∥ba(R)

= ‖T ‖C1

∥∥∥∥∥3∑

eℓ λℓ

∥∥∥∥∥ba(H)

which implies:∥∥∥∥∥T[

eℓ λℓ

]∥∥∥∥∥ba(H)

∥∥∥∥∥3∑

eℓ µℓ

∥∥∥∥∥ ≤ ‖T ‖ba∥∥∥∥∥

eℓ λℓ

∥∥∥∥∥

and since λ =

eℓ λℓ was an arbitrary element of ba(S, Σ; H), we conclude that

‖T H‖ ≤ ‖T ‖.

4.2. The norms of operators on the space of functions of bounded variation

Let I be an interval of R, which can be finite or infinite. Recall that the totalvariation of a K-valued function f on I is defined by

var(f, I) := sup

| f(bi)− f(ai) |

where the supremum is taken over all finite sets of points ai, bi ∈ I with a1 ≤ b1 ≤a2 ≤ b2 ≤ · · · ≤ an ≤ bn. If var(f, I) < ∞, f is said to be of bounded variationon I.

We denote by bv(I; K) the set of K-valued functions of bounded variation definedon I. Clearly (bv(I; R))C = bv(I; C) and (bv(I; R))H = bv(I; H).

The K-linear space bv(I; K) becomes a normed space with the norm

‖f‖bv(K) := var(f, I) .

Theorem 4.2. Let T : bv(I; R) −→ bv(I; R) be an R-linear operator. Then ‖T ‖ =‖T H‖.

Proof. We define a directed set F as the family of all finite collections Ijnj=1 of

disjoint subintervals of I with Ij ⊆ Li to mean that each Ij is the union ofsome of the sets Li. Writing Ij = (aj , bj), the total variation of f on I is the limitof a generalized sequence:

var(f, I) = limIj∈F

| f(bj)− f(aj) | .

The rest of the proof follows the proof of Theorem 4.1.

5. The case of spaces of continuous functions

As in the above sections, K will denote the fields R, C or the skew-field H. GivenS a compact space, the K-linear space C(S; K) consists of K-valued continuousfunctions. In the case K = H, C(S; H) is a bi-linear H-space. The norm in all thesespaces is given by the formula

‖f‖ := sup | f(s) | : s ∈ S .

It is well known that for the case K = C or R, C(S; K) is a Banach space and thisresult extends of course to the space C(S; H).

Denote by L(C(S; R)) the R-linear space whose elements are R-linear operatorson C(S; R).

Theorem 5.1. Let S be a Hausdorff compact, let T be an element of L(C(S; R))and take its H-extension T H. Then ‖T ‖L(C(S;R)) = ‖T H‖L(C(S;H)).

Proof. Consider the field of subsets Σ generated by the closed sets of S. Recallthat for the case K = R or C (see [DuSch]) a K-valued additive set function µdefined on Σ is said to be regular if for each E ∈ Σ and ǫ > 0 there is a set F ∈ Σsuch that F ⊂ E and a set G ∈ Σ such that E ⊂ G0 and | µ(C) |< ǫ for everyC ∈ Σ with C ⊆ G\F . Denote by rba(S; K) = rba(K) the K-linear space of regularbounded additive set functions. Of course all this directly extends onto the caseK = H, where rba(S; H) forms a bi-linear H-space. The norm is introduced by

‖µ‖rba(K) := var(µ) .

Again for the case K = C or R, in [DuSch, IV.6] it is shown that given f ∈ C(S; K),it is integrable with respect to every µ in rba(S; K). This fact allows one to definean isometric isomorphism between (C(S; K))∗ and rba(S; K), such that given β ∈(C(S; K))∗, its corresponding µ ∈ rba(S; K) is such that

β(f) =

f(s)µ(ds) , f ∈ C(S; K) .

Under this identification, it follows that (C(S; R))∗ can be seen as the space ofmeasures rba(S; R).

Following closely the proof of the above result, we extend it for K = H, but wehave to note the following: given µ ∈ rba(S; H), if we consider the left H-linearspace of µ-simple functions:

f =n∑

αj χBj ,

it gives rise to the notion of “left” integral∫

f(s)µ(ds) .

That is, we accept to multiply the above integral by quaternions on the leftside only. Then, under the identification which is made between rba(S; H) and(C(S; H))∗, the latter is a left H-linear space.For a given T : C(S; R) −→ C(S; R), we consider its adjoint operator

T ∗ : (C(S; R))∗ −→ (C(S; R))∗

given by T ∗(f) = f T for every f ∈ (C(S; R))∗.Then, considering T ∗ as an element of L(rba(S; R)) ⊂ L(ba(S; R)), it satisfies thehypothesis of Theorem 4.1, and we have that ‖T ∗‖ = ‖(T ∗)H‖.On the other hand, it is well known that ‖T ‖ = ‖T ∗‖ for a K-linear operator incase of K = R or C, and the same proof works for K = H. Then applying this factto the operator T H, it follows that ‖T H‖ = ‖(T H)∗‖. It suffices now to prove that(T H)∗ = (T ∗)H.Let us show that this is true even for a more general situation. Take V1, V2 twoK-linear spaces (with K = R or C) and let 〈·, ·〉V : V1 × V2 −→ K be a bilinearform. The triplet (V1, V2, 〈·, ·〉V ) is called a dual system. Given two dual systems

(V1, V2, 〈·, ·〉V ) and (W1, W2, 〈·, ·〉W ), let A : V1 −→ W1 be a linear operator. Ifthere exists an operator A∗ : W2 −→ V2 such that

〈A[v]; w〉W = 〈v; A∗[w]〉V , (5.1)

for any v ∈ V1 and w ∈W1, then the operator A∗ is called dual, or adjoint, to theoperator A with respect to the given dual systems. It is readily seen that if A∗

exists, it is unique.

All this remains true for K = H with the following changes. The spaces V1 and V2

must be respectively a left- and a right-H-linear spaces. The form 〈·, ·〉V shouldbe bi-linear in the following sense: it is left-H-linear in its first argument andright-H-linear in its second one.

Let us suppose that the given dual systems and the operator A are real, i.e., all theobjects involved are R-linear. Suppose additionally that there exists the adjointoperator A∗. We are going to prove that there exists the adjoint operator (AH)∗

of the H-extension of A and that the next formula holds:

(AH)∗ = (A∗)H .

Let h :=3∑

eℓ hℓ ∈ V H

1 and f :=3∑

fk ek ∈ W H

2 be arbitrary, then

〈AH[h] ; f〉W H = 〈AH

eℓ hℓ

fk ek〉W H = 〈3∑

eℓ A[hℓ] ;

fk ek〉W H

eℓ 〈A[hℓ] ;

fk ek〉W H =

eℓ 〈A[hℓ] ; fk〉W)

eℓ 〈hℓ ; A∗[fk]〉V)

On the other hand, we have that

〈h; (A∗)H[f ]〉V H = 〈3∑

eℓ hℓ ; (A∗)H

]〉V H =

eℓ 〈hℓ ;

A∗[fk] ek〉V H

eℓ 〈hℓ ; A∗[fk] 〉V)

Then one can conclude that

〈AH[h]; f〉W H = 〈h; (A∗)H[f ]〉V H .

Comparing now with (5.1) for K = H and together with the uniqueness of theadjoint operator, we obtain the desired result: (A∗)H = (AH)∗. In particular thisis true for the operator T above, which completes the proof of the theorem.

The next result is a quaternionic version of the theorem presented in [DuSch,V.8.11, p. 445]:

Theorem 5.2. Let (S, Σ, µ) be a positive measure space. Then there exists a compactHausdorff space S1 and an isometric isomorphism Λ : L∞(S, Σ, µ; H) −→ C(S1; H)such that it maps real functions (i.e., functions real µ-almost everywhere) into realfunctions, positive functions into positive functions, and quaternionic conjugatefunctions into quaternionic conjugate functions, i.e., Λf = Λf for every f ∈L∞(S, Σ, µ; H). Moreover, Λ is an algebraic isomorphism in the sense that if h(s) =f(s) g(s) µ-almost everywhere, then Λh = Λf · Λg.

The next statement is an immediate consequence of the last two theorems.

Corollary 5.3. Let T : L∞(Ω1; R) := L∞(Ω1, Σ1, µ1; R) −→ L∞(Ω2; R) :=L∞(Ω2, Σ2, µ2; R) be an R-linear operator and let T H : L∞(Ω1; H) −→ L∞(Ω2; H)be its H-extension. Then

‖T H‖L∞(Ω1;H)→L∞(Ω2;H) = ‖T ‖L∞(Ω1;R)→L∞(Ω2;R).

In other words, in Theorem 3.2 it is possible to take p = q =∞.

References

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[ShVa2] M.V. Shapiro, N.L. Vasilevski, Quaternionic ψ-hyperholomorphic functions,singular integral operators and boundary value problems. II. Algebras of singu-lar integral operators and Riemann-type boundary value problems. ComplexVariables. Theory and Applications 27 (1995), 67–96.

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M. Elena Luna-Elizarraras and Michael ShapiroDepartamento de MatematicasE.S.F.M. del I.P.N.07338 Mexico, D.F., Mexicoe-mail: eluna@esfm.ipn.mxe-mail: shapiro@esfm.ipn.mx

Hardy Algebras Associated withW∗-Correspondences

(Point Evaluation and Schur Class Functions)

Paul S. Muhly and Baruch Solel

1. Introduction

This is primarily an exposition of our work in [35] and [37] which builds on thetheory of tensor algebras over C∗-correspondences that we developed in [31]. Op-erator tensor algebras (and their w∗-analogues, which we call Hardy algebras) forma rich class of non-selfadjoint operator algebras that contains a large variety of op-erator algebras that have received substantial attention in the literature in recentyears.

Among these algebras are the classical disc algebra A(D), and its weak clo-sure, H∞(T); Popescu’s non-commutative disc algebras [45], and their weak clo-sures, the free semigroup algebras studied by Popescu [45] and Davidson and Pitts[19]; quiver algebras studied by us in [32] and by Kribs and Power in [25]; certainnest algebras; analytic crossed products, studied by Peters [41] and by McAseyand Muhly in [28]; and others. (We will describe the construction of tensor andHardy algebras and give many examples in Section 2.) The theory gives a commonapproach to the analysis of all these algebras and has its roots deeply embeddedin the model theory of contraction operators on the one hand and in classical ringtheory on the other.

In fact, the theory of contraction operators may be viewed as the theory ofcontractive representations of the disc algebra. The representation theory of thetensor algebras is a natural generalization of this theory that preserves many ofits features. The disc algebra may be viewed as an analytic generalization of thepolynomial algebra in one variable. The interplay between function theory andthe representation theory of the polynomial algebra has been one of the guiding

The first author was supported by grants from the U.S. National Science Foundation and from theU.S.-Israel Binational Science Foundation. The second author was supported by the U.S.-IsraelBinational Science Foundation and by the Fund for the Promotion of Research at the Technion.

222 Paul S. Muhly and Baruch Solel

beacons in model theory for decades [17]. The tensor algebras we analyze areoperator algebraic versions of algebras that generalize polynomial algebras andhave been of key importance in ring theory since 1947 [23] and, since 1972, havebeen a major focus of attention for analyzing finite-dimensional algebras (see [21]and [22]). (See [30] for an extended discussion of the connection between operatortensor algebras and the theory of finite-dimensional algebras.)

Recall that the disc algebra A(D) may be realized as the algebra of all an-alytic Toeplitz operators on l2(N) (or on H2(T)). Popescu generalizes A(D) byconsidering algebras of operators on the full Fock space over a Hilbert space H ofsome dimension, n say. Let F(H) = C ⊕H ⊕H⊗2 ⊕ · · · denote this Fock space.Then his non-commutative disc algebra of index n is the norm closed algebra gen-erated by the (left) creation operators. That is, his algebras are generated by theidentity operator and operators of the form λ(ξ)η := ξ ⊗ η, where ξ ∈ H andη ∈ F(H). The Fock space may also be written as l2(Fn

+) where Fn+ is the free

semigroup on n generators. In this realization, H may be identified with all thefunctions supported on the words of length one and for such a function ξ, λ(ξ) isjust convolution by ξ on l2(Fn

+). Observe that when n, the dimension of H , is one,then one recovers the disc algebra A(D) represented by analytic Toeplitz operatorson l2(N).

To construct more general tensor algebras one replaces the Hilbert spaceH by a correspondence E over some C∗-algebra (or von Neumann algebra) M .Roughly, a correspondence is a bimodule over M that is also equipped with anM -valued inner product. (For a precise definition see Section 2). When M = C acorrespondence over M is just a Hilbert space.

The tensor algebra associated with the correspondence E, T+(E), is generatedby creation operators on the Fock space F(E) = M ⊕ E ⊕ E⊗2 · · · together witha copy of M (formed by diagonal operators of multiplication, ϕ∞(a), a ∈ M).It follows from the results of [31] that (completely contractive) representations ofT+(E) are given by pairs (T, σ) where T : E → B(H) is a completely contractivemap and σ : M → B(H) is a C∗-representation of M that satisfy T (a · ξ · b) =σ(a)T (ξ)σ(b) for a, b ∈M and ξ ∈ E. (Note that we shall sometimes use ϕ for theleft multiplication on E; that is, a ·ξ may be written ϕ(a)ξ.) Such pairs, (T, σ), arecalled covariant representations of the correspondence E. Given (T, σ), one mayform the Hilbert space E ⊗σ H (see the discussion following Definition 2.1). Fora ∈ M , ϕ(a) ⊗ I then defines a bounded operator on this space. The “complete

contractivity” of T is equivalent to the assertion that the linear map T definedinitially on the balanced algebraic tensor product E⊗H by the formula T (ξ⊗h) :=T (ξ)h extends to an operator of norm at most 1 on the completion E ⊗σ H . Thebimodule property of T , then, is equivalent to the equation

T (ϕ(a) ⊗ I) = σ(a)T , (1)

for all a ∈ M , which means that T intertwines σ and the natural representationof M on E ⊗σ H – the composition of ϕ with Rieffel’s induced representation ofL(E) determined by σ.

Hardy Algebras Associated with W ∗-Correspondences 223

Thus we see that, once σ is fixed, the representations of T+(E) are pa-rameterized by the elements in the closed unit ball of the intertwining spaceη ∈ B(E ⊗σ H, H) | η(ϕ(·) ⊗ I) = ση and ‖η‖ ≤ 1. Reflecting on this leadsone ineluctably to the functional analyst’s imperative: To understand an algebra,view it as an algebra of functions on its space of representations. In our setting,then, we want to think about T+(E) as a space of functions on this ball. For rea-sons that will be revealed in a minute, we prefer to focus on the adjoints of theelements in this space. Thus we let Eσ = η ∈ B(H, E ⊗σ H) | ησ = (ϕ(·) ⊗ I)ηand we write D((Eσ)∗) for the set η ∈ B(E ⊗σ H, H) | η∗ ∈ Eσ, and ‖η‖ < 1.That is, D((Eσ)∗) is the norm-interior of the representation space consisting ofthose (T, σ) that are “anchored by σ”. One of our interests, then, is to understandthe kind of functions that elements X of T+(E) determine on D((Eσ)∗) via theformula

X(η∗) = σ × η∗(X),

where σ × η∗ is the representation of T+(E) that is determined by the pair (σ, T )

with T = η∗.In the special case when A = E = C and σ is the one-dimensional repre-

sentation of A on C, Eσ is also one-dimensional, so D((Eσ)∗) is just the openunit disc in the complex plane and, for X ∈ T+(E), X(η∗) is the ordinary valueof X at the complex number η. On the other hand, if A = E = C, and if σ isscalar multiplication on a Hilbert space H (the only possible representation of C

on H), then D((Eσ)∗) is the space of strict contraction operators on H and for

η∗ ∈ D((Eσ)∗)‖·‖

and X ∈ T+(E) = A(D), X(η∗) is simply the value of X at η∗

defined through the Sz.-Nagy-Foias functional calculus [39]. For another example,if A = C, but E = Cn, and if σ is scalar multiplication on a Hilbert space H , thenD((Eσ)∗) is the space of row contractions on H , (T1, T2, . . . , Tn), of norm less than1; i.e.

∑T ∗

i Ti ≤ rIH for some r < 1. In this case, X(η∗) is given by Popescu’sfunctional calculus [46].

In addition to parametrizing certain representations of T+(E), Eσ has anotherfundamental property: It is itself a C∗-correspondence - over the von Neumannalgebra σ(A)′. Indeed, it is not difficult to see that Eσ becomes a bimodule overσ(A)′ via the formulae: a · η = (IE ⊗ a)η and η · a = ηa, η ∈ Eσ, a ∈ σ(A)′.Further, if η and ζ are in Eσ, then the product η∗ζ lies in the commutant σ(A)′

and defines a σ(A)′-valued inner product 〈η, ζ〉 making Eσ a C∗-correspondence.In fact, since Eσ is a weakly closed space of operators, it has certain topologicalproperties making it what we call a W ∗-correspondence [35]. It is because Eσ is aW ∗-correspondence over σ(A)′ that we focus on it, when studying representationsof T+(E), rather than on its space of adjoints. While Eσ plays a fundamentalrole in our study of quantum Markov processes [33], its importance here – besidesproviding a space on which to “evaluate” elements of T+(E) – lies in the fact that acertain natural representation of Eσ generates the commutant of the representationof T+(E) obtained by “inducing σ up to” L(F(E)). (See Theorem 2.29.)

It is primarily because of this commutant theorem that we cast our work inthis paper entirely in terms of W ∗-correspondences. That is, we work with vonNeumann algebras M and W ∗-correspondences E over them. We still form theFock space F(E) and the tensor algebra T+(E) over E, but because F(E) is aW ∗-correspondence over M , the space L(F(E)) is a von Neumann algebra. Wecall the w∗-closure of T+(E) in L(F(E)) the Hardy algebra of E and denote itby H∞(E). This is our principal object of study. In the case when M = E = C,H∞(E) it the classical H∞(T) (viewed as analytic Toeplitz operators).

As we will see in Lemma 2.17, given a faithful normal representation σ ofM on a Hilbert space H , we may also evaluate elements in H∞(E) at points inD((Eσ)∗) (since the representation associated with a point in the open unit ballextends from T+(E) to H∞(E)). That is, elements in H∞(E) may be viewed asfunctions on D((Eσ)∗), also. Further, when H∞(E) is so represented, one maystudy the “value distribution theory” of these functions. In this context, we es-tablish two capstone results from function theory: The first, [35, Theorem 5.3]is presented as Theorem 3.2 below, generalizes the Nevanlinna-Pick interpolationtheorem. It asserts that given two k-tuples of operators in B(H) (where H is therepresentation space of σ), B1, B2, . . . , Bk, and C1, C2, . . . , Ck, and given pointsη1, η2, . . . , ηk in D((Eσ)∗), one may find an element X in H∞(E) of norm at mostone such that

BiX(η∗i ) = Ci,

for all i if and only if a certain matrix of maps, which resembles the classical Pickmatrix, represents a completely positive operator. This result captures numeroustheorems in the literature that go under the name of generalized Nevanlinna-Picktheorems. Our proof of the theorem (in [35]) uses a commutant lifting theorem thatwe proved in [31]. In the context of model theory, it was Sarason who introduced theuse of commutant lifting to prove the interpolation theorem ([49]). More recently,a number of authors have been studying interpolation problems in the context ofreproducing kernel Hilbert spaces. (See [1], [20], [2], [3] and [14].)

Our second capstone result is a generalization of Schwartz’s lemma (see The-orem 3.4). It follows from our Nevanlinna-Pick theorem that an element X inH∞(E) of norm at most one defines a “Pick-type” matrix of maps that representsa completely positive map. In fact, the matrix is defined using the values of X onD((Eσ)∗). Given an arbitrary operator-valued function Z on D((Eσ)∗), one maydefine a matrix of maps in a similar way. We say that Z is a Schur class operatorfunction if this matrix defines a completely positive map. (See Definition 4.2 fora precise statement.) Theorem 3.2 then shows that the function η∗ → X(η∗) is aSchur class operator function for X in the closed unit ball of H∞(E). In fact, weshow in Theorem 4.3 that every Schur class operator function arises in this wayand that every such function (with values in, say, B(E)) may be represented in theform Z(η∗) = A+B(I−L∗

ηD)−1L∗ηC where A, B, C and D are the entries of a 2×2

block matrix representing a coisometric operator V from E ⊕H to E ⊕ (Eσ ⊗H)(for some auxiliary Hilbert space H) with a certain intertwining property and Lη

is the operator from H to Eσ ⊗H that maps h to η ⊗ h. Borrowing terminologyfrom the classical function theory on the unit disc D, we call such a representationa realization of Z and we call the coisometry V a colligation. (In general, V is acoisometry but, under a mild assumption, it may be chosen to be unitary.)

These results, together with our work on canonical models in [36], representa generalization of some of the essential ingredients of a program that has beendeveloped successfully in model theory – the interaction between operator theoryand function theory on the unit disc D – and has been generalized in various waysto the polydisc and the ball in Cn. This program sets up (essentially) bijectivecorrespondences connecting the theory of unitary colligations (and their unitarysystems), the Sz.-Nagy-Foias functional model theory for contraction operatorsand the discrete-time Lax-Phillips scattering theory. Each theory produces a con-tractive operator-valued function (called the transfer function of the system, thecharacteristic operator function of the completely non-unitary contraction or thescattering function for the scattering system) from which one can recover the orig-inal object (the system or the contraction) up to unitary equivalence. For moredetails, see the works of Ball ([12]), Ball and Vinnikov ([15]), Ball, Trent andVinnikov ([14]) and the references there.

We shall not discuss the program in detail here but we note that Theorem 4.3below is the generalization, to our context, of Theorem 2.1 of [12] or Theorem 2.1of [14]. Here the elements of H∞(E) play the role of multipliers and the disc D inC is replaced by the open unit ball of (Eσ)∗.

We also note that the canonical models for contraction operators are re-placed, in our setting, by canonical models for representations of H∞(E). Thistheory was developed in [36] for completely non-coisometric representations (gen-eralizing results of Popescu in [44]) and it is shown there that the characteristicoperator function for such a representation has a realization associated with aunitary colligation.

In the next section we set the stage by defining our basic constructions,presenting examples and emphasizing the roles of duality and point evaluation inthe theory.

Section 3 deals with the Nevanlinna-Pick theorem and Section 4 with Schurclass operator functions.

2. Preliminaries: W∗-correspondences and Hardy algebras

We shall follow Lance [27] for the general theory of Hilbert C∗-modules that weuse. Let A be a C∗-algebra and let E be a right module over A endowed with abi-additive map 〈·, ·〉 : E×E → A (referred to as an A-valued inner product) suchthat, for ξ, η ∈ E and a ∈ A, 〈ξ, ηa〉 = 〈ξ, η〉a, 〈ξ, η〉∗ = 〈η, ξ〉, and 〈ξ, ξ〉 ≥ 0,with 〈ξ, ξ〉 = 0 only when ξ = 0. If E is complete in the norm ‖ξ‖ := ‖〈ξ, ξ〉‖1/2,the E is called a (right) Hilbert C∗-module over A. We write L(E) for the spaceof continuous, adjointable, A-module maps on E; that is every element of L(E) is

continuous and if X ∈ L(E), then there is an element X∗ ∈ L(E) that satisfies〈X∗ξ, η〉 = 〈ξ, Xη〉. The element X∗ is unique and L(E) is a C∗-algebra withrespect to the involution X → X∗ and the operator norm. If M is a von Neumannalgebra and if E is a Hilbert C∗-module over M , then E is said to be self-dual incase every continuous M -module map from E to M is given by an inner productwith an element of E. If E is a self-dual Hilbert module over M , then L(E) is aW ∗-algebra and coincides with all the bounded linear maps on E [40].

A C∗-correspondence over a C∗-algebra A is a Hilbert C∗-module E over Aendowed with a structure of a left module over A via a *-homomorphism ϕ : A→L(E). When dealing with a specific C∗-correspondence E over a C∗-algebra A,it will be convenient to suppress the ϕ in formulas involving the left action andsimply write aξ or a · ξ for ϕ(a)ξ. This should cause no confusion in context.

Having defined a left action on E, we are allowed to form balanced tensorproducts. Given two correspondences E and F over the C∗-algebra A one maydefine an A-valued inner product on the balanced tensor product E ⊗A F by theformula

〈ξ1 ⊗ η1, ξ2 ⊗ η2〉E⊗BF := 〈η1, ϕ(〈ξ1, ξ2〉E)η2〉F .

The Hausdorff completion of this bimodule is again denoted by E ⊗A F and iscalled the tensor product of E and F .

Definition 2.1. Let M be a von Neumann algebra and let E be a Hilbert C∗-moduleover M . Then E is called a Hilbert W ∗-module over M in case E is self-dual.The module E is called a W ∗-correspondence over M in case E is a self-dual C∗-correspondence over M such that the ∗-homomorphism ϕ : M → L(E) giving theleft module structure on E is normal.

It is evident that the tensor product of two W ∗-correspondences is again aW ∗-correspondence. Note also that, given a W ∗-correspondence E over M and aHilbert space H equipped with a normal representation σ of M , we may form theHilbert space E ⊗σ H (by defining 〈ξ1 ⊗ h1, ξ2 ⊗ h2〉 = 〈h1, σ(〈ξ1, ξ2〉)h2〉). Then,given an operator X ∈ L(E) and an operator S ∈ σ(N)′, the map ξ⊗h → Xξ⊗Shdefines a bounded operator on E ⊗σ H denoted by X ⊗ S. When S = I andX = ϕ(a), a ∈M , we get a representation of M on this space.

Observe that if E is a W ∗-correspondence over a von Neumann algebra M ,then each of the tensor powers of E, viewed as a C∗-correspondence over M in theusual way, is in fact a W ∗-correspondence over M and so, too, is the full Fock spaceF(E), which is defined to be the direct sum M ⊕E ⊕E⊗2 ⊕ · · · , with its obviousstructure as a right Hilbert module over M and left action given by the map ϕ∞,defined by the formula ϕ∞(a) := diaga, ϕ(a), ϕ(2)(a), ϕ(3)(a), . . . , where for alln, ϕ(n)(a)(ξ1 ⊗ ξ2 ⊗ · · · ξn) = (ϕ(a)ξ1) ⊗ ξ2 ⊗ · · · ξn, ξ1 ⊗ ξ2 ⊗ · · · ξn ∈ E⊗n. Thetensor algebra over E, denoted T+(E), is defined to be the norm-closed subalgebraof L(F(E)) generated by ϕ∞(M) and the creation operators Tξ, ξ ∈ E, definedby the formula Tξη = ξ ⊗ η, η ∈ F(E). We refer the reader to [31] for the basicfacts about T+(E).

Definition 2.2. Given a W ∗-correspondence E over the von Neumann algebra M ,the ultraweak closure of the tensor algebra of E, T+(E), in the w∗-algebra L(F(E)),will be called the Hardy Algebra of E, and will be denoted by H∞(E).

Example 2.3. If M = E = C then F(E) may be identified with H2(T). The tensoralgebra in this setting is isomorphic to the disc algebra A(D) and the Hardy algebrais the classical Hardy algebra H∞(T).

Example 2.4. If M = C and E = Cn, then F(E) may be identified with the spacel2(F

+n ) where F+

n is the free semigroup on n generators. The tensor algebra then iswhat Popescu refers to as the “non-commutative disc algebra” An and the Hardyalgebra is its w∗-closure. It was studied by Popescu ([47]) and by Davidson andPitts who denoted it by Ln ([19]).

Example 2.5. Let M be a von Neumann algebra and let α be a unital, injective,normal ∗-endomorphism on M . The correspondence E associated with α is equalto M as a vector space. The right action is by multiplication, the M -valued innerproduct is 〈a, b〉 = a∗b and the left action is given by α; i.e. ϕ(a)b = α(a)b. Wewrite αM for E. It is easy to check that E⊗n is isomorphic to αnM . The Hardyalgebra in this case is called the non-selfadjoint crossed product of M by α andwill be written M ⋊α Z+. This algebra is also called an analytic crossed product,at least when α is an automorphism. It is related to the algebras studied in [28]and [41]. If we write w for T1 (where 1 is the identity of M viewed as an elementof E), then the algebra is generated by w and ϕ∞(M) and every element X in thealgebra has a formal “Fourier” expression

X =∑

where bn ∈ ϕ∞(M). This Fourier expression is actually Ceasaro-summable to Xin the ultraweak topology on H∞(E) [35], but we do not need these details in thepresent discussion.

Example 2.6. Here we set M to be the algebra l∞(Z) and let α be the automorphismdefined by (α(g))i = gi−1. Write E for the correspondence αM as in Example 2.5.Another, isomorphic, way to describe E is to let M be the algebra D of all diagonaloperators on l2(Z), let U be the shift defined by Uek = ek−1 (where ek is thestandard basis), and set E = UD ⊆ B(l2(Z)). The left and right actions on E aredefined simply by operator multiplications and the inner product is 〈UD1, UD2〉 =D∗

1D2. It is easy to check that these correspondences are indeed isomorphic andthe Hardy algebra H∞(E) is (completely isometrically isomorphic to) the algebraU of all operators in B(l2(Z)) whose matrix (with respect to the standard basis) isupper triangular.

Example 2.7. Suppose that Θ is a normal, contractive, completely positive mapon a von Neumann algebra M . Then we may associate with it the correspondenceM ⊗Θ M obtained by defining the M -valued inner product on the algebraic tensorproduct M ⊗M via the formula 〈a ⊗ b, c ⊗ d〉 = b∗θ(a∗c)d and completing. (The

bimodule structure is by left and right multiplications.) This correspondence wasused by Popa ([43]), Mingo ([29]), Anantharam-Delarouche ([6]) and others tostudy the map Θ. If Θ is an endomorphism this correspondence is the one describedin example 2.5.

Example 2.8. Let M be Dn, the diagonal n × n matrices and E be the set of alln × n matrices A = (aij) with aij = 0 unless j = i + 1 with the inner product〈A, B〉 = A∗B and the left and right actions given by matrix multiplication. Thenthe Hardy algebra is isomorphic to Tn, the n×n upper triangular matrices. In fact,a similar argument works to show that, for every finite nest of projections N on aHilbert space H, the nest algebra algN (i.e., the set of all operators on H leavingthe ranges of the projections in N invariant) may be written as H∞(E) for someW ∗-correspondence E.

Example 2.9. (Quiver algebras) Let Q be a directed graph on the set V of vertices.For simplicity we assume that both V and Q are finite sets and view each α ∈ Qas an arrow from s(α) (in V ) to r(α) (in V ). Let M be C(V ) (a von Neumannalgebra) and E (or E(Q)) be C(Q). Define the M -bimodule structure on E asfollows: for f ∈ E, ψ ∈M and α ∈ Q,

(fψ)(α) = f(α)ψ(s(α)),

and(ψf)(α) = ψ(r(α))f(α).

The M -valued inner product is given by the formula

〈f, g〉(v) =∑

s(α)=v

f(α)g(α),

for f, g ∈ E and v ∈ V . The algebra H∞(E) in this case will be written H∞(Q)and is the σ-weak closure of T+(E(Q)). Viewing both algebras as acting on theFock space, one sees that they are generated by a set Sα : α ∈ Q of partialisometries (here Sα = Tδα where δα is the function in C(Q) which is 1 at α and0 otherwise) and a set Pv : v ∈ V of projections (i.e. the generators of ϕ∞(M))satisfying the following conditions.

(i) PvPu = 0 if u = v,(ii) S∗

αSβ = 0 if α = β(iii) S∗

αSα = Ps(α) and(iv)

∑r(α)=v SαS∗

α ≤ Pv for all v ∈ V .

These algebras were studied in [30] and [32], and also in [25] where they werecalled free semigroupoid algebras.

2.1. Representations

In most respects, the representation theory of H∞(E) follows the lines of therepresentation theory of T+(E). However, there are some differences that will beimportant to discuss here. To help illuminate these, we need to review some of thebasic ideas from [31, 32, 35].

A representation ρ of H∞(E) (or of T+(E)) on a Hilbert space H is completelydetermined by what it does to the generators. Thus, from a representation ρ weobtain two maps: a map T : E → B(H), defined by T (ξ) = ρ(Tξ), and a mapσ : M → B(H), defined by σ(a) = ρ(ϕ∞(a)). Analyzing the properties of T andσ one is lead to the following definition.

Definition 2.10. Let E be a W ∗-correspondence over a von Neumann algebra M .Then a completely contractive covariant representation of E on a Hilbert space His a pair (T, σ), where

1. σ is a normal ∗-representation of M in B(H).2. T is a linear, completely contractive map from E to B(H) that is continuous

in the σ-topology of [11] on E and the ultraweak topology on B(H).3. T is a bimodule map in the sense that T (SξR) = σ(S)T (ξ)σ(R), ξ ∈ E, and

S, R ∈M .

It should be noted that there is a natural way to view E as an operator space(by viewing it as a subspace of its linking algebra) and this defines the operatorspace structure of E to which the Definition 2.10 refers when it is asserted that Tis completely contractive.

As we noted in the introduction and developed in [31, Lemmas 3.4–3.6] andin [35], if a completely contractive covariant representation, (T, σ), of E in B(H)

is given, then it determines a contraction T : E⊗σ H → H defined by the formulaT (η ⊗ h) := T (η)h, η ⊗ h ∈ E ⊗σ H . The operator T satisfies

T (ϕ(·) ⊗ I) = σ(·)T . (2)

In fact we have the following lemma from [35, Lemma 2.16].

Lemma 2.11. The map (T, σ) → T is a bijection between all completely contrac-tive covariant representations (T, σ) of E on the Hilbert space H and contractive

operators T : E⊗σ H → H that satisfy equation (2). Given σ and a contraction Tsatisfying the covariance condition (2), we get a completely contractive covariant

representation (T, σ) of E on H by setting T (ξ)h := T (ξ ⊗ h).

The following theorem shows that every completely contractive representa-tion of the tensor algebra T+(E) is given by a pair (T, σ) as above or, equivalently,

by a contraction T satisfying (2).

Theorem 2.12. ([31, Theorem 3.10]) Let E be a W ∗-correspondence over a vonNeumann algebra M . To every completely contractive covariant representation,(T, σ), of E there is a unique completely contractive representation ρ of the tensoralgebra T+(E) that satisfies

ρ(Tξ) = T (ξ) ξ ∈ E and ρ(ϕ∞(a)) = σ(a) a ∈M.

The map (T, σ) → ρ is a bijection between the set of all completely contractivecovariant representations of E and all completely contractive (algebra) represen-

tations of T+(E) whose restrictions to ϕ∞(M) are continuous with respect to theultraweak topology on L(F(E)).

Definition 2.13. If (T, σ) is a completely contractive covariant representation of aW ∗-correspondence E over a von Neumann algebra M , we call the representationρ of T+(E) described in Theorem 2.12 the integrated form of (T, σ) and writeρ = σ × T .

Example 2.14. In the context of Example 2.4, M = C and E = Cn. Then, acompletely contractive covariant representation of E is simply given by a completelycontractive map T : E → B(H). Writing Tk = T (ek), where ek is the standardbasis in Cn, and identifying Cn⊗H with the direct sum of n copies of H, we maywrite T as a row (T1, T2, . . . , Tn). The condition that ‖T‖ ≤ 1 is the condition(studied by Popescu [45] and Davidson and Pitts [19]) that

∑TiT

∗i ≤ 1. Hence

representations of the non-commutative disc algebras are given by row contractions.

Example 2.15. Consider the setting of Example 2.9 and let V,Q, M and E be asdefined there. A (completely contractive covariant) representation of E is given bya representation σ of M = C(V ) on a Hilbert space H and by a contractive map

T : E⊗σ H → H satisfying (2) above. Write δv for the function in C(V ) which is 1on v and 0 elsewhere. The representation σ is given by the projections Qv = σ(δv)whose sum is I. For every α ∈ Q write δα for the function (on E) which is

1 at α and 0 elsewhere. Given T as above, we may define maps T (α) ∈ B(H)

by T (α)h = T (δα ⊗ h) and it is easy to check that T T ∗ =∑

α T (α)T (α)∗ andT (α) = Qr(α)T (α)Qs(α). Thus to every (completely contractive) representation ofthe quiver algebra T+(E(Q)) we associate a family T (α) : α ∈ Q of maps on Hthat satisfy

∑α T (α)T (α)∗ ≤ I and T (α) = Qr(α)T (α)Qs(α). Conversely, every

such family defines a representation by writing T (f ⊗ h) =∑

f(α)T (α)h. Thus,representations are indexed by such families. Note that, in fact, (σ×T )(Sα) = T (α)and (σ × T )(Pv) = Qv (where Sα and Pv are as in Example 2.9).

Remark 2.16. One of the principal difficulties one faces in dealing with T+(E) andH∞(E) is to decide when the integrated form, σ × T , of a completely contractivecovariant representation (T, σ) extends from T+(E) to H∞(E). This problem arisesalready in the simplest situation, namely when M = C = E. In this setting, T isgiven by a single contraction operator T (1) on a Hilbert space, T+(E) “is” thedisc algebra and H∞(E) “is” the space of bounded analytic functions on the disc.The representation σ × T extends from the disc algebra to H∞(E) precisely whenthere is no singular part to the spectral measure of the minimal unitary dilation ofT (1). We are not aware of a comparable result in our general context but we havesome sufficient conditions. One of them is given in the following lemma. It is notnecessary in general.

Lemma 2.17. ([35]) If ‖T‖ < 1 then σ × T extends to a σ-weakly continuousrepresentation of H∞(E).

Other sufficient conditions are presented in Section 7 of [35].

2.2. Duality and point evaluation

The following definition is motivated by condition (2) above.

Definition 2.18. Let σ : M → B(H)be a normal representation of the von Neumannalgebra M on the Hilbert space H. Then for a W ∗-correspondence E over M , theσ-dual of E, denoted Eσ, is defined to be

η ∈ B(H, E ⊗σ H) | ησ(a) = (ϕ(a)⊗ I)η, a ∈M.As we note in the following proposition, the σ-dual carries a natural structure

of a W ∗-correspondence. The reason to define the σ-dual using covariance condi-tion which is the “adjoint” of condition (2) is to get a right W ∗-module (insteadof a left W ∗-module) over σ(M)′.

Proposition 2.19. With respect to the actions of σ(M)′ and the σ(M)′-valued innerproduct defined as follows, Eσ becomes a W ∗-correspondence over σ(M)′: Fora, b ∈ σ(M)′, and η ∈ Eσ, a·η·b := (I⊗a)ηb, and for η, ζ ∈ Eσ, 〈η, ζ〉σ(M)′ := η∗ζ.

Example 2.20. If M = E = C, H is arbitrary and σ is the representation of C onH, then σ(M)′ = B(H) and Eσ = B(H).

Example 2.21. If Θ is a contractive, normal, completely positive map on a vonNeumann algebra M and if E = M ⊗Θ M (see Example 2.7 ) then, for every faith-ful representation σ of M on H, the σ-dual is the space of all bounded operatorsmapping H into the Stinespring space K (associated with Θ as a map from Mto B(H)) that intertwine the representation σ (on H) and the Stinespring repre-sentation π (on K). This correspondence was proved very useful in the study ofcompletely positive maps. (See [33], [38] and [34]). If M = B(H) this is a Hilbertspace and was studied by Arveson ([10]). Note also that, if Θ is an endomorphism,then this dual correspondence is the space of all operators on H intertwining σ andσ Θ.

We now turn to define point evaluation. Note that, given σ as above, theoperators in Eσ whose norm does not exceed 1 are precisely the adjoints of theoperators of the form T for a covariant pair (T, σ). In particular, every η in theopen unit ball of Eσ (written D(Eσ)) gives rise to a covariant pair (T, σ) (with

η = T ∗) such that σ × T is a representation of H∞(E). Given X ∈ H∞(E) wemay apply σ× T to it. The resulting operator in B(H) will be denoted by X(η∗).That is,

X(η∗) = (σ × T )(X)

where T = η∗.In this way, we view every element in the Hardy algebra as a (B(H)-valued)

function on D((Eσ)∗).

Example 2.22. Suppose M = E = C and σ the representation of C on some Hilbertspace H. Then H∞(E) = H∞(T) and (Example 2.20) Eσ is isomorphic to B(H).If X ∈ H∞(E) = H∞(T), so that we may view X with a bounded analytic functionon the open disc in the plane, then for S ∈ Eσ = B(H), it is not hard to check that

X(S∗), as defined above, is the same as the value provided by the Sz.-Nagy-FoiasH∞-functional calculus.

Note that, for a given η ∈ D(Eσ), the map X → X(η∗) is a σ-weakly contin-uous homomorphism on the Hardy algebra. Thus, in order to compute X(η∗), itsuffices to know its values on the generators. This is given in the following (easyto verify) lemma.

Lemma 2.23. Let σ be a faithful normal representation of M on H and for ξ ∈ Ewrite Lξ for the map from H to E⊗σ H defined by Lξh = ξ⊗ h. Then, for ξ ∈ E,a ∈M and η ∈ D(Eσ),

(i) (Tξ)(η∗) = η∗ Lξ, and

(ii) (ϕ∞(a))(η∗) = σ(a)

(Recall that η∗ is a map from E ⊗σ H to H.)

A formula for computing X(η∗), without referring to the generators, will bepresented later (Proposition 2.30).

Example 2.24. In the setting of Example 2.5 we may identify the Hilbert spaceE⊗σ H = αM ⊗σ H with H via the unitary operator mapping a⊗h (in αM ⊗σ H)to σ(a)h. Using this identification, we may identify Eσ with η ∈ B(H) : ησ(a) =σ(α(a))η, a ∈M.

Applying Lemma 2.23, we obtain w(η∗) = T1(η∗) = η∗ L1 = η∗ (viewed

now as an operator in B(H)). Thus, if X =∑

wnbn (as a formal series), withbn = ϕ∞(an) and η ∈ D(Eσ), then

X(η∗) =∑

(η∗)nσ(an)

with the sum converging in the norm on B(H). (In a sense, this equation assertsthat Ceasaro summability implies Abel summability even in this abstract setting.)

Example 2.25. Let D, U and E = UD be as in Example 2.6. Let σ be the identityrepresentation of D on H = l2(Z). The map V (UD⊗σh) = Dh (for D ∈ D, h ∈ H)is a unitary operator from E⊗σ H onto H such that, for every η ∈ Eσ, V η ∈ U∗Dand, conversely, for every D ∈ D, V ∗U∗D∗ lies in Eσ. We write ηD for V ∗U∗D∗.Recall that the Hardy algebra is U (the algebra of all upper triangular operators onH). Given X ∈ U we shall write Xn for the nth upper diagonal of X. A simplecomputation shows that, for D ∈ D with ‖D‖ < 1,

X(η∗D) =

∞∑

Un(U∗D)nXn.

Note here that, in [4], the authors defined point evaluations for operators X ∈ U .In their setting one evaluates X on the open unit ball of D and the values are alsoin D. Their formula (for what in [5] is called the right point evaluation) is

X∆(D) =

∞∑

Un(U∗D)nXnU∗n.

(One can also define a left point evaluation.) The apparent similarity of the twoformulas above may be deceiving. Note that both their point evaluation and ourscan be defined also for “block upper triangular” operators (acting on l2(Z, K) forsome Hilbert space K). But, in that case, the relation between the two formu-las is no longer clear. In fact, our point evaluation is multiplicative (that is,(XY )(η∗) = X(η∗)Y (η∗)) while theirs is not. On the other hand, their pointevaluation is “designed” to satisfy the property that, for X ∈ U and D ∈ D,(X −X∆(D))(U −D)−1 ∈ U ([4, Theorem 3.4]). For our point evaluation (in thegeneral setting), it is not even clear how to state such a property.

Example 2.26. (Quiver algebras) Let Q be a quiver as in Example 2.9 and writeE(Q) for the associated correspondence. We fix a faithful representation σ of M =C(V ) on H. As we note in Example 2.15, this gives a family Qv of projectionswhose sum is I (and, as σ is faithful, none is 0). Write Hv for the range of Qv.Then σ(M)′ = ⊕vB(Hv) and we write elements there as functions ψ defined on Vwith ψ(v) ∈ B(Hv). To describe the σ-dual of E we can use Example 3.4 in [35].

We may also use the description of the maps T in Example 2.15 because everyη in the closed unit ball of Eσ is T ∗ for some representation (σ, T ) of E. Usingthis, we may describe an element η of Eσ as a family of B(H)-valued operatorsη(β) : β ∈ Q−1 where Q−1 is the quiver obtained from Q by reversing all arrows.The σ(M)′-module structure of Eσ is described as follows. For η ∈ Eσ, ψ ∈ σ(M)′

and β ∈ Q−1,

(ηψ)(β) = η(β)ψ(s(β)),

(ψη)(β) = ψ(r(β))η(β).

The σ(M)′-valued inner product is given by the formula

〈η, ζ〉(v) =∑

s(β)=v

η(β)∗ζ(β),

for η, ζ ∈ Eσ and v ∈ V .Recall that the quiver algebra is generated by a set of partial isometries Sα

and projections Pv (see Example 2.9). If σ is given and η∗ = T lies in the open

unit ball of (Eσ)∗ and T is given by a row contraction (T (α)) (as in Example 2.15),then the point evaluation for the generators is defined by Sα(η∗) = T (α) = η(α−1)∗

and Pv(η∗) = Qv. For a general X ∈ H∞(Q), X(η∗) is defined by the linearity,multiplicativity and σ-weak continuity of the map X → X(η∗).

We turn now to some general results concerning the σ-dual. First, the term“dual” that we use is justified by the following result.

Theorem 2.27. ([35, Theorem 3.6]) Let E be a W ∗-correspondence over M and letσ be a faithful, normal representation of M on H. If we write ι for the identityrepresentation of σ(M)′ (on H) then one may form the ι-dual of Eσ and we have

(Eσ)ι ∼= E.

The following lemma summarizes Lemmas 3.7 and 3.8 of [35] and shows thatthe operation of taking duals behaves nicely with respect to direct sums and tensorproducts.

Lemma 2.28. Given W ∗-correspondences E,E1 and E2 over M and a faithful rep-resentation σ of M on H, we have

(i) (E1 ⊕ E2)σ ∼= Eσ

1 ⊕ Eσ2 .

(ii) (E1 ⊗ E2)σ ∼= Eσ

2 ⊗ Eσ1 .

(iii) F(E)σ ∼= F(Eσ).(iv) The map η⊗h → η(h) induces a unitary operator from Eσ⊗ιH onto E⊗σ H.(v) Applying item (iv) above to F(E) in place of E, we get a unitary operator U

from F(Eσ)⊗H onto F(E)⊗H.

Although H∞(E) was defined as a subalgebra of L(F(E)) it is often usefulto consider a (faithful) representation of it on a Hilbert space. Given a faithful,normal, representation σ of M on H we may “induce” it to a representation of theHardy algebra. To do this, we form the Hilbert space F(E)⊗σ H and write

Ind(σ)(X) = X ⊗ I, X ∈ H∞(E).

(in fact, this is well defined for every X in L(F(E)). Such representations werestudied by M. Rieffel in [48]). Ind(σ) is a faithful representation and is an home-omorphism with respect to the σ-weak topologies. Similarly one defines Ind(ι), arepresentation of H∞(Eσ). The following theorem shows that, roughly speaking,the algebras H∞(E) and H∞(Eσ) are the commutant of each other.

Theorem 2.29. [35, Theorem 3.9] With the operator U as in part (v) of Lemma 2.28,we have

U∗(Ind(ι)(H∞(Eσ)))U = (Ind(σ)(H∞(E)))′

and, consequently,

(Ind(σ)(H∞(E)))′′ = Ind(σ)(H∞(E)).

We may now use the notation set above to present a general formula for pointevaluation. For its proof, see [35, Proposition 5.1].

Proposition 2.30. If σ is a faithful normal representation of M on H, let ιH denotethe imbedding of H into F(Eσ)⊗H and write Pk for the projection of F(Eσ)⊗Honto (Eσ)⊗k ⊗H. Also, for η ∈ D(Eσ) and k ≥ 1, note that η⊗k lies in (Eσ)⊗k

and that L∗η⊗k maps (Eσ)⊗k ⊗H into H in the obvious way (and, for k = 0, this

is ιH). Then, for every X ∈ H∞(E),

X(η∗) =∞∑

L∗η⊗kPkU∗(X ⊗ I)UιH

where U is as defined in Lemma 2.28.

3. Nevanlinna-Pick Theorem

Our goal in this section is to present a generalization of the Nevanlinna-Pick The-orem. First, recall the classical theorem.

Theorem 3.1. . Let z1, . . . , zk ∈ C with |zi| < 1 and w1, . . . , wk ∈ C. Then thefollowing conditions are equivalent.

(1) There is a function f ∈ H∞(T) with ‖f‖ ≤ 1 such that f(zi) = wi for all i.

(1− wiwj

1− zizj

)≥ 0.

Since we are able to view elements of H∞(E) as functions on the open unitball of Eσ, it makes sense to seek necessary and sufficient conditions for finding anelement X ∈ H∞(E) with norm less or equal 1 whose values at some prescribedpoints, η1, . . . , ηk, in that open unit ball are prescribed operators C1, . . . , Ck inB(H). To state our conditions we need some notation. For operators B1, B2 inB(H) we write Ad(B1, B2) for the map, from B(H) to itself, mapping S to B1SB∗

2 .Also, for elements η1, η2 in D(Eσ), we let θη1,η2 denote the map, from σ(M)′ toitself, that sends a to 〈η1, aη2〉. Then our generalization of the Nevanlinna-Picktheorem may be formulated as follows.

Theorem 3.2. Let σ be a faithful normal representation of M on H. Fix η1,...,ηk∈Eσ with ‖ηi‖ < 1 and B1, . . . , Bk, C1, . . . , Ck ∈ B(H). Then the following condi-tions are equivalent

(1) There exists an X ∈ H∞(E) with ‖X‖ ≤ 1 such that BiX(η∗i ) = Ci for all i.

(2) The map from Mk(σ(M)′) into Mk(B(H)) defined by the k × k matrix((Ad(Bi, Bj)−Ad(Ci, Cj)) (id− θηi,ηj )

is completely positive.

Remark 3.3. If M = B(H) (and, then σ(M)′ = CI), condition (2) becomes(

BiB∗j − CiC

1− 〈ηi, ηj〉

)≥ 0.

This follows easily from a result of M.D. Choi ([18]).

For the complete proof of Theorem 3.2 we refer the reader to [35, Theorem5.3]. Here we just remark that in order to prove that (1) implies (2) one uses thecomplete positivity condition of (2) to construct a subspaceM⊆ F(Eσ)⊗H thatis invariant under Ind(ι)(H∞(Eσ))∗ and a contraction R that commutes with therestriction of Ind(ι)(H∞(Eσ))∗toM. Then it is possible to apply the commutantlifting theorem of [31, Theorem 4.4] to R∗ to get a contraction on F(Eσ)⊗H thatcommutes with Ind(ι)(H∞(Eσ)). An application of Theorem 2.29 completes theproof.

The following is a consequence of Theorem 3.2. It may be viewed as a gener-alization of the classical Schwartz’s lemma.

Theorem 3.4. Suppose an element X of H∞(E) has norm at most one and satisfiesthe equation X(0) = 0. Then for every η∗ ∈ D((Eσ)∗) the following assertions arevalid:

1. If a is a non-negative element in σ(M)′, and if 〈η, a · η〉 ≤ a, then

X(η∗)aX(η∗)∗ ≤ 〈η, a · η〉.2. If η⊗k denotes the element η ⊗ η ⊗ · · · ⊗ η ∈ E⊗k, then

X(η∗)〈η⊗k, η⊗k〉X(η∗)∗ ≤ 〈η⊗k+1, η⊗k+1〉.3. X(η∗)X(η∗)∗ ≤ 〈η, η〉.

We now illustrate how to apply Theorem 3.2 in various settings.

Example 3.5. When M = H = E = C, we obtain Theorem 3.1.

Example 3.6. If M = E = C and if H is arbitrary, then Eσ = B(H) and Theorem3.2 yields the following result.

Theorem 3.7. Given T1, . . . , Tk ∈ B(H), ‖Ti‖ < 1 and B1, . . . , Bk, C1, . . . , Ck inB(H). Then the following conditions are equivalent.

(1) There exists a function f ∈ H∞(T) with ‖f‖ ≤ 1 and Bif(Ti) = Ci.(2) The map defined by the matrix (φij) is completely positive where

φij(A) =∞∑

(BiTki AT ∗k

j B∗j − CiT

ki AT ∗k

j Bj).

Example 3.8. Assume M = B(H) = E. Then M ′ = CI and Eσ = C and Theorem3.2 specializes to the following.

Theorem 3.9. Given z1, . . . , zk ∈ D and B1, . . . , Bk, C1, . . . , Ck in B(H), then thefollowing conditions are equivalent.

(1) There exists G ∈ H∞(T) ⊗ B(H) with ‖G‖ ≤ 1 such that BiG(zi) = Ci forall i.

∗j − CiC

1− zizj

)≥ 0.

Example 3.10. Set M = B(H) and E = Cn(B(H)) (that is, E is a column of ncopies of B(H)). Then M ′ = CI, Eσ = Cn and Theorem 3.2 yields the followingtheorem due to Davidson and Pitts [19], Arias and Popescu [8] and Popescu [47].

Theorem 3.11. Given η1, . . . , ηk in the open unit ball of Cn and C1, . . . , Ck ∈B(H), then the following conditions are equivalent.

(1) There is a Y ∈ B(H) ⊗ Ln with ‖Y ‖ ≤ 1 such that (η∗i × id)(Y ) = Ci for

all i.

(I − CiC

1− 〈ηi, ηj〉

)≥ 0.

Moreover, if, for all i, the Ci all lie in some von Neumann subalgebra N ⊆B(H), then Y can be chosen in N ⊗ Ln.

Our final example of this section concerns interpolation for nest algebras.The first interpolation result for nest algebras was proved by Lance ([26]). It waslater generalized by Anoussis ([7]) and by Katsoulis, Moore and Trent ([24]). Arelated result was proved by Ball and Gohberg ([13]). The results we present belowrecapture the results of [24].

Theorem 3.12. Let N be a nest of projections in B(H) and fix B, C in B(H).Then the following conditions are equivalent.

(1) There exists an X ∈ AlgN with ‖X‖ ≤ 1 and BX = C.(2) For all projections N ∈ N , CNC∗ ≤ BNB∗.

The “vector version” of this theorem is the following.

Corollary 3.13. Let N be a nest in B(H) and fix u1, . . . , uk, v1, . . . , vk in H. Thenthe following conditions are equivalent.

(1) There exists X ∈ AlgN with ‖X‖ ≤ 1 and Xui = vi for all i.(2) For all N ∈ N ,

(〈N⊥vi, N

⊥vj〉)≤(〈N⊥ui, N

⊥uj〉),

where N⊥ denotes I −N .

These results are not immediate corollaries of Theorem 3.2 because, for ageneral nest N , AlgN is not of the form H∞(E). However, when N is finite, AlgNis a Hardy Algebra by Example 2.8. In this case, the conclusions are fairly straightforward computations. The case of general nests is then handled by approximationtechniques along the lines of [26] and [9]. Full details may be found in [35, Theorem6.8 and Corollary 6.9].

4. Schur class operator functions and realization

In this section we relate the complete positivity condition of Theorem 3.2 to theconcept of a Schur class function. As mentioned in the introduction, this may beviewed as part of a general program to find equivalences between canonical modeltheory, “non-commutative” systems theory and scattering theory. The results be-low are proved in [37].

We start with the following definition.

Definition 4.1. Let S be a set, A and B be two C∗-algebras and write B(A, B) forthe space of bounded linear maps from A to B. A function

K : S × S → B(A, B)

will be called a completely positive definite kernel (or a CPD-kernel ) if, for allchoices of s1, . . . , sk in S, the map

K(k) : (aij) → (K(si, sj)(aij))

from Mk(A) to Mk(B) is completely positive.

This concept of CPD-kernels was studied in [16] (see, in particular, Lemma3.2.1 there for conditions on K that are equivalent to being a CPD-kernel).

Definition 4.2. Let E be a Hilbert space and Z : D((Eσ)∗)→ B(E) be a B(E)-valuedfunction. Then Z is said to be a Schur class operator function if

K(η∗, ζ∗) = (id−Ad(Z(η∗), Z(ζ∗)) (id− θη,ζ)−1

is a CPD-kernel on D((Eσ)∗). (We use here the notation set for Theorem 3.2).

Note that, when M = E = B(E) and σ is the identity representation of B(E)on E , σ(M)′ is CIE , Eσ is isomorphic to C and D((Eσ)∗) may be identified withthe open unit ball D of C. In this case the definition above recovers the classicalSchur class functions. More precisely, these functions are usually defined as analyticfunctions Z from D into the closed unit ball of B(E) but it is known that this isequivalent to the positivity of the Pick kernel kZ(z, w) = (I − Z(z)Z(w)∗)(1 −zw)−1. The argument of [35, Remark 5.4] shows that the positivity of this kernelis equivalent, in this case, to the condition of Definition 4.2.

Note that it follows from Theorem 3.2 that every operator in the closed unitball of H∞(E) determines (by point evaluation) a Schur class operator function.In fact we have the following result whose proof may be found in [37].

Theorem 4.3. ([37]) Let E be a W ∗-correspondence over a von Neumann algebraM and let σ be a faithful normal representation of M on a Hilbert space E. For afunction Z : D((Eσ)∗)→ B(E), the following conditions are equivalent.

(1) Z is a Schur class operator function.(2) There is an X in the closed unit ball of H∞(E) such that X(η∗) = Z(η∗) for

all η ∈ D(Eσ).(3) (Realization) There is a Hilbert space H, a normal representation τ of N :=

σ(M)′ on H and operators A, B, C and D such that(i) A ∈ B(E),B ∈ B(H, E), C ∈ B(E , H) and D ∈ B(H, Eσ ⊗H).(ii) A, B, C and D intertwine the actions of N (on the relevant spaces).(iii) The operator

(A BC D

EEσ ⊗H

is a coisometry.(iv) For every η ∈ D(Eσ),

Z(η∗) = A + B(I − L∗ηD)−1L∗

where Lη : H → Eσ ⊗H is defined by Lηh = η ⊗ h.

Note that X in part (2) of the Theorem is not necessarily unique. (Although,as shown in [37], it is possible to choose σ such that the choice of X will be unique).

One may apply the techniques developed (in [37]) for the proof of the Theo-rem 4.3 to establish the following extension result.

Proposition 4.4. Every function defined on a subset Ω of the open unit ball of(Eσ)∗ with values in some B(E) such that the associated kernel (defined on Ω×Ω)is a CPD-kernel may be extended to a Schur class operator function (defined onall of D((Eσ)∗)).

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Paul S. MuhlyDepartment of MathematicsUniversity of IowaIowa City, IA 52242, USAe-mail: muhly@math.uiowa.edu

Baruch SolelDepartment of MathematicsTechnion32000 Haifa, Israele-mail: mabaruch@techunix.technion.ac.il

Notes on Generalized Lemniscates

Mihai Putinar

Abstract. A series of analytic and geometric features of generalized lemnis-cates are presented from an elementary and unifying point of view. A novelinterplay between matrix theory and elementary geometry of planar alge-braic curves is derived, with a variety of applications, ranging from a classicalFredholm eigenvalue problem and Hardy space estimates to a root separationalgorithm.

Mathematics Subject Classification (2000). Primary 47A56, 47A48; Secondary15A22, 14P05.

Keywords. lemniscate, rational embedding, determinantal variety, Schwarz re-flection, quadrature domain, Hardy space.

1. Introduction

The object of study in this article is the level set, with respect to a Hilbert spacenorm, of the resolvent of a matrix localized at a vector. We call these sets gen-eralized lemniscates in analogy with the classical lemniscates, that is level sets ofthe modulus of a polynomial. The latter class of domains is well known for its ap-plications to approximation theory and potential theory, see for instance [10, 22]and [30].

The origin of this study goes back to some determinantal functions relatedto the spectral theory of hyponormal operators. More specifically, if T ∈ L(H)is a linear bounded operator acting on the Hilbert space H and the commutator[T ∗, T ] = ξ ⊗ ξ is non-negative and rank-one, then the infinite determinant:

det[(T ∗ − z)−1(T − z)(T ∗ − z)(T − z)−1] =

1− ‖(T ∗ − z)−1ξ‖2was instrumental in understanding the fine structure of these operators. Its prop-erties were first investigated from the point of view of perturbation theory of

Paper supported by the National Science Foundation Grant DMS 0100367.

244 M. Putinar

symmetric operators and of scattering theory, see [25], [6]; for more references andlater related works see [24] and [28].

We have remarked in [27] that all rank-one self-commutator operators T asabove, subject to the additional condition

∞∨

T ∗ξ <∞,

are in a natural correspondence to Aharonov and Shapiro’s quadrature domains([2]). In particular, the spectrum of these finite type operators is precisely givenby the rational equation:

1− ‖(A− z)−1ξ‖2 < 0,

where A is the finite matrix obtained by compressing T ∗ to∨∞

k=0 T ∗ξ. This isa generalized lemniscate, in the terminology adopted below. This very correspon-dence between planar algebraic curves and matrices proved to be fruitful for betterunderstanding the nature of the defining equation of the boundary of a quadraturedomain (already investigated by Gustafsson [14]).

An effective exact reconstruction algorithm of a quadrature domains from apart of its moments was also derived from the same observation, [12]. Althoughclosely related to generalized lemniscates, we will not expand here these ideas.

The class of generalized lemniscates has emerged from such concepts andcomputations. We have tried below to simplify the access to these planar domainsand to make it independent of any sophisticated theory of semi-normal operators.Most of the operator theory and approximation theory aspects as well as impor-tant connections to extremal moment problems are left aside. They are partiallyexplained in the recent survey article [28], or in the more technical papers citedthere.

One of the aims of this essay is to connect in simple terms a variety of ideasof linear algebra, realization theory of linear systems, algebraic geometry and someclassical analysis on planar domains. We are well aware that this is only a firststep. The material below was freely borrowed and compiled into another form froma series of recent articles published in the last five years: [5], [18], [26], [29]. Thiswas done with the hope that the entire is more than its parts.

2. Realization theory

In this section we link the specific form of the algebraic equation of a generalizedlemniscate to a Hilbert space realization of it. This is in accord to the well-knownmatrix realization of transfer functions in linear systems theory.

Q(z, z) =d∑

αjkzjzk,

Notes on Generalized Lemniscates 245

be a Hermitian polynomial in (z, z), that is αjk = αkj , 1 ≤ j, k ≤ d. We willassume that the leading coefficient is non-zero, and normalized:

αdd = 1,

and we denote

P (z) =

αjdzj.

|P (z)|2 −Q(z, z) =d∑

[αjdαdk − αjk]zjzk.

The following result, proved in [17], can be taken as a starting point for ourdiscussion.

Theorem 2.1. The following conditions are equivalent:a) The matrix A(α) = (αjdαdk − αjk)d−1

j,k=0 is strictly positive definite;

b) There exists a linear transformation A of Cd with a cyclic vector ξ so that

P (A) = 0 andQ(z, z)

|P (z)|2 = 1− ‖(A− z)−1ξ‖2; (2.1)

c) There exist polynomials Qk(z) of degree k (exactly), 0 ≤ k < d, with theproperty

Q(z, z) = |P (z)|2 −d−1∑

|Qk(z)|2. (2.2)

In c), the Qk’s are uniquely determined if the leading coefficients are requiredto be positive.

Proof. Below we simply sketch the main steps of the proof. More details are in-cluded in [17].

a)⇒ b). Assume that A(α) is positive definite. Then there exist linearly indepen-dent vectors vk ∈ C

d, 0 ≤ k < d, satisfying:

〈vj , vk〉 = αjdαdk − αjk,

and consequently

|P (z)|2 −Q(z, z) = ‖V (z)‖2, (2.3)

where V (z) =∑d−1

j=0 vjzj is a vector-valued polynomial.

It follows that R : P1 → Pd, defined in terms of homogeneous coordinates inPd by R(z) = (P (z) : V (z)), is a rational map of degree d such that the image ofR spans Pd. Elementary arguments of linear algebra imply then the existence of amatrix A ∈ L(Cd) with minimal polynomial P (z) and cyclic vector ξ, such that:

V (z) = P (z)(A− z)−1ξ.

This proves assertion b).

246 M. Putinar

b)⇒ c). To achieve the decomposition (2.2) we orthonormalize the vectorsξ, Aξ, . . . , Ad−1ξ:

e0 =ξ

‖ξ‖ ,

e1 =Aξ − 〈Aξ, e0〉e0

‖ . . . ‖ ,

e2 =A2ξ − 〈A2ξ, e1〉e1 − 〈A2ξ, e0〉e0

‖ . . . ‖ ,

etc. Equivalently,

ξ = ‖ξ‖e0 = c0e0 (c0 > 0),

Aξ = c1e1 + 〈Aξ, e0〉e0 (c1 > 0),

A2ξ = c2e2 + 〈A2ξ, e1〉e1 + · · · (c2 > 0),

and so on. By rearranging the terms we obtain:

−P (z)(A− z)−1ξ = (P (A)− P (z))(A− z)−1ξ = T0(z)Ad−1ξ + · · ·+ Td−1(z)ξ

= T0(z)(cd−1ed−1 + 〈Ad−1ξ, ed−2〉ed−2 + · · · )+T1(z)(cd−2ed−2 + 〈Ad−2ξ, ed−3〉ed−3 + · · · ) + · · ·+ Td−1(z)c0e0

= cd−1T0(z)ed−1 + (cd−2T1(z) + 〈Ad−1ξ, ed−2〉T0(z))ed−2 + · · ·+(c0Td−1(z) + 〈Aξ, e0〉Td−2(z) + · · · )e0

= Q0(z)ed−1 + Q1(z)ed−2 + · · ·+ Qd−1(z)e0,

Qk(z) = cd−1−kTk(z) + O(zk−1).

Hence Qk(z) is a polynomial of degree k with leading coefficient cd−1−k > 0, and(2.2) now follows by inserting the above expression for P (z)(A − z)−1ξ into (2.1)and using that the ej are orthonormal.

c)⇒ a). If assertion c) is assumed to be true, then the vector-valued polynomial

V (z) = (Q0(z), Q1(z), . . . , Qd−1(z))

satisfies (2.3). Expanding V (z) along increasing powers of z gives V (z)=∑d−1

j=0 vjzj

where the vj are linearly independent vectors. Then (2.3) shows that A(α) is astrictly positive Gram matrix (associated to the vectors vj). Hence A(α) is strictlypositive definite, proving a).

It remains to prove the uniqueness of the decomposition (2.2). For this weobserve that there exists a simple algorithm of finding the polynomials Qk. Indeed,first observe that the coefficient of zd in Q(z, z) is P (z). Hence the polynomialFd−1(z, z) = |P (z)|2 − Q(z, z) has degree d − 1 in each variable. By assumptionthe coefficient γ1 of zd−1zd−1 in Fd−1 is positive, so that:

Fd−1(z, z) = γ11/2zd−1Qd−1(z) + O(zd−2, zd−2).

Therefore the polynomial Qd−1(z) is determined by Fd−1(z, z).

Proceeding by descending recurrence in k, (k < d − 1) we are led to thepolynomial

Fk(z, z) = Fk+1(z, z)− |Qk+1(z)|2which has as leading term a positive constant γk times zkzk. Then necessarily

Fk(z, z) = γk1/2zkQk(z) + O(zk−1, zk−1).

Thus Qk(z) is determined by Fk(z, z). And so on until we end by setting F0(z, z) =γ0 = |Q0(z, z)|2 > 0.

Definition 2.2. A generalized lemniscate is a bounded open set Ω of the complexplane, given by the equation:

Ω = z ∈ C; ‖(A− z)−1ξ‖ > 1 ∪ σ(A),

where A ∈ L(Cd) is a linear transformation and ξ is a cyclic vector of it, orequivalently:

Ω = z ∈ C; |P (z)|2 −d−1∑

|Qk(z)|2 < 0,

with polynomials P, Qk subject to the degree conditions: degP = d, degQk =k, 0 ≤ k ≤ d− 1.

Throughout this article the term generalized lemniscate refers both to therespective algebraic curve and the domain surrounded by it.

Henceforth we call the pair (A, ξ) the linear data of a generalized lemniscate.At this point we can easily make the link to the theory of determinantal curves

due to Moshe Livsic, Kravitsky, Vinnikov and their school. Specifically, startingwith a matrix A ∈ L(Cd) as above and a cyclic vector ξ of it, we can produce alinear pencil of matrices having the determinant equal to the polynomial Q above.Indeed,

∣∣∣∣ξ〈·, ξ〉 A− zA∗ − z I

∣∣∣∣ =∣∣∣∣

ξ〈·, ξ〉 − (A− z)(A∗ − z) A− z0 I

∣∣∣∣ =

det[ξ〈·, ξ〉 − (A− z)(A∗ − z)] =

|det(A− z)|2det[(A− z)−1ξ〈·, (A − z)−1ξ〉 − I] =

(−1)d|det(A− z)|2[1− ‖(A− z)−1ξ‖2].Thus we can state

Proposition 2.3. A generalized lemniscate Ω with linear data (A, ξ) is given by thedeterminantal equation:

Ω = z ∈ C; (−1)d

∣∣∣∣ξ〈·, ξ〉 A− zA∗ − z I

∣∣∣∣ > 0. (2.4)

We refer to [21], [35], [36] and the monograph [23] for the theory of deter-minantal curves. Again, we do not expand here the predictable implications offormula (2.4). Some of them have been considered by Alex. Shapiro [33].

248 M. Putinar

3. The rational embedding

The realization of a generalized lemniscate as the level set of the resolvent of amatrix has immediate geometric interpretations and consequences. One of them isthe derivation of a canonical rational embedding in an affine, or projective complexspace. Full proofs and a more detailed analysis of these aspects are contained in[17] Sections 4 and 5.

Henceforth we denote by C the Riemann sphere (that is the compactificationof the complex plane by one point at infinity). Equivalently, this is the projective

space of dimension one: C = P1(C). The projective space of dimension d will bedenoted by Pd(C) or simply Pd.

Let d be a positive integer, d > 1, let A be a linear transformation of Cd,

and assume that ξ ∈ Cd is a cyclic vector for A. Let us denote by:

R(z) = (A− z)−1ξ, z ∈ C \ σ(A),

the resolvent of A, localized at the vector ξ.

Lemma 3.1. The map R : C\σ(A) −→ Cd is one to one and its range is a smooth

complex curve.

A complete proof is contained in [17] Lemma 4.1. The main idea is to considerthe resolvent equation:

R(z)−R(w) = (z − w)(A − z)−1(A− w)−1ξ, z, w ∈ C \ σ(A).

Thus R(z) − R(w) = 0 for z = w. For the point at infinity we have R(∞) = 0 =R(z), for z ∈ C \ σ(A).

Moreover, the same resolvent equation shows that:

R′(z) = (A− z)−1R(z) = 0,

and similarly for the point at infinity we obtain:

limt→0

dtR(1/t) = − lim

t→0[t−2(A− t−1)−2ξ] = −ξ = 0.

Actually we can pass to projective spaces and complete the above curve asfollows. Let us denote by (z0 : z1) the homogeneous coordinates in P1, and by(u0 : u1 : . . . : ud) the homogeneous coordinates in Pd. Let z = z1/z0 in the affinechart z0 = 0 and w = (u1/u0, . . . , ud/u0) in the affine chart u0 = 0.

Let P (z) = det(A−z), so that P (z) is a common denominator in the rationalentries of the map R(z). Let us define, as in the preceding section, the function:

q(z, A)ξ = P (z)R(z) = (P (z)− P (A))R(z),

and remark that q(z, A) is a polynomial in z and A, of the form:

q(z, A) = −Ad−1 + O(Ad−2).

Actually we need for later use a more precise form of the polynomial q(z, A).We pause here to derive it by a series of elementary computations.

We have

P (w) − P (z)

w − z=

αkwk − zk

w − z=

k−1∑

zk−j−1wj =d−1∑

αkzk−j−1)wj =

T0(z)wd−1 + T1(z)wd−2 + · · ·+ Td−1(z),

where αd = 1 and

Tk(z) = αdzk + αd−1z

k−1 + · · ·+ αd−k+1z + αd−k.

Note that T0(z) = 1.Therefore we obtain, as in the previous section:

−q(z, A) = T0(z)Ad−1 + T1(z)Ad−2 + · · ·+ Td−1(z). (3.1)

Since ξ is a cyclic vector for A and dim∨∞k=0Akξ = d, we infer that q(z, A)ξ =

0 for all z ∈ C. In addition, for an eigenvalue λ of A (multiple or not), we have:

(A− λ)q(λ, A)ξ = P (λ)ξ = 0,

therefore q(λ, A)ξ is a corresponding (non-trivial) eigenvector.At this point we can define the completion of the map R as follows:

R(z0 : z1) =

(P (z1/z0) : q(z1/z0, A)ξ), z0 = 0,

(1 : 0 : . . . : 0), z0 = 0.(3.2)

By putting together these computations one obtains the following result.

Lemma 3.2. The map R : P1 −→ Pd is a smooth embedding, that is, R is one toone and its image is a smooth projective curve.

Note that R(P1) is a smooth rational curve of degree d in Pd and the rationalmap R has degree d. According to a classical result in algebraic geometry, R(P1)is projectively isomorphic to the rational normal curve of degree d in Pd obtainedas the range of the Veronese embedding

(z0 : z1) −→ (z0d : z0

d−1z1 : . . . : z1d).

See for details [13] pg. 178.Actually the cyclicity condition on ξ can be dropped, because the resolvent

(A− z)−1ξ has values in the cyclic subspace generated by ξ. Therefore, as a con-clusion of these computations we can state the following result.

Theorem 3.3. Let A be a linear transformation of Cd and let ξ be a non-zero vector

of Cd. Then the map R(z) = (A− z)−1ξ extends to a rational embedding:

R : P1 −→ Pd.

The range of R is contained in a linear subspace E of Pd of dimension equal todim ∨∞k=0 Akξ and the values R(z) span E as a linear space.

250 M. Putinar

Above, and throughout this note, by embedding we mean a (rational) mapwhich separates the points and the directions at every point. In particular thisimplies that R(P1) is a smooth rational curve.

Let us focus now on the geometry of the generalized lemniscate:

Ω = z ∈ C; ‖(A− z)−1ξ‖ > 1 ∪ σ(A).

The singular points a in the boundary of the bounded domain Ω are given by theequation 〈R′(a), R(a)〉 = 0. The proofs above show that ‖R′(a)‖ = 0, and on theother hand the Hessian H(a) at a of the defining equation ‖R(z)‖2 = 1 is:

H(a) =

(〈R′(a), R′(a)〉〈R′′(a), R(a)〉〈R(a), R′′(a)〉〈R′(a), R′(a)〉

In particular rankH(a) ≥ 1, which shows that a is either an isolated point ora singular double point of ∂Ω.

Our next aim is to study the reflection in the boundary of the domain Ωdefined above. More precisely, for a point s ∈ P1(C) we consider the multi-valuedSchwarz reflection in ∂Ω as the set of solutions z = r1(s), . . . , rd(s) of the equation:

〈R(s), R(z)〉 = 1. (3.3)

Proposition 3.4. The multi-valued reflection s → (rj(s))j=1d

satisfies:

a) All rj(s) ∈ Ω, 1 ≤ j ≤ d, for s ∈ P1(C) \ Ω;b) For an appropriate numbering of the rj ’s, r1(s) = s and rj(s) ∈ Ω, 2 ≤ j ≤ d,

for s ∈ ∂Ω.

Proof. Indeed, ‖R(s)‖ < 1 whenever s does not belong to Ω. Therefore ‖R(z)‖ > 1for every solution z of the equation (3.3). For s ∈ ∂Ω we obtain ‖R(s)‖ = 1, henceone solution of (16), say r1, satisfies r1(s) = s and all other solutions z satisfynecessarily ‖R(z)‖ > 1.

A rigidity result of the above rational embedding in the complement of thesphere, compatible to the reflections in the boundaries, is discussed in detail in[17] Section 5.

This is an appropriate moment to recall the definition of a quadrature domain,in the sense of Aharonov and Shapiro [2].

Definition 3.5. A bounded planar open set Ω is a quadrature domain if it isbounded by a real algebraic curve and the function z → z extends continuouslyfrom z ∈ ∂Ω to a meromorphic function in Ω.

This means, in line with Proposition 3.4, that one determination, say S1(z), ofthe Schwarz reflection satisfies S1(z) = z, z ∈ ∂Ω, and it does not have ramificationpoints inside Ω. Necessarily, it will have d poles a1, . . . , ad, d ≥ 1, there. Thenumber d is called the order of a quadrature domain and the terminology comesfrom the simple observation that, in this case there are d weights c1, c2, . . . , cd withthe property: ∫

f(z)dArea(z) = c1f(a1) + · · ·+ cdf(ad), (3.4)

for every integrable analytic function f in Ω, see [2] and [34]. If multiple poles ofS1(z) occur, then higher order derivatives of f evaluated there must be consid-ered, correspondingly. As a matter of fact, the existence of the quadrature formula(3.4), valid for all integrable analytic functions is equivalent to the above defini-tion of a quadrature domain. Since their discovery thirty years ago, [2] and [32],these domains have revealed a series of remarkable features, related to phenomenaof function and potential theory, fluid mechanics, moment problems and partialdifferential equations, [34].

The case d = 1 corresponds to a disk. By abuse of language we allow non-connected sets in the above definition. Thus, a disjoint union of disks is also aquadrature domain.

Quadrature domains are relevant for this survey because of the followingresult (which as a matter of fact was the origin of the whole project).

Theorem 3.6. A quadrature domain of order d is a generalized lemniscate of de-gree (d, d).

The original proof of this theorem was based on non-trivial results of thetheory of semi-normal operators ([27]). An elementary way to prove it was recentlydescribed in [18].

4. Fredholm eigenvalues

In this section we use a simple geometric feature of the multi-valued Schwarzreflection in the boundary of a generalized lemniscate and prove that a classicalproblem in potential theory does not have non-trivial solutions on this class ofdomains. It is worth mentioning that the similar picture on classical lemniscatesis quite different.

Let Ω be a bounded, simply connected domain of the complex plane andassume that the boundary of Ω is smooth.

Let u ∈ C(∂Ω) be a continuous function. The double layer potential of u,with respect to Ω is the harmonic function:

D(u)(z) =1

u(ζ)d arg(ζ − z).

An elementary computation shows that:

D(u)(z) = ℜ[

ζ − zdζ

u(ζ)ℜ[dζ

2πi(ζ − z)].

Whenever z belongs to Ω, respectively to its exterior, we mark the function Dby an index Di(z) respectively De(z). It is known that Di is a continuous function

on the closure Ω and that De is continuous on the Riemann sphere C minus Ω.

252 M. Putinar

Moreover, at each boundary point σ ∈ ∂Ω we have representations:

Di(u)(σ) =1

2u(σ) +

2KΩ(u)(σ),

De(u)(σ) = −1

2u(σ) +

2KΩ(u)(σ).

Thus, the jumping formula:

Di(u)(σ) −De(u)(σ) = u(σ), σ ∈ ∂Ω,

holds. Remark also that for the constant function u = 1 we have Di(1) = 1 andDe(1) = 0, hence KΩ(1) = 1.

The linear continuous transformation KΩ : C(∂Ω) −→ C(∂Ω) is the classicalNeumann-Poincare singular integral operator (in two real variables). In generalthis operator has better smoothness properties than the Hilbert transform.

Carl Neumann’s approach to the Dirichlet problem ∆f = 0 in Ω, f |∂Ω = u,was essentially the following:

Solve the equation 1/2(I + KΩ)v = u and then set f = Di(v) as a doublelayer potential of v.

Thus, knowing that the operator I + KΩ : C(∂Ω) −→ C(∂Ω) is invertiblesolves the Dirichlet problem for an arbitrary (originally convex) domain. Laterthis idea was applied by Poincare, Fredholm, Carleman to more general classes ofdomains, in any number of variables. For (historical) comments we refer to [20].

Particularly relevant for potential and function theory are the solutions ofthe Fredholm eigenvalue problem:

KΩu = 0.

They correspond to non-trivial solutions of the following matching problem: findanalytic functions, continuous up to the boundary f(z), z ∈ Ω, g(z), z ∈ C \Ω, g(∞) = 0, such that:

f(ζ) = g(ζ), ζ ∈ ∂Ω.

Non-trivial solutions exist on the lemniscates Ω = z ∈ C; |r(z)| < 1,where r is a rational function satisfying r(∞) =∞. Indeed, it is clear that f = rand g = 1/r solve the above matching problem. For more details see [9].

The following result is reproduced (with its entire proof) from [29].

Theorem 4.1. Let Ω be a connected and simply connected generalized lemniscateof degree d ≥ 2. Then kerKΩ = 0.

Proof. The proof is an adaptation of an argument, based on analytic continuation,from [9].

Ω = z ∈ C; ‖(A− z)−1ξ‖ > 1 ∪ σ(A),

as in the preceding sections, where A is a d× d matrix with cyclic vector ξ.

We denote by S(z) the d-valued Schwarz reflection, defined by the equation:

〈(A − z)−1ξ, (A− S(z))−1ξ〉 = 1, z ∈ C. (4.1)

Let a ∈ ∂Ω be a non-ramification point for S. Since ‖(A− a)−1ξ‖ = 1, thenall local branches Sj of S satisfy

‖(A− Sj(a))−1ξ‖ ≥ 1, 1 ≤ j ≤ d.

Denote S1(a) = a, so that ‖(A − S1(a))−1ξ‖ = 1. Since every other branch hasdifferent values Sj(a) = S1(a) we infer that:

‖(A− Sj(a))−1ξ‖ > 1, 2 ≤ j ≤ d.

Therefore Sj(a) ∈ Ω, 2 ≤ j ≤ d, just as we proved in Proposition 3.4.

Similarly, for every exterior point b of Ω we find that all branches satisfy

Sj(b) ∈ Ω, 1 ≤ j ≤ d.

If the function S(z) has no ramification points on C \ Ω, then Ω is the com-plement of a quadrature domain (in the terminology of Aharonov and Shapiro)and the statement is assured by Theorem 3.19 of [9].

On the contrary, if the algebraic function S(z) has ramification points in

C \Ω, then, by repeating the main idea in the proof of the same Theorem 3.19 of[9], there exists a Jordan arc α starting at and returning to a point a ∈ ∂Ω, such

that α \ a ⊂ C \Ω, having the property that the analytic continuation of S1(z)

along this arc returns to a at another branch, say S2(z), with S2(a) ⊂ Ω.

Assume by contradiction that the matching problem on ∂Ω has a non-trivialsolution: f ∈ A(Ω), g ∈ A(C \ Ω), g(∞) = 0, f(ζ) = g(ζ), ζ ∈ ∂Ω. Hence

f(S1(ζ)) = g(ζ), ζ ∈ ∂Ω.

Let us now let the point ζ traverse the curve α, and after returning to thepoint a, describe ∂Ω once. By analytic continuation along this path, the matchingcondition continues to hold, and becomes:

f(S2(ζ)) = g(ζ), ζ ∈ ∂Ω.

But now S2(ζ), ζ ∈ ∂Ω, remains “trapped” into a compact subset M ⊂ Ω.Thus, putting together the latter identities, we obtain:

max∂Ω|f | = max

∂Ω|g| = max

M|f |.

By the maximum principle, the function f should be a constant. Then g is aconstant, too. But g(∞) = 0, that is f and g are identically zero, a contradiction.

Apparently there are no other examples of domains Ω with non-zero elementsin the kernel of KΩ other than the level sets of moduli of rational functions.

254 M. Putinar

5. Root separation

The aim of this section is to show that the classical method of separating rootsdue to Hermite and later refined by Routh, Hurwitz, Schur, Cohn, Lienard andChipard, and many other authors, can be combined with the specific form of theequation of a generalized lemniscate Ω, to obtain matricial criteria for the rootlocation of an arbitrary polynomial, with respect to the domain Ω. The technicaltools we invoke are elementary: the expression of the defining equation of the do-main will be combined with some simple Hilbert space remarks; then Hermite’sseparation method (with respect to the half-space), or Schur’s criterion (with re-spect to the disk) will be used. Along the same lines, more powerful methods basedon the modern theory of the Bezoutiant can be exploited, [33].

ΩR = z ∈ C; Q(z, z) = 1− ‖R(z)‖2 < 0,be a generalized lemniscate associated to a rational function

R : C −→ Pd, R(z) = (A− z)−1ξ,

as considered in Section 3.

To start we remark that a point α belongs to C\ΩR if and only if, by defini-tion, ‖R(α)‖ < 1. In its turn, the latter condition is equivalent to |〈R(α), v〉| < 1for all unit vectors v ∈ C

d, or at least for the vectors of the form v = R(β)/‖R(β)‖,where β is not a pole of at least one, or a common zero of all, entries of R. Notethat in the last formula R(α) depends rationally on the root α. Schur’s criterionof separation with respect to the unit disk can then be applied, see for instance[3]. As a matter of fact the proof below allows us to consider slightly more generalrational functions.

Theorem 5.1. Let R : C −→ Pd be a rational function satisfying limz→∞ R(z) = 0,and let Π ⊂ C be the set of all poles and common zeroes of R.

Then a monic polynomial f has all its roots α1, . . . , αn in the open set C\ΩR

if and only if, for every β ∈ C \Π, the polynomial

Fβ(X) =

(X − 1−Q(αj , β)√1−Q(β, β)

) (5.1)

has all its roots in the unit disk.

Proof. Let f be a polynomial with all roots α1, α2, . . . , αn in the set C \Ω. Then‖R(αi)‖ < 1 for all i, 1 ≤ i ≤ n. Consequently, if β ∈ C \Π we obtain:

|〈R(αi), R(β)〉| < ‖R(β)‖,which is exactly condition in the statement.

Conversely, if the condition holds for all β ∈ C \ Π, then by reversing thepreceding argument we find that ‖R(αi)‖ < 1, 1 ≤ i ≤ n.

Note that the polynomial Fβ(X) is a symmetric function of the roots αj , 1 ≤j ≤ n, hence its coefficients are rational functions of c1, . . . , cn. Therefore Schur’scriterion will involve only rational combinations of the coefficients c1, . . . , cn.

Specifically, if F (z) is a polynomial with complex coefficients of degree d, wedefine the associated polynomials:

F (z) = F (z), F ∗(z) = zdF (1

then the inertia of the bilinear form:

GF (X, Y ) =F ∗(X)F

∗(Y )− F (X)F (Y )

1−XY,

gives full information about the root location of F with respect to the unit disk.That is, if GF has d+ positive squares and d− negative squares, then the polynomialF has exactly d+ roots in the unit disk, d− roots outside the closed disk, andd− d+ − d− roots lie on the unit circle.

Variations of the above result are readily available: for instance one can re-place the rational map R(z) by a polynomial map, or instead of Fβ one can considerthe polynomial involving the squares of the roots of Fβ , and so on.

If we want to have more information about the root location of the polynomialf(z) = (z − α1) · · · (z − αn), then the scalar products 〈R(αj), v〉, with v a fixed

unit vector, can be replaced by an expression such as 〈R(αj), r(αj)〉, where r(z)is a vector-valued rational function, of norm less than one in a large disk, wherethe roots are first estimated to be. Then, by counting parameters, the degree of rcan be chosen to be dependent on n, the degree of f .

In order to state such a result, we make the following notation: for r : C −→C

d a vector-valued rational map, let

Fr(X) =

(X − 〈R(αj), r(αj)〉). (5.2)

Note that this polynomial in X depends rationally on the entries αj and issymmetrical in them. We also denote below by |A| the cardinality of the set A andby V (h) the zero set of the function h. We have then

Corollary 5.2. In the conditions of Theorem 2.2, let U = tD be a disk centered atthe origin, that contains all the roots of the polynomial f(z).

Let r : C −→ Cd be a rational map of degree less than or equal to s on each

entry, satisfying ‖r(z)‖ ≤ 1, z ∈ U, where we assume: (2s + 1)d > dn.

Then, with the above notations, we have:

|V (f) ∩ ΩR| = maxr|V (Fr) \D|,

|V (f) \ ΩR| = minr|V (Fr) ∩D|.

256 M. Putinar

Proof. Let d+ = |V (f) \ ΩR| and d− = |V (f) ∩ ΩR|.Since ‖r(αj)‖ ≤ 1 for all j, 1 ≤ j ≤ n, we have 〈r(αj), r(αj)〉) ≤ ‖R(αj)‖.

Therefore, the polynomial Fr has at least d+ zeroes in the unit disk and at mostd− zeroes outside its closure.

To see that these bounds are attained, we remark that, due to the degreeassumption, the map r(z) can be chosen to have prescribed values at every pointαi, 1 ≤ j ≤ n. Thus we can choose the values r(αj) so that〈R(αj), r(αj)〉 = ‖R(αj)‖.

Going into another direction, it is easy to establish sufficient criteria for theroots of the polynomial f to be all contained in the exterior of Ω. Let us denotethe defining rational map by R(z) = (R1(z), . . . , Rd(z)).

Corollary 5.3. In the conditions of Theorem 5.1, let ai, 1 ≤ i ≤ d, be positivenumbers satisfying a2

1 + a22 + · · ·+ a2

d = 1.

Define the polynomials:

Fi(X) =

(X − Ri(αj)

ai), 1 ≤ i ≤ d. (5.3)

If the roots of each Fi, 1 ≤ i ≤ d, are contained in the unit disk, then theroots of f are contained in C \ Ω.

Proof. It is sufficient to remark that, under the assumption for the roots of Fj , foreach fixed j, 1 ≤ j ≤ n, we have

‖R(αj)‖2 ≤d∑

‖Ri(αj)‖2 <

a2i = 1.

For more details and a couple of examples see [26].

6. The reproducing kernel

Let R(z) = (A − z)−1ξ be a vector-valued rational function attached as beforeto a matrix A and its cyclic vector ξ ∈ C

d. The complement of the associatedgeneralized lemniscate is a subset of the Riemann sphere

G = z ∈ C; ‖R(z)‖ < 1.As we have seen in Section 3, the map R : G −→ Bd is a smooth rational embeddingof G into the unit ball Bd of C

There are several reasons, simplicity and rationality being one of them, toconsider the positive definite kernel

K(z, w) =1

1− 〈R(z), R(w)〉 , z, w ∈ G.

The theory of such kernels is well understood but we will not invoke deep resultsabout them; see [1], [4] for further applications of these kernels. We confine our-selves to reproduce from [26] and [5] an identification, as topological vector spaces,between the Hardy space of G and the reproducing Hilbert space with kernel K.This identification holds under a generic smoothness assumption and has someinteresting function theoretic consequences.

Lemma 6.1. Assume the domain G is simply connected and with smooth boundary.Then there are positive constants C1, C2 such that

λjλk

1− zjzk≤

λjλk

1− 〈R(zj), R(zk)〉 ≤ C2

λjλk

1− zjzk,

for every N ∈ N and choice of points zj ∈ G, λj ∈ C, 1 ≤ j ≤ N.

The proof, based on the (Fredholm) analysis of the singular integral operator

f →∫

f(w)dw

1− 〈R(z), R(w)〉 ,

is contained in [5].By passing for instance to the unit disk via a conformal mapping one deduces

the following result.

Theorem 6.2. Let G = z ∈ C; ‖R(z)‖ < 1 be a smooth, simply connecteddomain. The Hilbert space with reproducing kernel K(z, w) coincides as a set, butnot isometrically in general, with the Hardy space of G.

This fact was used in [5] to prove that a bounded analytic function along ananalytic arc, smoothly attached to the unit sphere Bd, admits a bounded extensionto the Schur class of Bd. This is a slight improvement of the known results of ex-tending bounded analytic functions, from an attached disk, to a bounded analyticfunction defined on the whole ball.

7. Examples

The transition from the defining equation of a generalized lemniscate to its lineardata is not totally trivial. Few concrete examples of this sort are known in fulldetail. We list below a couple of such elementary computations (appearing in [17]).

7.1. Domains corresponding to a nilpotent matrix.

A simple and basic example of a generalized lemniscate, which in general is nota quadrature domain, can be obtained as follows. Let us consider the nilpotentmatrix A and the cyclic vector ξ:

⎛⎝

0 1 00 0 10 0 0

⎞⎠ , ξ =

⎛⎝

⎞⎠ ,

258 M. Putinar

where a, b, c are complex numbers, c = 0. A simple computation shows that:

‖(A− z)−1ξ‖2 = |az

z3|2 + | b

z2|2 + | c

Therefore the equation of the associated domain is:

|z|6 < |az2 + bz + c|2 + |bz2 + cz|2 + |cz2|2.According to Proposition 3.4, the multi-valued reflection in the boundary of

this domain maps the exterior completely into its interior.The rational embedding associated to this example is:

R(1 : z) = (−z3 : az2 + bz + c : bz2 + cz : cz2).

Similarly one can compute without difficulty the corresponding objects asso-ciated to a nilpotent Jordan block and an arbitrary cyclic vector of it. For instancethe nilpotent n×n-Jordan block and the vector ξ = (0, 0, . . . , 0,−1) give preciselythe Veronese embedding:

R(1 : z) = (zn : 1 : z : . . . : zn−2 : zn−1).

7.2. The Limacon

This class of curves exhaust all quadrature domains of order two, with a doublepoint.

Let us consider the conformal mapping z = w2 + bw, where |w| < 1 andb ≥ 2. Then it is well known that z describes a quadrature domain Ω of order 2,whose boundary has the equation:

Q(z, z) = |z|4 − (2 + b2)|z|2 − b2z − b2z + 1− b2 = 0.

The Schwarz function of Ω has a double pole at z = 0, whence the associated2× 2-matrix A is nilpotent. Moreover, we know that:

|z|4‖(A− z)−1ξ‖2 = |z|2‖(A + z)ξ‖2 = Q(z, z).

Therefore

‖(A + z)ξ‖2 = (2 + b2)|z|2 + b2z + b2z + b2 − 1,

or equivalently: ‖ξ‖2 = 2 + b2, 〈Aξ, ξ〉 = b2 and ‖Aξ‖2 = b2 − 1.Consequently the linear data of the quadrature domain Ω are:

(0 b2−1

(b2−2)1/2

), ξ =

(b2−1)1/2

( b2−2b2−1 )1/2

This shows in particular that the linear data (A, ξ) of a quadrature domainof order two, with a double node, is subject to very rigid conditions.

The associated rational embedding can easily be computed from the defini-tion:

R(1 : z) = (−z2 :b2

(b2 − 1)1/2z +

b2 − 1

(b2 − 1)1/2: (

b2 − 2

b2 − 1)1/2z).

7.3. Quadrature domains with two distinct nodes

a) Suppose that Ω is a quadrature domain with the quadrature distribution:

u(f) = af(0) + bf(1),

where we choose the constants a, b to be positive numbers. Then P (z) = z(z − 1)and

z(z − 1)(A− z)−1ξ = −Aξ + ξ − zξ.

Therefore the equation of the boundary of Ω is:

Q(z, z) = |z(z − 1)|2 − ‖Aξ − ξ + zξ‖2.Accordingly we obtain:

‖ξ‖2 =a + b

π, 〈Aξ, ξ〉 =

Let us denote ‖Aξ‖2 = c. Then the defining polynomial becomes:

Q(z, z) = |z(z − 1)|2 − π−1(a|z − 1|2 + b(|z|2 − 1))− c.

The constant c actually depends on a, b, via, for instance, the relation Area(Ω)= a + b, or, whenever a = b, the fact that Q(1/2, 1/2) = 0. The latter are calledspecial points of a quadrature domain and were studied in [15].

We can choose an orthonormal basis with respect to which we have:

(0 α0 1

), ξ =

The matricial elements α, β, γ are then subject to the relations:

|β|2 + |γ|2 = π−1(a + b), αβγ + |γ|2 = π−1b, |α|2|γ|2 + |γ|2 = c.

An inspection of the arguments shows that the above system of equationshas real solutions α, β, γ given by the formulas:

α2 =(πc− b)2

π(a + b)c− b2,

β2 =a−2

π(a− b) + π2c,

γ2 =π(a + b)c− b2

π(a− b) + π2c.

Let us remark that, if a = b > π/4, the constant c is effectively computable,as mentioned earlier, and becomes:

This again illustrates the special nature of the linear data (A, ξ) of a quad-rature domain. A simple computation shows that the corresponding canonicalembedding of the domain Ω is:

R(1 : z) = (z(z − 1) : β(1 − z)− αγ : γz).

260 M. Putinar

We remark that in both of the above examples, the matrix A and the vectorξ are uniquely determined, as soon as we require that A is upper triangular.

b) In complete analogy, we can treat the case of two nodes with equal weights asfollows.

Assume that the nodes are fixed at ±1. Hence P (z) = z2 − 1. The definingequation of the quadrature domain Ω of order two with these nodes is:

Q(z, z) = (|z + 1|2 − r2)(|z − 1|2 − r2)− c,

where r is a positive constant and c ≥ 0 is chosen so that either Ω is a union oftwo disjoint open disks (in which case c = 0), or Q(0, 0) = 0, as a special pointproperty [15]. A short computation yields:

Q(z, z) = z2z2 − 2rzz − z2 − z2 + α(r),

α(r) =

(1 − r2)2, r < 10, r ≥ 1

Exactly as in the preceding two situations, the identification

|P (z)|2(1− ‖(A− z)−1ξ‖2) = Q(z, z) (7.1)

leads to (for example) the following simple linear data:

( √2r0

), A =

⎛⎝

2r√1−α(r)√

1−α(r)√2r

⎞⎠ .

7.4. Domains with rotational symmetry

One of the best understood classes of quadrature domains, and by extension,generalized lemniscates, is that of domains Ω invariant under a finite group ofrotations

Ω = ǫΩ,

where ǫ is a primitive root of order n of unity.

Assume that the equation of Ω has the form:

Q(z, z) = |P (z)|2 −d−1∑

|Qk(z)|2,

with degP = d, degQk = k. In view of the uniqueness of the above decompositionof Q, see Theorem 2.1, we infer:

|P (ǫz)| = |P (z)|and

|Qk(ǫz)| = |Qk(z)|, 0 ≤ k ≤ d− 1.

Assuming for instance that n = d, that is the order of the domain matchesthe symmetry order, we obtain:

Q(z, z) = |zd − ad|2 −d−1∑

ck|zk|2,

where a ∈ C, ck > 0, 0 ≤ k ≤ d− 1.

Among these domains only those for which a = 0 are quadrature domains.Computations along these lines around the uniqueness question, whether the quad-rature data determine Ω, were carried out in [14], [15] and more recently in [7].

8. Disjoint unions of disks

In view of the preceding discussion, the constructive passage from an algebraicequation of a generalized lemniscate to its linear data is interesting and possiblyuseful for applications. We treat below a simple situation where the linear datacan be recurrently computed.

A generic class of quadrature domains with positive weights in the associatedquadrature formula is offered by the disjoint unions of disks. On fluid mechanicalgrounds any quadrature domain with positive weights has its roots in a disjointunion of disks. Although these sets are not connected, their equation is obviouslywithin reach. We show below, following [18], how to compute inductively the as-sociated linear data.

Lemma 8.1. Let Di = D(ai, ri), 1 ≤ i ≤ n, be disjoint disks and let

Q(z, w) =

[(z − ai)(w − ai)− r2i ],

be the polarized equation defining their union. Then the matrix (−Q(ai, aj))ni,j=1

is positive semidefinite.

Proof. Let Ω = ∪ni=1D(ai, ri). Since the union is disjoint, Ω is a quadrature domain

with nodes at a1, a2, . . . , an. Let P (z) be the monic polynomial vanishing at thesepoints. According to Theorem 3.6, one can write:

Q(z, w) = P (z)P (w)−d−1∑

|Qk(z)|2,

with polynomials Qk of exact degree k, respectively.

Q(ai, aj) = −d−1∑

Qk(ai)Qk(aj),

and this shows that the matrix in the statement is positive semidefinite.

262 M. Putinar

A more detailed analysis shows that the matrix (−Q(ai, aj))ni,j=1 is actually

strictly positive definite, see [18].We denote the same disjoint union of disks Ωn = ∪n

i=1D(ai, ri), and weconsider the addition of an external disjoint disk; let Ωn+1 = ∪n+1

i=1 D(ai, ri) be theenlarged set. At each stage we have a finite-dimensional Hilbert space K, a cyclicvector ξ ∈ K and an operator A ∈ L(K) which provide the linear data of thesesets. We write accordingly, the equation of a disjoint union of k disks as:

Qk(z, w) = Pk(z)Pk(w)[1− 〈(Ak − z)−1ξk, (Ak − w)−1ξk〉],where Ak ∈ L(Kk) has cyclic vector ξk, dim Kk = k, k = n, n + 1, and thepolynomial Pk has degree k and annihilates Ak.

Our aim is to understand the structure of the matrix An+1 and its cyclic vec-tor ξn+1 as functions of the previous data (An, ξn) and the new disk D(an+1, rn+1).

Henceforth we assume that the closed disks D(ai, ri) are still disjoint. In order tosimplify notation we suppress for a while the index n +1, e.g. a = an+1, r = rn+1

etc. The following computations are based on standard realization techniques inlinear systems theory.

Due to the multiplicativity of the defining equation for disjoint domains wefind:

[1− 〈(An − z)−1ξn, (An − w)−1ξn〉][1−r2

(z − a)(w − a)]

= 1− 〈(A− z)−1ξ, (A− w)−1ξ〉.Equivalently,

〈(An − z)−1ξn, (An − w)−1ξn〉+r2

(z − a)(w − a)

= 〈 r

z − a(An − z)−1ξn,

w − a(An − w)−1ξn〉+ 〈(A− z)−1ξ, (A− w)−1ξ〉.

Thus, for each z avoiding the poles, the norm of the vector

f(z) =

((An − z)−1ξn

rz−a

)∈ Kn ⊕C

equals that of the vector

g(z) =

z−a (An − z)−1ξn

(A− z)−1ξ

)∈ Kn ⊕K.

And moreover, the same is true for any linear combination

‖λ1f(z1) + · · ·+ λrf(zr)‖ = ‖λ1g(z1) + · · ·+ λrg(zr)‖.Because the span of f(z), z ∈ C, is the whole space Kn⊕C, there exists a uniqueisometric linear operator V : Kn⊕C −→ Kn⊕K mapping f(z) to g(z). We write,corresponding to the two direct sum decompositions

(B βC γ

where B : Kn −→ Kn, β ∈ Kn, C : Kn −→ Kn+1, γ ∈ K. Since V f(z) = g(z)for all z, we find by coefficient identification:

B = r(An − a)−1, β = (An − a)−1ξn.

The isometry condition V ∗V = I written at the level of the above 2 × 2matrix yields the identities:

⎧⎨⎩

r2(A∗n − a)−1(An − a)−1 + C∗C = I,

r(A∗n − a)−1(An − a)−1ξn + C∗γ = 0,

‖(An − a)−1ξn‖2 + ‖γ‖2 = 1.(8.1)

In particular we deduce that (A∗n − a)−1(An − a)−1 ≤ r−2 and since this

operator inequality is valid for every radius which makes the disks disjoint, we canenlarge slightly r and still have the same inequality. Thus, the defect operator

∆ = [I − r2(A∗n − a)−1(An − a)−1]1/2 : Kn −→ Kn (8.2)

is strictly positive.The identity C∗C = ∆2 shows that the polar decomposition of the matrix

C = U∆ defines without ambiguity an isometric operator U : Kn −→ K. SincedimK = dim Kn +1 we will identify K = Kn⊕C, so that the map U becomes thenatural embedding of Kn into the first factor. Thus the second line of the isometryV becomes

(C γ) =

(∆ d0 δ

): Kn ⊕C −→ Kn ⊕C = K,

where d ∈ Kn, δ ∈ C. We still have the freedom of a rotation of the last factor,and can assume δ ≥ 0. One more time, equations (8.1) imply

r (∆ξn −∆−1ξn),

δ = [1− ‖(An − a)−1ξn‖2 − ‖d‖2]1/2.(8.3)

From relation V f(z) = g(z) we deduce:(

∆ d0 δ

)((An − z)−1ξn

rz−a

)= (A− z)−1ξ.

This shows that δ > 0 because the operator A has the value a in its spectrum.At this point straightforward matrix computations lead to the following exact

description of the couple (A, ξ) = (An+1, ξn+1) (by restoring the indices):

An+1 =

(∆An∆−1 −δ−1∆(An − an+1)∆

−1d0 an+1

), ξ =

(∆−1ξn

−δrn+1

). (8.4)

It is sufficient to verify these formulas, that is:(

∆(An − z)∆−1 −δ−1∆(An − a)∆−1d0 a− z

)(∆ d0 δ

)((An − z)−1ξn

rz−a

(∆−1ξn

−δr

264 M. Putinar

And this is done by direct multiplication:

∆ξn + ∆(An − z)∆−1 rd

z − a−∆(An − a)∆−1 rd

z − a= ∆−1ξn,

which is equivalent to the known relation dr = ∆ξn −∆−1ξn.Summing up, we can formulate the transition laws of the linear data of a

disjoint union of disks.

Proposition 8.2. Let D(ai, ri), 1 ≤ i ≤ n + 1, be a disjoint family of closed disks,and let Ωk = ∪k

i=1D(ai, ri), 1 ≤ k ≤ n + 1.The linear data (Ak, ξk) of the quadrature domain Ωk can be inductively ob-

tained by the formula (8.4), with the aid of the definitions (8.2), (8.3).

Remark that letting r = rn+1 → 0 we obtain ∆ → I and d → 0, which isconsistent with the fact that Ωn+1 → Ω, in measure, in case such a limit domainΩ is given.

Acknowledgment

I would like to thank Daniel Alpay and Victor Vinnikov for sharing the joy of thegeneralized lemniscate game. I am much indebted to the anonymous referee for acareful and critical reading of the manuscript.

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266 M. Putinar

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Mihai PutinarMathematics DepartmentUniversity of CaliforniaSanta Barbara, CA 93106,USAe-mail: mputinar@math.ucsb.edu

One-Sided Tangential Interpolationfor Hilbert-Schmidt Operator Functionswith Symmetries on the Bidisk

M.C.B. Reurings and L. Rodman

Abstract. One-sided tangential interpolation problems for functions with sym-metries having values in the set of Hilbert–Schmidt operators and defined onthe bidisk are studied. General solutions are described as well as solutionswith the minimal scalar and operator-valued norms. Two types of symme-tries are considered: (a) componentwise symmetries that operate separatelyon each component of a general point in the bidisk; (b) interchange symmetrythat interchanges the two components of a general point in the bidisk. Appli-cations are made to multipoint tangential interpolation problems of specialform.

Mathematics Subject Classification (2000). 47A56, 47A57.

Keywords. Tangential interpolation, symmetries, Hilbert–Schmidt operators.

1. Introduction

The letters H and G, with or without subscripts, designate complex Hilbert spaces;all Hilbert spaces in the present paper are assumed to be separable, and all oper-ators are assumed to be bounded and linear. The inner product and the norm inH are denoted by 〈·, ·〉H and ‖ · ‖H, respectively. If X , Y are selfadjoint operatorson a Hilbert space, we write X ≥ Y or Y ≤ X to mean that X − Y is positivesemidefinite.

Denote by HSG→H the (separable) Hilbert space of Hilbert-Schmidt opera-

tors acting from G into H, and let HSG→H(Dd) be the Hardy space of HSG→H-valued functions which are analytic in the polydisk Dd, defined by

Dd = (z1, z2, . . . , zd) ∈ Cd : |zi| < 1, i = 1, 2, . . . , d.

Research of the first author was supported by the Netherlands Organization for Scientific Re-search (NWO). Research of the second author was supported in part by an NSF grant.

268 M.C.B. Reurings and L. Rodman

In this paper we focus on the bidisk D2. The space HSG→H(D2) is a separableHilbert space with the inner product

〈G, H〉HSG→H(D2) =1

(2π)2

∫ 2π

Trace(H(eit1 , eit2)∗G(eit1 , eit2)) dt1 dt2.

The space HSG→H(D2) is also a Hilbert module with respect to the trace classoperator-valued inner product

[G, H ]HSG→H(D2) =1

(2π)2

∫ 2π

H(eit1 , eit2)∗G(eit1 , eit2) dt1 dt2.

The Hilbert module property means, in particular, that

[GA, HB]HSG→H(D2) = B∗[G, H ]HSG→H(D2)A

for any choice of operators A, B : G → G. Note that every H ∈ HSG→H(D2) canbe written as a power series

H(z1, z2) =∞∑

j1,j2=0

Hj1,j2zj11 zj2

2 , (z1, z2) ∈ D2, (1.1)

where Hj1,j2 ∈ HSG→H are such that

‖H‖HSG→H(D2) :=

⎛⎝

∞∑

j1,j2=0

‖Hj1,j2‖2HSG→H

⎞⎠

In this paper we study certain tangential interpolation problems with thesolution set restricted to some subspace of HSG→H(D2). The subspace is specifiedby a given symmetry. The data set of these interpolation problems consists of aquintuple of operators

Ω = A1, A2, B1, B2, B−, (1.2)

where Ai : Hi → Hi(i = 1, 2), B1 : H → H1, B2 : H1 → H2, B− : G → H2 areoperators such that

σ(A1) ∪ σ(A2) ⊂ D. (1.3)

(Here and elsewhere σ(A) stands for the spectrum of an operator A.) It willbe assumed throughout the present paper that the spaces H1 and H2 are finite-dimensional. (Extending the results, if possible, to infinite-dimensional H1 andH2 seems to be a challenging problem, and is beyond the scope of this paper.)

The interpolation problems in question are formulated as follows:

Interpolation with Symmetries on Bidisk 269

Problem 1.1. Tangential Interpolation – TIGiven a data set (1.2), find all functions H ∈ HSG→H(D2) satisfying the interpo-lation condition

(2πi)2

|ζ|=1

(ζIH2 −A2)−1B2 ·

|ξ|=1

(ξIH1 −A1)−1B1H(ξ, ζ) dξ

)dζ = B−.

Problem 1.2. TI with Operator Norm constraint – ONormGiven a data set (1.2) and an operator Υ : G → G, where Υ ≥ 0, find all functions

H ∈ HSG→H(D2) satisfying (1.4) and the operator-valued norm constraint

[H, H ]HSG→H(D2) ≤ Υ. (1.5)

Problem 1.3. TI with Scalar Norm constraint – SNormGiven a data set (1.2) and a positive number γ, find all functions H ∈ HSG→H(D2)satisfying the interpolation condition (1.4) and the norm constraint

‖H‖HSG→H(D2) ≤ γ. (1.6)

Substituting (1.1) into the left-hand side of (1.4) leads to the explicit expres-sion of this integral in terms of the Taylor coefficients of the function H :

∞∑

j1,j2=0

Aj22 B2A

j11 B1Hj1,j2 = B−. (1.7)

It follows from (1.7) that a necessary condition for the problem TI to be solvable is

Ran B− ⊆ spanRan Aj22 B2A

j11 B1; j1, j2 = 0, 1, . . . . (1.8)

Here and elsewhere in the paper we denote by Ran X the range of an operator X .It turns out that (1.8) is also sufficient for solvability of TI (see [9]).

Problems of the type TI, ONorm, and SNorm are called one-sided (left) tan-gential problems. An analogous one-sided (right) tangential problem can be formu-lated by having in (1.4) the unknown function H on the left side of the integrandrather than on the right side of the integrand as it stands in (1.4). The resultsfor the right problem are completely analogous to the results for the left problem,and can be obtained from each other essentially by passing to adjoint operators.For this reason, right problems will not be considered here. Two-sided tangentialinterpolation problems are more involved; they include (1.4), the correspondingcondition for the right problem, and a third two sided condition (Nudelman for-mulation [16]); see, for example, [11], where the two-sided problem is studied indepth for rational matrix-valued functions. In the context of the present paper,the two-sided problems seem to be intractable, and therefore are not studied here.

Tangential interpolation problems on the unit disk, with or without normconstraints, have been studied extensively in the literature. We only mention herethe books [11], [14] and [15]. It is of special interest to study Hilbert spaces ofanalytic operator functions as sets in which the interpolants are sought. This is

done in [2], with the standard Hardy space H2 of the unit disk as the Hilbert spaceof matrix functions.

In recent years, the research on tangential interpolation was extended tofunctions of several variables, using a variety of techniques, and producing a wealthof results. It would take us too far a field to quote many works in this area. Weonly mention that the study of tangential interpolation in Hardy spaces of operatorfunctions of several variables was started in [1] and continued in [9]. The first paperdiscusses the one-sided tangential interpolation problem for matrix-valued Hardyfunctions on the bidisk, whereas the subject of the second paper is one-sidedinterpolation for Hilbert-Schmidt operator-valued functions on the polydisk. Thetwo-sided generalization of the latter can be found in [8].

In [9] the problems TI, ONorm and SNorm are solved in case H1 = H2 andB2 = IH1 . In [8] those problems are solved in their general form, because they areparticular cases of the two-sided tangential interpolation problems studied in thatpaper.

In the present paper we will study Problems TI, ONorm, SNorm under addi-tional symmetry conditions on the solutions. Tangential interpolation problems forsymmetric functions of one variable has already been studied, motivated by appliedproblems in which symmetry of interpolant functions is inherent in the problem (e.g., periodic problems [6]). See [12] and [13] for rational matrix functions and [3],[4], [5], [7] and [10] for matrix-valued H2-functions of one variable, and [6] for ap-plications in structured positive extension problems. Tangential interpolation withsymmetries in one variable was applied in [7] to two point interpolation problems;see Section 9 for more information on multipoint problems.

Two type of symmetries are considered in the present paper for Problems TI,ONorm, SNorm: (a) componentwise symmetries that operate separately on eachcomponent of a general point in D2, and when restricted to each component, thesymmetry is represented by a conformal involutive mapping of the unit disk; (b)interchange symmetry that interchanges the two components of a general point inD2. It is a challenging open problem to extend if possible the results obtained hereto more general classes of symmetries of the bidisk.

Besides the introduction, the paper consists of eight sections. We formulatethe interpolation problems which are determined by componentwise symmetriesand prove some preliminary results on these problems in the next section. In Sec-tion 3 we recall results from [8] and [9] that will be used later on. The main resulton solution of the interpolation problems with componentwise symmetries and itsproof are given in Section 4. A more transparent formulation of the main result, aswell as the treatment of interpolation problems with componentwise symmetriesunder additional norm constraints, is possible provided the componentwise sym-metries are specialized so that one of them is the map z → −z. This is done inSections 5 and 6. The interpolation problems with the interchange symmetry, withor without additional norm constraints, are formulated in Section 7, and their so-lutions are described in Section 8. Finally, the last section is devoted to multipointapplications of the main results.

2. Interpolation with componentwise symmetries

In this section we formulate the interpolation problems TI, ONorm, and SNormin a subspace of HSG→H(D2) which is defined by componentwise symmetries.

We denote by HSG→H(J1, J2, s1, s2) the class of all H ∈ HSG→H(D2) whichsatisfy the symmetry relation

J1H(s1(z1), s2(z2))J2 = H(z1, z2), ∀ (z1, z2) ∈ D2, (2.1)

where J1 : H → H and J2 : G → G are two fixed signature operators, i.e.,

Ji = J−1i = J∗

i , i = 1, 2,

and where s1 and s2 are two conformal mappings from D onto D of the forms

sj(z) =ωj − z

1− zωj, ωj ∈ D fixed (j = 1, 2). (2.2)

It is easily seen that except for the case sj(z) = z(j = 1, 2), formula (2.2) presentsthe general form of conformal involutive mappings from D onto itself. Also,HSG→H(J1, J2, s1, s2) is clearly a (closed) subspace of HSG→H(D2). We considerthe following interpolation problems in this subspace.

Problem 2.1. TI with Componentwise Symmetries – CSymLet be given a data set (1.2) satisfying (1.8), signature operators J1 and J2 andtwo mappings s1 and s2 of the form (2.2). Find all functions

H ∈ HSG→H(J1, J2, s1, s2)

satisfying the interpolation condition (1.4).

Problem 2.2. CSym with Operator Norm constraint – ONorm-CSymLet be given a data set (1.2), J1, J2, s1 and s2 as in Problem CSym. Let also begiven an operator

Υ : G → G, Υ ≥ 0.

Find all H ∈ HSG→H(J1, J2, s1, s2) satisfying the interpolation condition (1.4) andthe operator-valued norm constraint (1.5).

Problem 2.3. CSym with Scalar Norm constraint – SNorm-CSymLet be given a data set (1.2), J1, J2, s1 and s2 as in Problem CSym. Let also

be given a positive number γ. Find all H ∈ HSG→H(J1, J2, s1, s2) satisfying theinterpolation condition (1.4) and the norm constraint (1.6).

As part of the solution of problems CSym, ONorm-CSym, SNorm-CSym,criteria for existence of solutions will be given.

Note that if H ∈ HSG→H(J1, J2, s1, s2) satisfies (1.4), then it also satisfies

(2πi)2

|ζ|=1

(ζIH2 −A2)−1B2 ·

·(∫

|ξ|=1

(ξIH1 −A1)−1B1J1H(s1(ξ), s2(ζ))J2 dξ

)dζ = B−.

We will write this equation in the form of a left-sided interpolation condition. Forthis purpose, we need the following easily proved lemma.

Lemma 2.4. Let A be an operator on a Hilbert space H such that σ(A) ⊂ D. Let

T ∈ HSG→H(D) and let s be a conformal involution mapping D onto itself. Then∫

(zIH −A)−1T (s(z)) dz =

(zIH − s(A))−1T (z) dz.

Hence, applying this lemma twice to (2.3) and then multiplying both sidesof the equality on the right with J2, gives us that H also satisfies

(2πi)2

|ζ|=1

(ζIH2 − s2(A2))−1B2 ·

·(∫

|ξ|=1

(ξIH1 − s1(A1))−1B1J1H(ξ, ζ) dξ

)dζ = B−J2,

where in accordance to (2.2),

sj(Aj) = (ωjIHj −Aj)(IHj − ωjAj)−1 (j = 1, 2).

The interpolation conditions (1.4) and (2.4) can be written together as

(2πi)2

|ζ|=1

(ζIH2⊕H2 −A2)−1

·(∫

|ξ|=1

(ξIH1⊕H1 −A1)−1

B1H(ξ, ζ) dξ

)dζ = B−,

(A1 00 s1(A1)

), A2 =

(A2 00 s2(A2)

), B2 =

(B2 00 B2

), B− =

B−J2

Analogously to Problem TI a necessary and sufficient condition for the exis-tence of solutions H ∈ HSG→H(D2) of (2.5) is

j11 B1; j1, j2 = 0, 1, . . .. (2.7)

If a data set Ω satisfies this inclusion, together with (1.3), then we call Ω C-admissible. Note that if a data set Ω is C-admissible, then it also satisfies (1.8).We will assume throughout the first part of the paper that Ω is C-admissible. Itwill be proved that the C-admissibility of Ω is also sufficient for the solvability ofCSym, i.e., for the existence of solutions H ∈ HSG→H(J1, J2, s1, s2) of (2.5). As afirst result we obtain:

Proposition 2.5. Let Ω be a C-admissible data set. Then the following equivalenceshold true.

(a) H ∈ HSG→H(J1, J2, s1, s2) solves CSym if and only if H satisfies (2.5).

(b) H ∈ HSG→H(J1, J2, s1, s2) solves ONorm-CSym if and only if H satisfies(2.5) and the norm constraint (1.5).

(c) H ∈ HSG→H(J1, J2, s1, s2) solves SNorm-CSym if and only if H satisfies(2.5) and the norm constraint (1.6).

We solve Problems CSym, ONorm-CSym and SNorm-CSym in two steps. Thefirst step is to solve the problems without the symmetry condition, which amountsto Problems TI, ONorm and SNorm with the data Ω = A1,A2,B1,B2,B−. Thisstep was already done in [8] and [9]; in the next section we will recall results fromthese papers adapted to the present setting. Then we will show how to select amongall the solutions those which also satisfy the symmetry constraint (2.1), i.e., those

H which belongs to the subspaceHSG→H(J1, J2, s1, s2). In view of Proposition 2.5,solution of Problems CSym, ONorm-CSym and SNorm-CSym will be obtained.

3. Tangential interpolation without symmetries

In this section we will describe, using the results of [8] and [9], all functions

H ∈ HSG→H(D2) which satisfy the left-sided interpolation condition (2.5) withor without one of the norm constraints (1.5) or (1.6). Note that we will apply theresults of [8] and [9] not to the original data set, but rather to the set

Ω = A1,A2,B1,B2,B−,where Ai,Bi and B− are defined as in (2.6). Hence, Ai maps Hi⊕Hi into Hi⊕Hi

(for i = 1, 2), and

B1 : H → H1 ⊕H1, B2 : H1 ⊕H1 → H2 ⊕H2 and B− : G → H2 ⊕H2.

Because Ω is assumed to be C-admissible, this Ω exactly has the properties it musthave in order to be able to apply the results in [8] and [9], namely σ(A1)∪σ(A2) ⊂D and (2.7).

Before we can formulate the results of [8] and [9], we have to introduce somenotations. First, let ℓ2(H) be the space defined by

ℓ2(H) =

⎧⎨⎩hj∞j=0 : hj ∈ H and

∞∑

‖hj‖2H <∞

⎫⎬⎭

and let E(z) : ℓ2(H)→ H be the operator defined by

E(z) = ( IH zIH z2IH · · · ). (3.1)

Note that for every x ∈ ℓ2(H), we have

‖E(·)x‖H(D) = ‖x‖ℓ2(H), (3.2)

whereH(D) is the Hardy space of allH-valued functions f(z) =∑∞

j=0 fjzj, fj ∈ H,

which are analytic on the unit disk and satisfy∑∞

j=1 ‖fj‖2H <∞. The space H(D)is equipped with the inner product

〈f, g〉H(D) =1

∫ 2π

〈f(eit), g(eit)〉H dt.

Making use of (1.1) we represent the function H as

H(z1, z2) =

∞∑

zj1Fj(z2) = E(z1)F (z2),

Fj(z) =

∞∑

Hjℓzℓ and F (z) =

⎛⎜⎝

F0(z)F1(z)

⎞⎟⎠ =

∞∑

⎛⎜⎝

⎞⎟⎠ zℓ.

For a given j, the function Fj belongs toHSG→H(D) whereas the function F clearly

belongs to HSG→ℓ2(H)(D). The following lemma holds true, see Lemma 2.1 in [9].

Lemma 3.1. The operator F (z)→ E(z1)F (z) is a unitary operator from

HSG→ℓ2(H)(D) onto HSG→H(D2),

with respect to the operator-valued inner products

[·, ·]HSG→ℓ2(H)(D) and [·, ·]HSG→H(D2).

More precisely, every function H ∈ HSG→H(D2) admits a representation of theform

H(z1, z2) = E(z1)F (z2), (3.3)

where F is a uniquely defined function in HSG→ℓ2(H)(D) such that

[H, H ]HSG→H(D2) = [F, F ]HSG→ℓ2(H)(D). (3.4)

Note that from (3.4) it follows that also

‖H‖HSG→H(D2) = ‖F‖HSG→ℓ2(H)(D).

Now introduce the operator

2πiB2

(zIH1⊕H1 −A1)−1

B1E(z) dz : ℓ2(H) → H2, (3.5)

and let P : H2⊕H2 → H2⊕H2 be the unique positive semidefinite solution of theStein equation

P−A2PA∗2 = B+B

∗+, (3.6)

so P is given by

∞∑

Ak2B+B

∗k2 (3.7)

(the convergence here is in the operator norm). We will denote by P[−1] the Moore–Penrose pseudoinverse of P, i.e., the operator uniquely defined by the relations

P[−1]PRanP = PRanPP[−1] = P[−1], P[−1]P = PP[−1] = PRanP,

where PRanP is the orthogonal projection on the range of P. (The existence of P[−1]

is guaranteed by the finite-dimensionality of H2.) Further, let µ be an arbitrarypoint on the unit circle and define Θ as the following function on D:

Θ(z, µ) = Iℓ2(H) +(z−µ)B∗+(IH2⊕H2 − zA∗

2)−1P[−1](µIH2⊕H2 −A2)

−1B+. (3.8)

Thus, Θ takes values in the algebra of operators acting on ℓ2(H).

Theorem 3.2. Let µ and ν be arbitrary points on the unit circle. Then the functionΘ defined by (3.8) satisfies

Iℓ2(H)−Θ(z, µ)Θ(ω, µ)∗

= (1− zω)B∗+(IH2⊕H2 − zA∗

2)−1P[−1](IH2⊕H2 − ωA2)

−1B+

Θ(z, µ)Θ(µ, ν) = Θ(z, ν). (3.10)

This theorem is well known, see for example [3], where it is proved in thematrix case (i.e., when all Hilbert spaces involved are assumed to be finite-dimen-sional); in the present case the proof is the same.

Now we can give a description of all solutions of (2.5), with and without thenorm constraints (1.5) and (1.6).

Theorem 3.3. Let Ω = A1,A2,B1,B2,B− be a data set such that

σ(A1) ∪ σ(A2) ⊂ D

and (2.7) is satisfied. Then the following statements hold true.

(a) There exists an H ∈ HSG→H(D2) which satisfies (2.5). Moreover, all such Hare described by the formula

H(z1, z2) = Hmin(z1, z2) + E(z1)Θ(z2, µ)f(z2), (3.11)

where µ is a fixed point on the unit circle and Hmin is the minimal norm solution,with respect to the norm ‖ · ‖HSG→H(D2), given by

Hmin(z1, z2) = B∗1(IH1⊕H1 − z1A

∗2(IH2⊕H2 − z2A

−1P[−1]B−, (3.12)

and f is a free parameter in HSG→ℓ2(H)(D). The sum in (3.11) is orthogonal withrespect to the operator-valued inner product [·, ·]HSG→H(D2), as well as with respect

to the inner product 〈·, ·〉HSG→H(D2). The norm of Hmin is equal to

‖Hmin‖2HSG→H(D2) = Trace(B∗−P[−1]

B−).

(b) There exists a solution H ∈ HSG→H(D2) of (2.5) satisfying (1.5) if and only if

Υ−B∗−P[−1]

B− ≥ 0.

Moreover, all such H are described by formula (3.11), where the parameter f ∈HSG→ℓ2(H)(D) satisfies the operator-valued norm constraint

[f, f ]HSG→ℓ2(H)(D) :=1

∫ 2π

f(eit)∗f(eit) dt ≤ Υ−B∗−P[−1]

but otherwise is free.

(c) There exists a solution H ∈ HSG→H(D2) of (2.5) satisfying (1.6) if and only if

γ2 ≥ Trace(B∗−P[−1]

B−).

Moreover, all such H are described by the formula (3.11), where the parameter

f ∈ HSG→ℓ2(H)(D) satisfies the norm constraint

‖f‖2HSG→ℓ2(H)(D) :=1

∫ 2π

Trace(f(eit)∗f(eit) dt ≤ γ2 − Trace(B∗−P[−1]

B−),

For the proof of this theorem we refer to [9].Since the sum in (3.11) is orthogonal and Θ is inner (which follows from

(3.9)), any solution H of (2.5) satisfies

‖H‖2HSG→H(D2) = ‖Hmin‖2HSG→H(D2) + ‖f‖2HSG→ℓ2(H)(D).

So Hmin is the only solution with ‖Hmin‖2HSG→H(D2)= Trace(B∗

−P[−1]B−).

4. Solution of CSym

We will start this section with a lemma which states that H belongs to the subspaceHSG→H(J1, J2, s1, s2) if and only if the function F in the representation (3.3) of Hsatisfies a certain symmetry condition. If X and Y are operators, and s : D→ D afunction, we denote by HSG→H(X, Y, s) the subspace of HSG→H(D) that consists

of all functions f ∈ HSG→H(D) satisfying Xf(s(z))Y = f(z) for every z ∈ D.

Lemma 4.1. The following are equivalent:

(1) The function H belongs to HSG→H(J1, J2, s1, s2).(2) H admits a representation (3.3) for some function

F ∈ HSG→ℓ2(H)(J1, J2, s2),

where J1 : ℓ2(H) −→ ℓ2(H) is the operator defined by

∫ 2π

E(eit)∗J1E(s1(eit)) dt. (4.1)

The operator J1 satisfies

J1E(s1(z)) = E(z)J1, (4.2)

J1 = J−11 and J1J∗1 =

1− |ω1|2(Iℓ2(H) − ω1T

∗)(Iℓ2(H) − ω1T ), (4.3)

where T ∈ L(ℓ2(H)) is the backward block shift operator represented by theoperator matrix

⎛⎜⎜⎜⎝

0H IH 0H · · · · · ·0H IH 0H · · · · · ·

0H IH 0H. . .

. . .. . .

⎞⎟⎟⎟⎠ . (4.4)

Proof. To show that J1 is bounded, we take two arbitrary vectors x = xj∞j=0 and

y = yj∞j=0 in ℓ2(H) and compute 〈J1x, y〉ℓ2(H). Then, because of (4.1) we have

〈J1x, y〉ℓ2(H) = 〈 1

∫ 2π

E(eit)∗J1E(s1(eit))x dt, y〉ℓ2(H).

By using (3.1) we can write out the first term in the inner product explicitly,namely

∫ 2π

E(eit)∗J1E(s1(eit))x dt =

⎛⎜⎜⎜⎜⎜⎝

∑∞j=0

∫ 2π

0 (s1(eit))jJ1xj dt

∑∞j=0

∫ 2π

0e−it(s1(e

it))jJ1xj dt∑∞

∫ 2π

0 e−2it(s1(eit))jJ1xj dt

⎞⎟⎟⎟⎟⎟⎠

〈J1x, y〉ℓ2(H) =1

∞∑

⟨ ∞∑

∫ 2π

e−kit(s1(eit))jJ1xj dt, yk

∞∑

⟨ ∞∑

∫ 2π

(s1(eit))jJ1xj dt, eiktyk

⟨∫ 2π

∞∑

(s1(eit))jxj dt, J∗

∞∑

eiktyk

⟨∫ 2π

E(s1(eit))x dt, J∗

1 E(eit)y

∫ 2π

〈E(s1(eit))x, J∗

1 E(eit)y〉H dt = 〈E(s1(·))x, J∗1 E(·)y〉H(D).

With Cauchy’s inequality it now follows that

〈J1x, y〉ℓ2(H) ≤ ‖E(s1(·))x‖H(D)‖J∗1 E(·)y‖H(D).

Making use of equality (3.2) and taking into account that J21 = IH, we conclude

that ‖J1E(·)y‖H(D) = ‖y‖ℓ2(H). Setting

eiτ = s1(eit), eit = s1(e

iτ ), dt =1− |ω1|2|1− eiτω1|2

we get

‖E(s1(·))x‖2H(D) =1

∫ 2π

〈E(s1(eit))x, E(s1(e

it))x〉H dt

∫ 2π

〈E(eiτ )x, E(eiτ )x〉H1− |ω1|2|1− eiτω1|2

∫ 2π

〈E(eiτ )x, E(eiτ )x〉H dτ = C‖E(·)x‖2H(D) = C‖x‖ℓ2(H),

C = maxτ∈[0,2π]

[1− |ω1|2|1− eiτω1|2

So we have derived the inequality

〈J1x, y〉ℓ2(H) ≤ C‖x‖ℓ2(H)‖y‖ℓ2(H),

which implies that ‖J1‖ ≤ C. Hence J1 is bounded.

Furthermore, since

E(z)E(ζ)∗ =IH

1− zζ, (4.5)

it follows by Cauchy’s formula that

E(z)J1 =1

∫ 2π

J1E(s1(eit))

1− ze−itdt =

|ζ|=1

J1E(s1(ζ))

ζ − zdζ = J1E(s1(z)),

which proves (4.2). Since s1(s1(z)) = z, z ∈ D, it follows from (4.2) that

E(s1(z))J1 = J1E(z)

and therefore, that

E(z)J21 = J1E(s1(z))J1 = J2

1 E(z) = E(z).

Thus, for every x ∈ ℓ2(H),

E(z)J21x = E(z)x

which implies by Lemma 3.1, that J21 = Iℓ2(H), which proves the first equality in

(4.3).

Next, using (4.1), (4.2), (4.5) and the equality

1− s1(z)s1(ζ)∗ =(1− |ω1|2)(1 − zζ)

(1− zω1)(1− ζω1),

we get

E(z)J1J∗1 = J1E(s1(z))J∗1 =

∫ 2π

J1E(s1(z))E(s1(eit))∗J∗

1 E(eit) dt

∫ 2π

E(eit)

1− s1(z)s1(eit)∗dt =

1− zω1

2π(1− |ω1|2)

∫ 2π

eit − ω1

eit − zE(eit) dt

=1− zω1

2πi(1− |ω1|2)

|ζ|=1

(ζ − ω1)

(ζ − z)ζE(ζ) dζ.

Making use of equalities

E(z)− E(0)

z= E(z)T and zE(z) = E(z)T ∗,

which follow readily from (3.1) and (4.4), we obtain by the residue theorem

E(z)J1J∗1 =

1− zω1

1− |ω1|2(

z − ω1

zE(z) +

=1− zω1

1− |ω1|2(

E(z)− ω1E(z)− E(0)

=1− zω1

1− |ω1|2E(z)(Iℓ2(H) − ω1T )

1− |ω1|2E(z)(Iℓ2(H) − ω1T

∗)(Iℓ2(H) − ω1T ),

which proves the second relation in (4.3).Finally, let H be of the form (3.3). Then, on account of (4.2),

J1H(s1(z1), s2(z2))J2 = J1E(s1(z1))F (s2(z2))J2 = E(z1)J1F (s2(z2))J2

and thus, H belongs to HSG→H(J1, J2, s1, s2) if and only if

E(z1)J1F (s2(z2))J2 = E(z1)F (z2),

which is equivalent to

J1F (s2(z2))J2 = F (z2),

by Lemma 3.1. By definition, this means that F ∈ HSG→ℓ2(H)(J1, J2, s2).

We now present solution of the Problem CSym.

Theorem 4.2. Let Ω be an C-admissible data set. Then the problem CSym admits asolution H ∈ HSG→H(J1, J2, s1, s2). Moreover, all such H are given by the formula

H(z1, z2) = H(z1, z2) + E(z1)Θ(z2, µ)f(z2), (4.6)

where µ is an arbitrary point on the unit circle, H is given by

H(z1, z2) =1

2(Hmin(z1, z2) + J1Hmin(s1(z1), s2(z2))J2)

(here Hmin is defined by (3.12)) and where f ∈ HSG→ℓ2(H)(D) is such that

Θ(·, µ)f ∈ HSG→ℓ2(H)(J1, J2, s2). (4.7)

Here Θ is defined by (3.8).

Proof. Since sj(sj(zj)) = zj , we have

2(Hmin(z1, z2) + J1Hmin(s1(z1), s2(z2))J2 ∈ HSG→H(J1, J2, s1, s2),

and therefore the function H given by (4.6) belongs to HSG→H(J1, J2, s1, s2) ifand only if

E(z1)Θ(z2, µ)f(z2) ∈ HSG→H(J1, J2, s1, s2).

This happens, in view of Lemma 4.1, precisely when f ∈ HSG→ℓ2(H)(D) is such

that (4.7) holds. So H given by (4.6) is indeed in HSG→H(J1, J2, s1, s2).On the other hand, by Theorem 3.3, the function Hmin satisfies (2.5), and

therefore so does the function

J1Hmin(s1(z1), s2(z2))J2.

Indeed, first note that from the equality

sj(sj(zj)) = zj, zj ∈ D,

and from Lemma 2.4 it follows that condition (2.5) is equivalent to

(2πi)2

|ζ|=1

(ζIH2⊕H2 − s2(A2))−1

·(∫

|ξ|=1

(ξIH1⊕H1 − s1(A1))−1

B1H(s1(ξ), s2(ζ)) dξ

)dζ = B−.

Next, let J1 and J2 be the operators defined by

(0 IH1

), J2 =

(0 IH2

). (4.9)

It follows from the special structure of the matrices A1, A2, B1 and B2 that

JiAiJi = si(Ai), J1B1 = B1J1, J2B2 = B2J1, J2B− = B−J2. (4.10)

So multiplying (4.8) on the left with J2 and making use of the equalities in (4.10)lead to

(2πi)2

|ζ|=1

(ζIH2⊕H2 −A2)−1

·(∫

|ξ|=1

(ξIH1⊕H1 −A1)−1

B1J1H(s1(ξ), s2(ζ)) dξ

)dζ = B−J2.

Multiplying both sides of this equality on the right with J2 gives us that also thefunction J1H(s1(ξ), s2(ζ))J2 satisfies (2.5).

Since the set of functions satisfying (2.5) is obviously convex, we obtain that

2(Hmin(z1, z2) + J1Hmin(s1(z1), s2(z2))J2) (4.11)

also satisfies (2.5), and by Proposition 2.5 the function (4.11) solves CSym. Itremains to prove that E(z1)Θ(z2, µ)f(z2) is a general solution of the homogeneousequation (2.5), i.e., in which B− = 0. But this statement follows immediately fromTheorem 3.3.

5. Solution of CSym in case ω1 = 0

In this section we will give a description of all solutions of CSym under the addi-tional assumption that ω1 = 0. It follows from (4.3) that J1 is a signature matrixin this case. This makes it possible to replace (4.7), which is a condition on bothΘ and f , by a condition on the free parameter f only. In addition, the assumptionω1 = 0 will be used in the next section to study the CSym problem with normrestrictions.

We assume ω1 = 0 throughout this section. Several lemmas will be needed.

Lemma 5.1. Let P be given by (3.7) and let P[−1] be its Moore–Penrose pseudoin-verse. Then

(1− |ω2|2)J2PJ2 = (IH2⊕H2 − ω2A2)P(IH2⊕H2 − ω2A∗2), (5.1)

(1− |ω2|2)P[−1] = (IH2⊕H2 − ω2A∗2)J2P[−1]J2(IH2⊕H2 − ω2A2). (5.2)

Proof. The symmetry relation (5.1) holds in view of the special structure of thematrices A1, A2, B1 and B2. Indeed, it follows from (4.2), (4.10) and Lemma 2.4that

J2B+ =1

2πiJ2B2

(zIH1⊕H1 −A1)−1

B1E(z) dz

2πiB2J1

(zIH⊕H −A1)−1

B1E(z) dz

2πiB2

(zIH⊕H − J1A1J1)−1J1B1E(z) dz

2πiB2

(zIH1⊕H1 − s1(A1))−1

B1J1E(z) dz

2πiB2

(zIH1⊕H1 −A1)−1

B1J1E(s1(z)) dz

2πiB2

(zIH1⊕H1 −A1)−1

B1E(z)J1 dz = B+J1.

Now use [7, Lemma 4.2].

Lemma 5.2. Let µ be an arbitrary point on the unit circle and let P[−1] be thepseudoinverse of the matrix P given by (3.7). Then the operator-valued function Θgiven in (3.8) satisfies

J1Θ(s2(z), µ)J1 = Θ(z, s2(µ)). (5.3)

Proof. In order to show that Θ satisfies the symmetry relation (5.3), let us considerthe matrix-valued function

D(z, ν) = (ν − z)(IH2⊕H2 − zA∗2)

−1P[−1](νIH2⊕H2 −A2)−1, (5.4)

where ν is an arbitrary fixed point on the unit circle. Note the following threeequalities (see [7, formula(4.20)])

IH2⊕H2 − s2(z)s2(A2)∗ =

1− |ω2|21− zω2

(IH2⊕H2 − ω2A∗2)

−1(IH2⊕H2 − zA∗2),

νIH2⊕H2 − s2(A2) = −(1− νω2)(s2(ν)IH2⊕H2 −A2)(IH2⊕H2 − ω2A2)−1,

(1− zω2)(s2(z)− ν) = (1 − νω2)(s2(ν)− z). (5.5)

Using (4.10), (5.2) and (5.5), we complete the proof as in the proof of [7, Lemma4.3]. The infinite-dimensional context of the present paper is not an obstacle (incontrast with the entirely finite-dimensional context of [7]), since we assume thatH1 and H2 are finite-dimensional.

Now we can describe all solutions of CSym in case ω1 = 0.

Theorem 5.3. Let Ω be an C-admissible data set. Then the problem CSym admits asolution H ∈ HSG→H(J1, J2, s1, s2). Moreover, all such H are given by the formula(4.6), where µ is an arbitrary point on the unit circle, f is a free parameter in

HSG→ℓ2(H)(J1, J2, s2), and J1 is the matrix defined by

J1 = Θ(µ, s2(µ))J1. (5.6)

Proof. In view of Theorem 4.2 and its proof we only have to show that Θ(·, µ)f ∈HSG→ℓ2(H)(J1, J2, s2) if and only if f ∈ HSG→ℓ2(H)(J1, J2, s2). Well then, first

assume that Θ(·, µ)f belongs to HSG→ℓ2(H)(J1, J2, s2). Hence we have the equality

J1Θ(s2(z2), µ)f(s2(z2))J2 = Θ(z2, µ)f(z2).

Using Lemma 5.2 and then (3.10) we can write the left-hand side of this equality as

Θ(z2, s2(µ))J1f(s2(z2))J2 = Θ(z2, µ)Θ(µ, s2(µ))J1f(s2(z2))J2,

which is equal to Θ(z2, µ)J1f(s2(z2))J2. So we have the equality

Θ(z2, µ)J1f(s2(z2))J2 = Θ(z2, µ)f(z2).

Since detΘ ≡ 0 it follows that

J1f(s2(z2))J2 = f(z2),

i.e., that f ∈ HSG→ℓ2(H)(J1, J2, s2).Reversing these arguments proves the other implication.

6. Solution of ONorm-CSym and SNorm-CSym in case ω1 = 0

If we also take ω2 equal to zero, then the representation (4.6) is orthogonal withrespect to [·, ·]HSG→H(D2) and to 〈·, ·〉HSG→H(D2). In this case the minimal norm

solution Hmin of (2.5) is in the class HSG→H(J1, J2, s1, s2), which means that

Hmin = H. Hence, Hmin is also the minimal norm solution of CSym. In generalthis is not the case. However, to solve ONorm-CSym and SNorm-CSym we needthe minimal norm solution of CSym. In this section we will construct the minimalnorm solution of CSym in case ω1 = 0 and ω2 = 0. We will assume ω1 = 0 andω2 = 0 throughout this section.

To do so, note that according to Lemma 3.1 and Lemma 4.1 every solutionH of CSym has a representation H(z1, z2) = E(z1)F (z2), such that

F ∈ HSG→ℓ2(H)(J1, J2, s2)

[H, H ]HSG→H(D2) = [F, F ]HSG→ℓ2(H)(D), ‖H‖HSG→H(D2) = ‖F‖HSG→ℓ2(H)(D).

Moreover, we know from [9] that F is a solution of the interpolation condition

|z2|=1

(z2IH2⊕H2 −A2)−1

B+F (z2) dz2 = B−. (6.1)

So H is the minimal norm solution of CSym if and only if F is the minimal normsolution of the following problem.

Problem 6.1. Sym on the Unit Disk - Sym-UD

Find all F ∈ HSG→ℓ2(H)(J1, J2, s2) satisfying the interpolation condition (6.1).

The minimal norm solution of this problem is constructed in [7] in case Gand H are finite-dimensional. However, this construction (together with its proof)in [7] also holds true under the present setting.

A necessary and sufficient condition for the existence of solutions of Sym-UD is

Ran B− ⊆ spanRan Aj2B+; j1, j2 = 0, 1, . . ..

This inclusion holds true, because in [8] (Lemma 4.3) it is proved that the right-hand side of this inclusion is equal to the right-hand side of (2.7), which holds trueby assumption.

We know from Theorem 3.3 in [9] that all solutions of Sym-UD without thesymmetry condition are parametrized by

F (z2) = Fmin(z2) + Θ(z2, µ)h(z2), (6.2)

Fmin(z2) = B∗+(IH2⊕H2 − z2A

−1P[−1]B−

and where Θ is given by (3.8). Further, h is free parameter in HSG→ℓ2(H)(D2).Moreover, the sum in (6.2) is orthogonal with respect to both inner products[·, ·]HSG→ℓ2(H)(D) and 〈·, ·〉HSG→ℓ2(H)(D).

In [7, Theorem 5.3] a description is given of all the solutions F of Sym-UD

which are in the set HSG→ℓ2(H)(J1, J2, s2). Namely,

F (z2) = F (z2) + Θ(z2, µ)h(z2), (6.3)

F (z2) =1

2(Fmin(z2) + J1Fmin(s2(z2))J2,

and h is a free parameter in HSG→ℓ2(H)(J1, J2, s2). Also in this case the represen-

tation in (6.3) is not orthogonal in general and hence F is not the minimal normsolution of Sym-UD. However, the minimal norm solution can be constructed fromFmin, which is done in Section 6 of [7]. The following theorem is proved there.

Theorem 6.2. Let Ω be an C-admissible data set and let z0 be the fixed point of s2,i.e.,

z0 =1 +√

1− |ω2|2ω2

Then all functions F ∈ HSG→ℓ2(H)(J1, J2, s2) satisfying the interpolation condition(6.1) are parametrized by the formula

F (z2) = Fmin(z2) + Θ(z2, µ)h(z2), (6.4)

Fmin(z2) =1

ω2(z2 − z0)((1− z2ω2)Fmin(z2) +

√1− |ω2|2J1Fmin(s2(z2))J2) (6.5)

and h is a free parameter in HSG→ℓ2(H)(J1, J2, s2). The function F of the form(6.4) satisfies the norm constraint (1.5), resp. (1.6), if and only if the correspondingparameter h is subject to

[h, h]HSG→ℓ2(H)(D) ≤ Υ− [Fmin, Fmin]HSG→ℓ2(H)(D),

‖h‖2HSG→ℓ2(H)(D) ≤ γ2 − ‖Fmin‖2HSG→ℓ2(H)(D).

Moreover, the decomposition (6.4) is orthogonal with respect to [·, ·]HSG→ℓ2(H)(D)

and 〈·, ·〉HSG→ℓ2(H)(D).

Because the decomposition (6.4) is orthogonal, it follows that Fmin is the

minimal norm solution of Sym-UD. It is possible to compute the norm of Fmin

explicitly. We will do this following the approach of Section 6 in [7].

First note that

J1Fmin(s2(z2))J2 = (1− z2ω2)B∗+(IH2⊕H2 − z2A

· P[−1](IH2⊕H2 − ω2A2)−1

B−.(6.6)

Here we used (4.10), Lemma 5.1 and the first equality in (5.5). Substituting Fmin

and (6.6) into (6.5) gives us the following expression for Fmin :

Fmin(z2) =1− z2ω2

z2 − z0B

∗+(IH2⊕H2 − z2A

−1P[−1]

· (z0IH2⊕H2 −A2)(IH2⊕H2 − ω2A2)−1

It is obvious that Fmin is also a solution of Sym-UD without the symmetry condi-

tion, so it is possible to write Fmin in the form of (6.2). This means that there is

an h ∈ HSG→ℓ2(H)(D) such that

Fmin(z2) = Fmin(z2) + Θ(z2, µ)h(z2). (6.8)

Formula (6.40) in [7] gives us that this equality holds true for

h(z2) = h(z2) :=

√1− |ω2|2

ω2(z2 − z0)T,

T = ω2B∗+(IH2⊕H2 − µA

−1P[−1](µIH2⊕H2 −A2)(IH2⊕H2 − ω2A2)−1

Because Θ is inner (this follows from (3.9)) and the sum in (6.8) is orthogonalwith respect to [·, ·]HSG→ℓ2(H)(D) and 〈·, ·〉HSG→ℓ2(H)(D) we have

[Fmin, Fmin]HSG→ℓ2(H)(D) = [Fmin, Fmin]HSG→ℓ2(H)(D) + [h, h]HSG→ℓ2(H)(D),

‖Fmin‖2HSG→ℓ2(H)(D) = ‖Fmin‖2HSG→ℓ2(H)(D) + ‖h‖2HSG→ℓ2(H)(D).

We know from Lemma 3.2 in [9] that

[Fmin, Fmin]HSG→ℓ2(H)(D) = B∗−P[−1]

‖Fmin‖2HSG→ℓ2(H)(D)= Trace(B∗

−P[−1]B−),

and from formula (6.41) in [7] we know that

[h, h]HSG→ℓ2(H)(D) =

√1− |ω2|2

2(1 +√

1− |ω2|2)T ∗T

‖h‖2HSG→ℓ2(H)(D) =

√1− |ω2|2

2(1 +√

1− |ω2|2)Trace(T ∗T ).

Hence,

[Fmin, Fmin]HSG→ℓ2(H)(D) = B∗−P[−1]

B− +

√1− |ω2|2

2(1 +√

1− |ω2|2)T ∗T,

‖Fmin‖2HSG→ℓ2(H)(D) = Trace(B∗−P[−1]

B−) +

√1− |ω2|2

2(1 +√

1− |ω2|2)Trace(T ∗T ).

Further,

T ∗T = |ω2|2B∗−(IH2⊕H2 − ω2A

−1(P[−1] −A∗2P[−1]

A2)(IH2⊕H2 − ω2A2)−1

see (6.16) in [7], so

[Fmin, Fmin]HSG→ℓ2(H)(D) = B∗−(P[−1] +

√1− |ω2|2

2(1 +√

1− |ω2|2)(IH2⊕H2 − ω2A

· (P[−1] −A∗2P[−1]

A2)(IH2⊕H2 − ω2A2)−1)B−,

‖Fmin‖2HSG→ℓ2(H)(D) = Trace(B∗−(P[−1]

√1− |ω2|2

2(1 +√

1− |ω2|2)(IH2⊕H2 − ω2A

· (P[−1] −A∗2P[−1]

A2)(IH2⊕H2 − ω2A2)−1)B−).

Now we will return to the original problems CSym, ONorm-CSym, andSNorm-CSym. We remarked before that

H(z1, z2) = E(z1)F (z2)

is the minimal norm solution of CSym if and only if F is the minimal norm solutionof Sym-UD. So we have proved that the minimal norm solution of CSym, denoted

by Hmin, is given by

Hmin(z1, z2) =1− z2ω2

z2 − z0E(z1)B

∗+(IH2⊕H2 − z2A

−1P[−1]

· (z0IH2⊕H2 −A2)(IH2⊕H2 − ω2A2)−1

According to Lemma 4.3 in [8] this is equal to

Hmin(z1, z2) =1− z2ω2

z2 − z0B

∗1(IH1⊕H1 − z1A

∗2(IH2⊕H2 − z2A

−1P[−1]

· (z0IH2⊕H2 −A2)(IH2⊕H2 − ω2A2)−1

Moreover, we have that

[Hmin, Hmin]HSG→H(D2) = [Fmin, Fmin]HSG→ℓ2(H)(D),

‖Hmin‖HSG→H(D2) = ‖Fmin‖HSG→ℓ2(H)(D).

All this leads to the following theorem.

Theorem 6.3. Let Ω = A1, A2, B1, B2, B− be an C-admissible data set. Then thefollowing statements hold true.

(a) Problem CSym has a solution H ∈ HSG→H(J1, J2, s1, s2). Moreover, all so-lutions H are described by the formula

H(z1, z2) = Hmin(z1, z2) + E(z1)Θ(z2, µ)h(z2), (6.10)

where µ is a fixed point on the unit circle and Hmin is the minimal normsolution, with respect to the norm ‖ · ‖HSG→H(D2), given by (6.9) and h is

a free parameter in HSG→ℓ2(H)(J1, J2, s2). The sum in (6.10) is orthogonalwith respect to the operator-valued inner product [·, ·]HSG→H(D2), as well as

with respect to 〈·, ·〉HSG→H(D2).

(b) Problem ONorm-CSym has a solution H ∈ HSG→H(J1, J2, s1, s2) if and onlyif the inequality

Υ− [Hmin, Hmin]HSG→H(D2) ≥ 0

holds true. In this case, all solution H are described by formula (6.10), where

the parameter h ∈ HSG→ℓ2(H)(J1, J2, s2) satisfies the operator-valued normconstraint

[h, h]HSG→ℓ2(H)(D) ≤ Υ− [Hmin, Hmin]HSG→H(D2),

but otherwise is free.(c) Problem SNorm-CSym has a solution H ∈ HSG→H(J1, J2, s1, s2) if and only

if the inequality

γ2 ≥ ‖Hmin‖2HSG→H(D2)

holds true. In this case, all solutions H are described by the formula (6.10),

where the parameter h ∈ HSG→ℓ2(H)(J1, J2, s2) satisfies the norm constraint

‖h‖2HSG→ℓ2(H)(D) ≤ γ2 − ‖Hmin‖2HSG→H(D2)

We conclude this section with a remark (pointed out to us by the referee)that a linear fractional change of variables allows one to reduce the general caseof conformal involutive mappings to the case when ω1 = 0. Indeed, let

sj(z) =ωj − z

1− zωj, (j = 1, 2),

and let

σ(z) = −z and τj(z) =z +

1−√

1−|ωj|2ωj

1 + z1−√

1−|ωj |2ωj

(j = 1, 2)

be two automorphisms of the unit disk. It is readily checked that

τ−1j sj τj = σ.

Therefore, a function H(z1, z2) satisfies (2.1) if and only if the function G(z1, z2) :=H(τ1(z1), τ2(z2)) satisfies

J1G(−z1,−z2)J2 = G(z1, z2), ∀ (z1, z2) ∈ D2.

Furthermore, H satisfies interpolation condition (1.4) if and only if G satisfies

(2πi)2

(zI − τ2(A2))−1B2

|ζ|=1

(ζI − τ1(A1))−1B1G(ζ, z)

)dz = B−,

by Lemma 2.4. Thus, one can use formula (6.10) (applied for G) to obtain a general

formula for H ∈ HSG→H(J1, J2, s1, s2) satisfying (1.4). However, the properties oforthogonality and minimal norm of Theorem 6.3 generally are not preserved underthe above change of variables.

7. Interpolation with interchange symmetry

In this section we formulate our basic interpolation problems IT, ONorm andSNorm in a subspace ofHSG→H(D2) which is defined by another type of symmetry.

Definition 7.1. We denote by HSG→Hsym (D2) the class of all H ∈ HSG→H(D2) which

satisfy the interchange symmetry relation

H(z1, z2) = H(z2, z1) (7.1)

for all z1, z2 ∈ D.

It is easily seen that HSG→Hsym (D2) is a subspace of HSG→H(D2). In this sub-

space we consider the following interpolation problems.

Problem 7.2. TI with Interchange Symmetry – ISym

Let be given a data set (1.2) satisfying (1.8). Find all H ∈ HSG→Hsym (D2) satisfying

the interpolation condition (1.4).

Problem 7.3. ISym with Operator Norm Constraint – ONorm-ISym

Let be given a data set (1.2) as in Problem ISym. Let also be given an operator

Υ : G → G, Υ ≥ 0. Find all H ∈ HSG→Hsym (D2) satisfying the interpolation condition

(1.4) and the operator-valued norm constraint (1.5).

Problem 7.4. ISym with Scalar Norm Constraint – SNorm-ISym

Let be given a data set (1.2) as in Problem ISym. Let also be given positive number

γ. Find all H ∈ HSG→Hsym (D2) satisfying the interpolation condition (1.4) and the

norm constraint (1.6).

As in the case of the CSym problems, we will proceed to set up a secondinterpolation problem using the symmetry. To make this approach work, additionalhypotheses are needed. Namely, we assume that H1 = H2 and that there exist

operators A1 : H1 → H1, A2 : H1 → H1 and B2 : H1 → H1 such that

σ(A1) ∪ σ(A2) ⊂ D

and the equality

(λIH1 −A2)−1B2(µIH1 −A1)

−1 = (µIH1 − A2)−1B2(λIH1 − A1)

−1 (7.2)

holds true for all λ and µ on the unit circle. (These hypotheses will be assumedthroughout this section.) In particular, (7.2) holds if B2 = I and A1A2 = A2A1

(then take B2 = I, A2 = A1, A1 = A2.)

Then every H ∈ HSG→Hsym (D2) satisfying (1.4) also satisfies the equality

(2πi)2

|ζ|=1

|ξ|=1

(ξIH1 − A2)−1B2(ζIH1 − A1)

−1B1H(ζ, ξ) dξ dζ = B−.

Interchanging the order of integration and replacing ξ by ζ and vice versa givesthe following interpolation condition:

(2πi)2

|ζ|=1

(ζIH1 − A2)−1B2

·(∫

|ξ|=1

(ξIH1 − A1)−1B1H(ξ, ζ) dξ

)dζ = B−.

The interpolation conditions (1.4) and (7.3) can be written together as

(2πi)2

|ζ|=1

(ζIH1⊕H1 −A2)−1

·(∫

|ξ|=1

(ξIH1⊕H1 −A1)−1

B1H(ξ, ζ) dξ

)dζ = B−,

), A2 =

), B1 =

), B− =

(B−B−

As before, a necessary condition for (7.4) to have solutions is that the oper-ators in (7.5) satisfy (2.7). We will call a data set

Ω = A1, A2, B1, B2, B−

I-admissible if it satisfies (2.7) (with A1,A2,B1,B2 and B− as in (7.5)) togetherwith (1.3). In the next section we will assume that Ω is I-admissible. We obtainan analogue of Proposition 2.5:

Proposition 7.5. Let Ω be an I-admissible data set. Then the following equivalenceshold true.

(a) H ∈ HSG→Hsym (D2) solves ISym if and only if H satisfies (7.4).

(b) H ∈ HSG→Hsym (D2) solves ONorm-ISym if and only if H satisfies (7.4) and the

(c) H ∈ HSG→Hsym (D2) solves SNorm-ISym if and only if H satisfies (7.4) and the

We will solve Problems ISym, ONorm-ISym and SNorm-CSym in the nextsection. These problems will be solved in a similar pattern as Problems CSym,ONorm-CSym and SNorm-CSym. First we will describe all the solutions of ISym.Then we will determine the minimal norm solution of ISym, which makes it possibleto solve Problems ONorm-ISym and SNorm-ISym.

8. The solution of ISym, ONorm-ISym and SNorm-CSym

Note that the special structure of the operators in (2.6) is not used in Theorem 3.3.Hence, if we assume that Ω is I-admissible, then the statements (a), (b) and (c)in Theorem 3.3, after replacing (2.5) by (7.4), also hold true for the operatorsA1,A2,B1,B2 and B− defined by (7.5). This means that all solutions of (7.4),without the symmetry constraint, are described by (3.11) and that the minimalsolution is given by (3.12). In the next theorem, which is an analogue of Theo-rem 4.2, we give a parametrization of all solutions of ISym, hence all solutions of(7.4) also satisfying (7.1).

Theorem 8.1. Let Ω be an I-admissible data set such that (7.2) is satisfied. Then

the problem ISym admits a solution H ∈ HSG→Hsym (D2). Moreover, all solutions H

are given by the formula

H(z1, z2) = H(z1, z2) + E(z1)Θ(z2, µ)f(z2), (8.1)

where µ is an arbitrary point on the unit circle, H is given by

H(z1, z2) =1

2(Hmin(z1, z2) + Hmin(z2, z1)) (8.2)

(here Hmin is defined by (3.12)) and where f ∈ HSG→ℓ2(H)(D) is such that

E(z1)Θ(z2, µ)f(z2) = E(z2)Θ(z1, µ)f(z1). (8.3)

Proof. It is easy to check that H is in HSG→Hsym (D2), so the function H given by

(8.1) is an element of HSG→Hsym (D2) if and only if the free parameter f satisfies

(8.3). So the only thing we have to show is that H indeed is a solution of ISym.

First consider H. We know from Theorem 3.3 that Hmin is a solution of (7.4).From this it follows that also the function H↔

min, defined by

H↔min(z1, z2) = Hmin(z2, z1),

is a solution of (7.4). Indeed, first note that we can write

(ηIH1⊕H1 −A2)−1

B2(ξIH1⊕H1 −A1)−1

in the form((ζIH1 − A2)

−1B2(ξIH1 −A1)−1 0

0 (ζIH1 − A2)−1B2(ξIH1 − A1)

Because of condition (7.2) this is equal to(

(ξIH1 − A2)−1B2(ζIH1 − A1)

−1 0

0 (ξIH1 −A2)−1B2(ζIH1 −A1)

This proves the equality

(ζIH1⊕H1 −A2)−1

B2(ξIH1⊕H1 −A1)−1 = J1(ξIH1⊕H1 −A2)

−1B2

· (ζIH1⊕H1 −A1)−1J1,

where J1 is as in (4.9). This implies that Hmin also satisfies

(2πi)2

|ζ|=1

|ξ|=1

J1(ξIH1⊕H1 −A2)−1

B2(ζIH1⊕H1 −A1)−1J1

·B1Hmin(ξ, ζ) dξ dζ = B−,

which is equivalent (by changing the order of integration and replacing ξ by ζ andvice versa) to

(2πi)2

|ζ|=1

|ξ|=1

J1(ζIH1⊕H1 −A2)−1

B2(ξIH1⊕H1 −A1)−1J1

·B1Hmin(ζ, ξ) dξ dζ = B−.

Now we multiply this equality on the left by J1 and note that

J1B1 = B1 and J1B− = B−.

Hence also H↔min satisfies (7.4). Because the solution set of (7.4) is convex, it follows

that H defined by (8.2) also satisfies (7.4). As a consequence of Proposition 7.5

we have that H is a solution of ISym.It remains to show that E(z1)Θ(z2, µ)f(z2) is a solution of (7.4) with B− = 0.

But this immediately follows from Theorem 3.3.

The representation (4.6) in Theorem 4.2 was not orthogonal, so the minimalnorm solution of CSym had to be constructed to solve ONorm-CSym and SNorm-CSym. In the case of ISym we do not have to do this, because it turns out thatthe representation (8.1) is orthogonal. This is the following lemma.

Lemma 8.2. Let Ω be an I-admissible data set such that (7.2) is satisfied. Then thesum in (8.1) is orthogonal with respect to [·, ·]HSG→H(D2) and 〈·, ·〉HSG→H(D2).

Proof. We have to prove that

[H, G]HSG→H(D2) = 0, 〈H, G〉HSG→H(D2) = 0 (8.5)

where G is of the form

G(z1, z2) = E(z1)Θ(z2, µ)f(z2)

with f a free parameter in HSG→ℓ2(H)(D) such that (8.3) is satisfied, i.e. such thatG(z1, z2) = G(z2, z1). We will only prove the first equality, because the second oneis a direct consequence of it.

Note that

[H, G]HSG→H(D2) =1

2[Hmin, G]HSG→H(D2) +

2[H↔

min, G]HSG→H(D2).

First we will show that the first term on the right-hand side is equal to zero. In[9] it is proved that Hmin admits the representation

Hmin(z1, z2) = E(z1)Fmin(z2),

where Fmin is given by

Fmin(z2) = B+(IH1⊕H1 − z2A∗2)

−1P[−1]B−.

Hence, with Lemma 3.1 in the present paper and Lemma 3.7 in [3] we see that

[Hmin, G]HSG→H(D2) = [Fmin, Θ(·, µ)f ]HSG→H(D2) = 0.

Next, remark that H↔min admits the factorization

H↔min(z1, z2) = E(z2)Fmin(z1)

and that

G(z1, z2) = E(z2)Θ(z1, µ)f(z1).

So again with the same lemmas as above we see that also

[H↔min, G]HSG→H(D2) = [Fmin, Θ(·, µ)f ]HSG→H(D2) = 0.

This concludes the proof of the lemma.

This lemma shows that H is the minimal norm solution of ISym. It turns

out that H is equal to Hmin, which is the minimal norm solution of (7.4). We willprove this statement by showing that

‖H‖HSG→H(D2) = ‖Hmin‖HSG→H(D2).

Lemma 8.3. Let Ω be an I-admissible data set such that (7.2) is satisfied. Then the

norm of the operator H defined by (8.2) is given by

‖H‖2HSG→H(D2) = Trace(B∗−P[−1]

B−).

Proof. We can split up ‖H‖2HSG→H(D2)in four terms as follows:

‖H‖2HSG→H(D2) =1

(〈Hmin, Hmin〉HSG→H(D2) + 〈H↔

min, H↔min〉HSG→H(D2)

+ 〈Hmin, H↔min〉HSG→H(D2) + 〈H↔

min, Hmin〉HSG→H(D2)

Recall from Theorem 3.3 that

〈Hmin, Hmin〉HSG→H(D2) = Trace(B∗−P[−1]

B−).

In the proof of the previous lemma we have remarked that H↔min admits the rep-

resentation

H↔min(z1, z2) = E(z2)Fmin(z1),

so it follows from Lemma 3.1 that

〈H↔min, H

↔min〉HSG→H(D2) = 〈Fmin, Fmin〉HSG→H(D2) = Trace(B∗

−P[−1]B−).

The last equality follows from Lemma 3.2 in [9]. Since

〈H↔min, Hmin〉HSG→H(D2) = 〈Hmin, H↔

min〉HSG→H(D2),

we only have to compute 〈Hmin, H↔min〉HSG→H(D2).

Well then, by definition we have that 〈Hmin, H↔min〉HSG→H(D2) is equal to

(2π)2

∫ 2π

Trace(B

∗−P[−1]Φ(eit1 , eit2)P[−1]

dt1 dt2.

Here Φ is given by

Φ(eit1 , eit2) = (IH1⊕H1 − e−it1A2)−1

B2(IH1⊕H1 − e−it2A1)−1

·B∗1(IH1⊕H1 − eit1A

∗2(IH1⊕H1 − eit2A

Using (8.4) and the equality J1B1 = B1 we can write Φ as

Φ(eit1 , eit2) = J1(IH1⊕H1 − e−it2A2)−1

B2(IH1⊕H1 − e−it1A1)−1

·B∗1(IH1⊕H1 − eit1A

∗2(IH1⊕H1 − eit2A

It follows from Lemma 4.3 in [8] that

B2(IH1⊕H1 − e−it1A1)−1

B1 = B+E(eit1)∗,

Φ(eit1 , eit2) = J1(IH1⊕H1 − e−it2A2)−1

B+B∗+(IH1⊕H1 − eit2A

Now making use of (3.6), we find

Φ(eit1 , eit2) = J1(IH1⊕H1 − e−it2A2)−1(P−A2PA

∗2)(IH1⊕H1 − eit2A

= J1P(IH1⊕H1 − eit2A∗2)

−1 + J1(eit2IH1⊕H1 −A2)

−1A2P,

which implies that

〈Hmin, H↔min〉HSG→H(D2) =

∫ 2π

Trace((B∗−P[−1]J1

P(IH1⊕H1 − eit2A

+ (eit2IH1⊕H1 −A2)−1

P[−1]B−) dt2

= Trace(B∗−P[−1]J1PP[−1]

= Trace(B∗−P[−1]J1PRanPB−).

We obtain from (3.7) the equality

Ran P = spanRan Al2B+; l = 0, 1, 2, . . ..

Further, Lemma 4.3 in [8] gives us that

spanRan Aj22 B2A

j11 B1; j1, j2 = 0, 1, . . . = spanRan A

l2B+; l = 0, 1, 2, . . ..

Because of the assumption that Ω is I-admissible, this equality implies that

B− ⊆ Ran P,

hence PRanPB− = B−. So

〈Hmin, H↔min〉HSG→H(D2) = Trace(B∗

−P[−1]J1B−) = Trace(B∗−P[−1]

B−).

Substituting all the values we have found into (8.6) proves the lemma.

This lemma shows that the norm of H is equal to the norm of Hmin. It is

obvious that H is also a solution of (7.4) without the symmetry condition, hence

from the remark after Theorem 3.3 it follows that H = Hmin.

We now present the solution of Problems ONorm-ISym and SNorm-ISym.

Theorem 8.4. Let Ω = A1, A2, B1, B2, B− be an I-admissible data set such that(7.2) is satisfied. Then the following statements hold true.

(a) Problem ISym has a solution H ∈ HSG→Hsym (D2). Moreover, all solutions H are

described by the formula (3.11), where µ is a fixed point on the unit circle andHmin is the minimal norm solution, with respect to the norm ‖ · ‖HSG→H(D2),

given by (3.12), and where f is a free parameter in HSG→ℓ2(H)(D) such that(8.3) holds. The sum in (3.11) is orthogonal with respect to the operator-valuedinner product [·, ·]HSG→H(D2), as well as with respect to the inner product

〈·, ·〉HSG→H(D2).

(b) Problem ONorm-ISym has a solution H ∈ HSG→Hsym (D2) if and only if

Υ−B∗−P[−1]

B− ≥ 0.

In this case, all solutions H are described by formula (3.11), where the pa-

rameter f ∈ HSG→ℓ2(H)(D) satisfies the operator-valued norm constraint

[f, f ]HSG→ℓ2(H)(D) ≤ Υ−B∗−P[−1]

and (8.3), but otherwise is free.

(c) Problem SNorm-ISym has a solution H ∈ HSG→Hsym (D2) if and only if

B−).

In this case, all solutions H are described by the formula (3.11), where the

parameter f ∈ HSG→ℓ2(H)(D) satisfies the norm constraint

‖f‖2HSG→ℓ2(H)(D) ≤ γ2 − Trace(B∗−P[−1]

and (8.3), but otherwise is free.

9. Multipoint interpolation

The basic two-sided tangential interpolation problems for matrix functions of onevariable in the Nudelman formulation, as studied in [11] for rational matrix func-tions, may be thought of as one-point interpolation. This is because the interpo-lation conditions for these problems can be expressed in terms of equations suchthat each equation involves tangential values of the unknown interpolant functionand its first several derivatives at one point only (however, this point may bedifferent from one equation to another); see Chapter 16 in [11]. In contrast, themultipoint interpolation is stated in terms of equations such that each equationmay involve in an essential way linear combinations of tangential values of theunknown interpolant function and its first several derivatives at several points.

There are few results available concerning multipoint interpolation, see [7], [10].There, the multipoint (actually, two point) interpolation was studied using resultson bitangential interpolation with symmetries.

In the present framework, we shall derive results on certain multipoint in-terpolation problems for operator functions on the bidisk, by utilizing a similarapproach of using theorems on interpolation with componentwise symmetries ob-tained in previous sections. The data set for these problems consist of an orderedset of six operators

Ωm = A1, A2, B1,1, B1,2, B2, B−, (9.1)

Ai : Hi → Hi, for i = 1, 2,

B1,j : H → H1, (for j = 1, 2), B2 : H1 → H2, B− : G → H2

are operators such that σ(A1) ∪ σ(A2) ⊂ D. (The subscript m stands for “multi-point”). The interpolation condition is:

(2πi)2

|ζ|=1

(ζI −A2)−1B2

|ξ|=1

(ξI −A1)−1B1,1H(ξ, ζ) dξ

(2πi)2

|ζ|=1

(ζI − s2(A2))−1B2

·(∫

|ξ|=1

(ξI − s1(A1))−1B1,2H(ξ, ζ) dξ

)dζ = B−,

where the mappings sj : D −→ D are given by

sj(z) =ωj − z

1− zωj, ωj ∈ D fixed (j = 1, 2). (9.3)

Problem 9.1. Multipoint Tangential Interpolation – MTI

Given a data set (9.1), find all functions H ∈ HSG→H(D2) satisfying the interpo-lation condition (9.2).

Using Lemma 2.4, rewrite (9.2) in the form

(2πi)2

|ζ|=1

(ζI −A2)−1B2

|ξ|=1

(ξI −A1)−1B1,1H(ξ, ζ) dξ

(2πi)2

|ζ|=1

(ζI −A2)−1B2

·(∫

|ξ|=1

(ξI −A1)−1B1,2H(s1(ξ), s2(ζ)) dξ

)dζ = B−,

and letting

B1 = (B1,1 B1,2) , G(z1, z2) =

(H(z1, z2)

H(s1(z1), s2(z2))

): G −→ H⊕H,

we see that (9.4) is the interpolation condition for a CSym problem, with theunknown interpolant function G(z1, z2), where

(0 II 0

), J2 = I.

Theorem 4.2 now gives:

Theorem 9.2. The problem MTI admits a solution H ∈ HSG→H(D2) if and onlyif the inclusion

j11 B1; j1, j2 = 0, 1, . . . (9.5)

holds true, where

(A1 00 s1(A1)

), A2 =

(A2 00 s2(A2)

(B1,1 B1,2

B1,2 B1,1

), B2 =

(B2 00 B2

), B− =

(B−B−

In this case, all solutions H are given by the following formula, where the unimod-ular number µ is arbitrary but fixed in advance:

H(z1, z2) = [I 0]

[(G1(z1, z2)G2(z1, z2)

(G2(s1(z1), s2(z2))G1(s1(z1), s2(z2))

+ E(z1)Θ(z2, µ)f(z2) ,

where f ∈ HSG→ℓ2(H⊕H) is a parameter subject to the only condition that

Θ(·, µ)f ∈ HSG→ℓ2(H⊕H)(J1, I, s2),

∫ 2π

E(eit)∗(

0 II 0

)E(s1(e

it)) dt.

The function

(G1(z1, z2)

G2(z1, z2)

)in (9.7) is given by the formula

(G1(z1, z2)

G2(z1, z2)

∗1(I − z1A

∗2(I − z2A

−1P[−1]B−,

where P[−1] is the Moore–Penrose pseudoinverse of the unique solution P of theequation

P−A2PA∗2 = B+B

∗+, B+ =

2πiB2

(zI −A1)−1

B1E(z) dz.

The function Θ in (9.7) is given by

Θ(z, µ) = Iℓ2(H⊕H) + (z − µ)B∗+(IH2⊕H2 − zA∗

2)−1P[−1](µIH2⊕H2 −A2)

To study the problem MTI under additional norm constraints, we shall usea different approach, similar to that of [7, Section 7].

Consider an auxiliary interpolation problem given by the next two simulta-neous equations:

(2πi)2

|ζ|=1

(ζI −A2)−1B2

|ξ|=1

(ξI −A1)−1B1,1H(ξ, ζ) dξ

)dζ = D, (9.8)

(2πi)2

|ζ|=1

(ζI − s2(A2))−1B2

·(∫

|ξ|=1

(ξI − s1(A1))−1B1,2H(ξ, ζ) dξ

)dζ = B− −D,

where D is an auxiliary operator. Letting

(A1 00 s1(A1)

), A2 =

(A2 00 s2(A2)

), B2 =

(B2 00 B2

), B− =

B− −D

(9.10)

we rewrite equations (9.8) and (9.9) in the form

(2πi)2

|ζ|=1

(ζI − A2)−1

|ξ|=1

(ξI − A1)−1

B1H(ξ, ζ) dξ

)dζ = B−. (9.11)

Invoking a part of Theorem 3.3, we obtain:

Proposition 9.3. For a fixed D, assume that

Ω := A1, A2, B1, B2, B−is a C-admissible data set. Let P[−1] be the Moore–Penrose pseudoinverse of theunique positive semidefinite solution P of the Stein equation

P− A2PA∗2 = B+B

∗+, B+ =

2πiB2

(zI − A1)−1

B1E(z) dz. (9.12)

(a) There exists a function H ∈ HSG→H(D2) satisfying (9.8) and (9.9) and suchthat

[H, H ]HSG→H(D2) ≤ Υ (9.13)

if and only if

Υ− B∗−P[−1]

B− ≥ 0. (9.14)

Moreover, all such H are described by formula (3.11), with A1, A2, B1, B2,

and B− are replaced with A1, A2, B1, B2, and B−, respectively, and where

the parameter f ∈ HSG→ℓ2(H)(D) is subject only to the operator-valued normconstraint

[f, f ]HSG→ℓ2(H)(D) ≤ Υ−B∗−P[−1]

(b) There exists a solution H ∈ HSG→H(D2) satisfying (9.8) and (9.9) and suchthat

‖H‖HSG→H(D2) ≤ γ (9.15)

if and only if

B−). (9.16)

Moreover, all such H are described by formula (3.11), with A1, A2, B1, B2,

and B− are replaced with A1, A2, B1, B2, and B−, respectively, and where

the parameter f ∈ HSG→ℓ2(H)(D) is subject only to the norm constraint

‖f‖2HSG→ℓ2(H)(D) ≤ γ2 − Trace(B∗−P[−1]

B−).

To guarantee that Ω is C-admissible regardless of D, we assume that

spanRan Aj22 B2A

j11 B1; j1, j2 = 0, 1, . . . = H2 ⊕H2. (9.17)

As it follows from Proposition 9.3, there exists a function H ∈ HSG→H(D2) satis-fying the interpolation condition (9.2) and such that (9.13) holds true if and onlyif (9.14) holds true for some D.

We now analyze when (9.14) holds true for some D. Let

(I 0I I

(I I0 I

where I stands for the identity operator on H2. Then

B∗−P[−1]

B− =(

D∗ B∗−)

P[−1]

(DB−

). (9.18)

Partition

P[−1] =

(α ββ∗ γ

), α, β, γ : H2 −→ H2.

Denoting by α[−1] the Moore–Penrose pseudoinverse of the positive definite oper-ator α, we have (cf. formula (7.15) in [7])

α[−1]αα[−1] = α[−1], αα[−1]β = β. (9.19)

The first equality in (9.19) is a part of the definition of the Moore–Penrose pseu-doinverse, whereas the second equality in (9.19) is a consequence of the posi-

tive semidefiniteness of P[−1]. Indeed, it follows that Ran β ⊆ Ran α, and sinceI − αα[−1] is an orthogonal projection that maps the range of α to zero, we have

(I − αα[−1])β = 0,

which is equivalent to the second equality in (9.19). Using (9.19), write

(D∗ B∗

P[−1]

(DB−

= (D∗ + B∗−β∗α[−1])α(D + α[−1]βB−) + B∗

−(γ − β∗α[−1]β)B−.

Comparing with (9.18), we conclude that the inequality

B∗−P[−1]

B− ≤ Υ (9.20)

holds true for some D if and only if

B∗−(γ − β∗α[−1]β)B− ≤ Υ, (9.21)

and in this case (9.20) holds true for D = −α[−1]βB−. We have proved the firstpart of the following theorem; the second part can be proved analogously.

Theorem 9.4. Let there be given a data set (9.1), and assume that condition (9.17)holds true.

(a) Let Υ : G → G be a positive semidefinite operator. Then there exists

a function H ∈ HSG→H(D2) satisfying the interpolation condition (9.2) and theoperator norm constraint [H, H ]HSG→H(D2) ≤ Υ if and only if (9.21) holds true.

(b) For a given γ > 0, there exists an H ∈ HSG→H(D2) satisfying the inter-polation condition (9.2) and the norm constraint ‖H‖HSG→H(D2) ≤ γ if and onlyif the inequality

Trace(B∗

−(γ − β∗α[−1]β)B−)≤ γ

holds true.

Using the above approach and Theorem 3.3, one can derive in a straightfor-ward manner a description of all solutions in the class HSG→H(D2) of the interpo-lation problem (9.2) with either the operator or scalar norm constraint. However,such a description is cumbersome, because the extraneous operator D is involved,and therefore we leave it out.

Acknowledgment. We thank V. Bolotnikov for several useful discussions concerningthe paper.

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[9] D. Alpay, V. Bolotnikov, and L. Rodman, One-sided tangential interpolation foroperator-valued Hardy functions on polydisks, Integral Equations and Operator The-ory 35 (1999), 253–270.

[10] D. Alpay, V. Bolotnikov, and L. Rodman, Two-sided residue interpolation in matrixH2 spaces with symmetries: conformal conjugate involutions, Linear Algebra Appl.351/352 (2002), 27–68.

[11] J.A. Ball, I. Gohberg, and L. Rodman, Interpolation of Rational Matrix Functions,Birkhauser Verlag, Basel, 1990.

[12] J.A. Ball, I. Gohberg, and L. Rodman, Two sided tangential interpolation ofreal rational matrix functions, Operator Theory: Advances and Applications, 64,Birkhauser Verlag, Basel, 1993, 73–102.

[13] J.A. Ball and J. Kim, Bitangential interpolation problems for symmetric rationalmatrix functions, Linear Algebra and Appl. 241/243 (1996), 133–152.

[14] H. Dym, J Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces andInterpolation, CBMS Reg. Conf. Ser. in Math, Vol. 71, Amer. Math. Soc., 1989.

[15] C. Foias and A.E. Frazho, The Commutant Lifting Approach to Interpolation Prob-lems, Birkhauser Verlag, Basel, 1990.

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M.C.B. ReuringsDepartment of MathematicsCollege of William and MaryWilliamsburg VA 23187-8795, USAe-mail: mcreurin@math.wm.edu

L. RodmanDepartment of MathematicsCollege of William and MaryWilliamsburg VA 23187-8795, USAe-mail: lxrodm@math.wm.edu

Favard’s Theorem Modulo an Ideal

Franciszek Hugon Szafraniec

Abstract. Recently a treatise [2] dealing with the three term recurrence re-lation for polynomials orthogonal on algebraic sets was born. Because thepaper is pretty sizeable in volume and rich in essence a kind of assistance inthe matter as well as some invitation to it has become a need.

Mathematics Subject Classification (2000). Primary 42C05, 47B25; Secondary47B15.

Keywords. Polynomials in several variables, orthogonal polynomials, threeterm recurrence relation, Favard’s theorem, ideal of polynomials, algebraicset, Zariski’s topology, symmetric operator, selfadjoint operator, joint spec-tral measure, product polynomials, Krawchouk polynomials, Charlier poly-nomials.

The ideology

The celebrated three term recurrence relation for polynomials pn∞n=0 orthogonalon the real line is the formula

Xpn = αnpn+1 + βnpn + γnpn−1, n = 0, 1, . . . ,

which comes out from applying ‘orthogonality’ to the fact that the sequencepn∞n=0 forms a basis (cf. the Glossary section for this as well as for other items)ordered by deg pn = n. Another form of this formula (apparently not equivalent,in general), appearing so often in the literature,

pn+1 = (AnX + Bn)pn + Cnpn−1, n = 0, 1, . . . ,

which, in contrast to the previous one, is ready to perform the recurrence, requiresthe meaning of ‘orthogonality’ to be more decisive. What usually authors do onthis occasion is they assume

the measure realizing orthogonality has infinite support,

This work was supported by the KBN grant 2 P03A 037 024.

302 F.H. Szafraniec

which leaves the other case out of interest. This is not a great offence providedone can avoid any of the two following events:

1o demanding some finite sets of polynomials to be orthogonal,2o creating a logical inconsistency in the so called Favard’s theorem.

The above sins appear in a number of publications dealing with the general theoryof orthogonal polynomials on the real line (and on the unit circle too). Not to men-tion the sinners an example of a fairly acceptable approach which sees the problemand tries not to fall into those discrepancies is [1]. Regardless of someone neglectsmeasures having finite support or does not, the problem becomes of essential sig-nificance if orthogonality of polynomials in several variables is treated: one cannot just simply say supports like spheres, for instance, are of minor significance.

A substantial part of the paper [2] deals with quasi-orthogonality, whichelucidates the appearing algebraic issues in the main and enables us to pass toorthogonality in a rather smooth way. This is done in the second part of [2] andhere we offer excerpts from that.

A cross-section of [2]

Bases with respect to an ideal

Let V be a proper ideal in Pd. Denote by Pd/V the quotient algebra (i.e., Pd/Vis the algebra of all cosets p + V , p ∈ Pd) and by ΠV : Pd −→ Pd/V the quotient

mapping (i.e., ΠV (p)def

= p+V , p ∈ Pd). It will be convenient to extend the equalityof two (scalar) polynomials modulo the ideal V to matrix polynomials. Given

two matrix polynomials P = [pkl]mk=1

nl=1 and Q = [qkl]

nl=1, we write P

=Q ifpkl − qkl ∈ V for all k, l. A set B ⊂ Pd is said to be linearly V -independent, ifΠV (B) is a linearly independent subset of Pd/V and ΠV |B is injective. We saythat F ⊂ Pd is a linear V -span of B, if B ⊆ F and ΠV (F ) = linΠV (B). Finally,B is said to be a (linear) V -basis of F , if B is linearly V -independent and F is alinear V -span of B. Clearly, B is a V -basis of F if and only if B ⊆ F , ΠV (B) is abasis of ΠV (F ) and ΠV |B is injective.

dV (k) =

⎧⎨⎩

dim ΠV

(P〈0]

)= 1 for k = 0,

dim ΠV

(P〈k]

)− dimΠV

(P〈k−1]

)for k 1,

κV = supj 0 : dV (j) = 0 ∈ N ∪ ∞;notice that κ0 =∞ and

d0(k) = cardα ∈ Nd : |α| = k =

(k + d− 1

), k 0.

We say that a sequence Yknk=0 (0 n ∞) of column polynomials is acolumn representation of a non-empty subset B of Pd if every element of B is

Favard’s Theorem Modulo an Ideal 303

an entry of exactly one column Yi and for every k = 0, . . . , n, entries of Yk arepairwise distinct elements of B. Thus, in particular, we can identify V -bases withtheir column representations: a sequence Yknk=0 of column polynomials is calleda V -basis of F if Yknk=0 is a column representation of a V -basis of F .

After the above modification we say sequence PkκV

k=0 of column polynomialsis a rigid V -basis of Pd, if PkκV

k=0 is a column representation of a V -basis of Pd

such that for every k ∈ 0, κV , Pk ⊆ P〈k]d and ℓ(Pk) = dV (k). If PkκV

k=0 is a rigid V -

basis of Pd, then, by Propositions 6 of [2], for every k ∈ 0, κV ,⋃k

i=0 Pi is a V -basis

of P〈k]d and the degree of each member of Pk is equal to k. Moreover, if PkκV

k=0 andQkκV

k=0 are rigid V -basis of Pd and⋃κV

i=0 Pi =⋃κV

i=0 Qi, then for every k ∈ 0, κV ,the columns Pk and Qk are identical up to an arrangement of entries.

Having all these purposeful definitions done we may start to build up (aselected fragment of) the theory.

Orthogonalization

Given a linear functional L : Pd → C, we define the set

q∈Pd

p ∈ Pd; L(pq) = 0.

It is clear that VL is an ideal in Pd such that VL ⊆ kerL. The latter inclusion andthe definition of VL imply that VL is the greatest ideal contained in kerL, andthat VL is a proper ideal if and only if L is non-zero. If L is a Hermitian linear

functional (that is if L(p∗) = L(p) for all p ∈ Pd), then VL is a ∗-ideal.A sequence Qknk=0 (finite or not) of column polynomials is said to be L-

orthonormal if L(QiQ∗j ) = 0 for all i = j, and L(QkQ∗

k) is the identity matrix

for every k ∈ 0, n. Notice that each L-orthonormal sequence Qknk=0 is linearlyV -independent for any ideal V contained in VL.

Because measures orthogonalizing polynomials in several variables are notalways available positive definite functionals serve as substitutes for them. So alinear functional L : Pd → C is said to be positive definite if L(p∗p) ≥ 0 forevery p ∈ Pd. Applying the Cauchy-Schwarz inequality to the semi-inner product(p, q) → L(pq∗) on Pd, we get

VL = p ∈ Pd; L(pp∗) = 0.The following relates bases modulo an ideal to positive definiteness of the orthog-onalizing functional.

Proposition 1 ([2], Proposition 32). If L : Pd → C is a non-zero linear functional,then the following conditions are equivalent

(i) L is positive definite,(ii) VL is a ∗-ideal and there is a rigid VL-basis of Pd, which is L-orthonormal,(iii) VL is a ∗-ideal and there is a VL-basis of Pd, which is L-orthonormal,(iv) there is a basis B of Pd such that L(pp∗) ∈ 0, 1 and L(qr∗) = 0 for all

p, q, r ∈ B such that q = r.

304 F.H. Szafraniec

If (iv) holds we call B semi-orthonormal with respect to L (because the map(p, q) → L(pq∗) is a semi-inner product on Pd). Notice the difference betweenorthonormality of column polynomials and semi-orthonormality of a basis; thoughthey merge in Proposition 1 they concern different notions.

The three term relation and Favard’s theorem

Consider the following two conditions:

(A) Qknk=0 is a rigid V -basis of Pd, which is L-orthonormal, and V ⊆ kerL;(B) Qknk=0 is such that for every k ∈ 0, n, ℓ(Qk) ≤ dV and deg Qk k and

there exists a system [Ak,j , Bk,j ]κV

k=0dj=1 of scalar matrices for which, with

convention A−1,jdef

= 1 and Q−1def

= 0, the following relation holds

=Ak,jQk+1 + Bk,jQk + A∗k−1,jQk−1, j = 1, . . . , d, k ∈ 0, κV ; (3tr)

in the case κV < +∞ take QκV +1def

Notice that if (A) holds for some linear functional L on Pd then apparently

L|V = 0, L(Q0) = Q0 and L(Qk) = 0 for all k ∈ 1, κV (1)

and, in view of Proposition 1, L is positive definite.Now we can restate Theorem 36 of [2] as follows

Theorem 2. Let V be a proper ∗-ideal in Pd and Qknk=0 be a sequence, with nfinite or not, of real column polynomials such that Q0 = 1. Then

1o (A) for some L : Pd → C implies κV = n, V = VL and (B) to hold;2o Conversely, (B) with κV = n implies the matrix [A∗

k,1, . . . , A∗k,d]∗ to be injec-

tive, for every k ∈ N in the case κV =∞ or for every k = 0, 1, . . . , κV −1 oth-erwise, and to exist L such that (A) holds, which must necessarily be unique,positive definite and satisfy (1).

We ought to notice that the matrices [Ak,j ] and [Bk,j ] are of the size whichis suitable to perform the operations on the column polynomials Qk, which arerequired in (3tr).

Remark. Theorem 2 is stated in a slightly different way than the correspondingresult of [2]. In particular, we make more explicit the existence of L as it is inthe classical Favard’s theorem; the existence is a matter of Step 3 in the proof ofTheorem 16 therein and it relies upon the fact that, under the circumstances ofTheorem 2,

Pd = V ∔ lin

∞⋃

This condition determines L satisfying (1) uniquely.

Remark. The three term relation (3tr) contemplated here is in its symmetric formwhich is ready to apply operator methods; the not necessarily symmetric case isconsider in full detail in Sections 5 and 7 of [2].

Set ideals and the three term relation

For p ∈ Pd we define as usually Zpdef

= x ∈ Rd; p(x) = 0. Then the set idealinduced by ∆ ⊂ Rd

I(∆)def

= p ∈ Pd; ∆ ⊂ Zp, ∆ ⊂ Rd;

is a ∗-ideal; warning I(ZX2) = (X2) in P1. Moreover, I(∆) = I(∆ z), where

∆z def

=⋂Zp : p ∈ Pd and ∆ ⊆ Zp

is the Zariski closure of ∆.

As referring to the quotient construction we have, cf. [2], for Zp = ∅

dimPd/I(Zp) =

κI(Zp)∑

dI(Zp)(k) = cardZp. (2)

Notice that for d = 1, it follows from (2) that κI(Zp) + 1 = cardZp.

For a positive Borel measure on Rd, µ = 0, define the moment functional Lµ as

Lµ(p)def

p dµ, p ∈ Pd.

Then, cf. [2, Proposition 38]

VLµ = I(supp µ) = I(supp µz), (3)

and, by (2),

supp µ finite if and only if κVLµ<∞, equivalently: dimPd/VLµ <∞. (4)

The following shows that set ideals are the proper to deal with in the case whenorthogonalization is in terms of measure rather than by functionals.

Proposition 3 ([2], Proposition 41). Let V be a proper ∗-ideal in Pd and let ∆ ⊆ Rd

be non-empty.

(i) If a sequence QkκV

k=0 of real column polynomials (with Q0 = 1) satisfies thecondition (B) of Theorem 2 with κV = n, and L defined by (1) is a momentfunctional induced by a measure µ, then V = I(suppµ).

(ii) If V = I(∆), then there exists a rigid V -basis QkκV

k=0 of Pd composed of realcolumn polynomials (with Q0 = 1), orthonormalized by some measure µ (withsupp µ = ∆) and satisfying the condition (B) of Theorem 2 with κV = n.

Thus as far as properties modulo a set ideal are concerned (and this happenswhen we want to think of orthogonality with respect to a measure, cf. Proposition3 below) it is enough to restrict ourselves to supports being Zariski closed.

On the existence of orthogonalizing measures; the multiplication operators

A way to get orthonormalizing measures is to use spectral theorem for the multi-plication operators. It goes in our ‘modulo an ideal’ situation much like usually;let us describe it briefly. Assume that V is a proper ∗-ideal in Pd and QkκV

k=0 isa sequence of real column polynomials (with Q0 = 1) satisfying the condition (B)of Theorem 2. By this theorem, the sequence QkκV

k=0 and the positive definite

306 F.H. Szafraniec

linear functional L defined by (1) fulfil the condition (A) of Theorem 2. Then thespace Pd/V is equipped with the inner product 〈 · ,−〉L given by

Pd/V × Pd/V ∋ (q + V, r + V ) → 〈q + V, r + V 〉L def

= L(qr∗) ∈ C.

Define the multiplication operators MX1 , . . . , MXdon Pd/V via MXj (q + V ) =

Xjq + V for q ∈ Pd and j ∈ 1, d. It is easily seen that (MX1 , . . . , MXd) ∈

s (Pd/V )d is a cyclic commuting d-tuple with the cyclic vector X0 + V , wherePd/V is equipped with the inner product 〈·,−〉L. If the d-tuple (MX1 , . . . , MXd

)has an extension to an d-tuple T = (T1, . . . , Td) of spectrally commuting selfad-joint operators acting possibly in a larger Hilbert space, then the functional L isinduced by the measure µ(·) = 〈E(·)(X0 + V ), X0 + V 〉, where E stands for thejoint spectral measure of T . Hence µ orthonormalizes the sequence QkκV

k=0 and,by Theorem 2, V = VLµ . The following proposition sheds more light on this.

Proposition 4. Let V be a proper ∗-ideal in Pd, L : Pd → C be a linear functionalsuch that V ⊆ kerL, and QkκV

k=0 be an L-orthonormal sequence of real columnpolynomials (with Q0 = 1), which is a rigid V -basis of Pd. Then for every p ∈ Pd,(p) ⊆ V if and only if p(MX1 , . . . , MXd

) = 0.

Orthogonality on the real line revised

Favard’s theorem

The three term recurrence relation is always meant as

xpn(x) = αnpn+1(x) + βnpn(x) + γnpn−1(x) (5)

to hold for all x ∈ R. Critical values of one of the coefficients αn or γn causedifferent kind of trouble. If some γn = 0 and no αi = 0 as i = 0, 1, . . . , n − 1,there is no functional establishing orthogonality of pn∞n=0, cf. (9); if L existed,L(pnXpn−1) = 0, this would violate the fact that αn−1 = 0. On the other hand,if some αn = 0, the same happens with L(Xpnpn+1) = 0, like in (8), also therecurrence can not be performed anymore and even worse the relation for thatparticular n is no longer an equality between involved polynomials.

The symmetric form of (5)

xpn(x) = anpn+1(x) + bnpn(x) + an−1pn−1(x),

which is now controlled by one sequence an∞n=0 of parameters, has the sameweak point as before if some aN = 0. Nevertheless, the exceptional case of Favard’stheorem is included in the following as well.

Corollary 5 (Favard’s Theorem, complete version). Let p be in P1 and pkNk=0 ⊂P1 be a sequence, with N finite or not, such that deg pn = n and p0 = 1. Then

1o if pkNk=0 is a (p)-basis of P1 and L(pmpn) = δm,n, m, n ∈ 1, N , for somepositive functional L such that Zp ⊂ kerL, then N = cardZp − 1, (p) =

q ∈ P1; L(|q|2) = 0 and for n ∈ 0, N

xpn(x) = anpn+1(x) + bnpn(x) + an−1pn−1(x), x ∈ Zp,

a−1def

= 1, p−1def

= 0 and pN+1 = 0 if N < +∞;(6)

2o Conversely, if a sequence pkNk=0 with N = cardZp− 1 satisfies the relation

(6) then an = 0 for all n if N = +∞ and for n ∈ 0, N − 1 otherwise, andthere exists a unique positive definite functional L such that pkNk=0 is a

(p)-basis of P1 and L(pmpn) = δm,n, m, n ∈ 1, N .

It is clear that a measure which represents the functional L has finite supportif and only if p = 0; if this happens the measure is unique and cardinality of itssupport is equal to cardZp. Favard’s theorem, as commonly stated, leaves thatcase aside.

Krawtchouk polynomials

With 0 < p < 1 and N ∈ N the Krawtchouk polynomials Kn( · ; p, N) are usuallydefined from 0 up to N by

Kn(x; p, N)def

(−n,−x−N

∣∣ 1p

∞∑

(−n)(k)(−x)(k)

(−N)(k)

k!, (7)

where (a)(0)def

= 1 and (a)(k)def

= (a)(k−1)(a+k−1) is the Pochhammer symbol. Theysatisfy the three term recurrence relation (having fixed the parameters p and N

set Kndef

= Kn( · ; p, N))

XKn + p(N − n)Kn+1 − (p(N − n) + (1− p)n)Kn + (1− p)nKn−1 = 0. (8)

Inserting in (7)Kn = ((−N)(n)p

n)−1kn,

after making proper cancellation, we can define for all n the polynomials kn as

(−N + i)(n−i)(−n)(i)(−X)(i)pn−i

and derive the recurrence relation which now takes the form, cf. [3],

Xkn − kn+1 − (p(N − n) + n(1− p))kn − np(1− p)(N + 1− n)kn−1 = 0. (9)

Defining the functional L by

L(p)def

)px(1− p)N−xp(x), p ∈ P1

we come to the relation (again for all n)

L(kmkn) = (−1)nn!(−N)(n) (p(1− p))n

which says the polynomials knNn=0 are orthogonal (and so are KnNn=0).The normalization

=((−1)nn!(−N)(n) (p(1− p))

n )−1/2kn, n = 0, 1, . . . , N (10)

308 F.H. Szafraniec

turns (9) into the symmetric form (with K−1def

XKn +√

p(1− p)(n + 1)(N − n)Kn+1 − (p(N − n) + n(1− p))Kn

p(1 − p)n(N − n + 1)Kn−1 = 0. (11)

and this can be carried out for 0 ≤ n ≤ N exclusively. However (11) does nothold 1 for n = N (compute its left-hand side at x = N + 1, for instance) and both(8) and (9) do not hold either. Nevertheless the left-hand side of (11) is in the ideal((−X)(N+1)) which is the right excuse to include Krawtchouk polynomials into ourgeneral framework. Thus all of it and Theorem 2 in particular can be illustratedin this ‘negligible’ case. More precisely, referring to the notation so far we have:L defined by (8), VL = ((−X)(N+1)), κV = N , Qi = Ki for i = 0, 1, . . . , N + 1

(notice it follows from (10) automatically all Ki = 0 for i ≥ N +1) and the relation(11) to hold modulo the ideal ((−X)(N+1)).

Notice that deg kn = n and the leading coefficient is (−1)n

n! for all n; kn ∈((−X)(N+1)) as n ≥ N + 1, in particular kN+1 =

(−X)(N+1)

N ! . Thus the semi-

orthogonal basis B appearing in Proposition 1, (iv) can be chosen as Kn if n =0, 1, . . . , N and kn otherwise.

An illustrative though simple case

Product polynomials

By product polynomials in this context we mean roughly orthogonal polynomialswhich are tensor product of other orthogonal polynomials. Let us be more precise

and describe that in the case d = 2. So for i = 1, 2 let p(i)n ∞n=0 be a sequence

of orthonormal polynomials with respect to a Borel measure µi, or, equivalently,with respect to a functional Li and let

Xp(i)n = a(i)

n p(i)n+1 + b(i)

n p(i)n + a

(i)n−1p

(i)n−1

be its (formal) three term recurrence relation. The measure µdef

= µ1⊗µ2 corresponds

to the functional L extended from L(p(1)⊗p(2))def

= L1(p(1))L2(p

(2)) for p(1) ∈ C[X1]

and p(2) ∈ C[X2]. It is clear that the polynomials pm,ndef

= p(1)n ⊗p

(2)n are orthonormal

with respect to µ1⊗µ2 or, equivalently, with respect to L. Grouping pm,n∞m,n=0

according to their degree as

= [p(1)n ⊗ p

(2)0 , p

(1)n−1 ⊗ p

(2)1 , . . . , p

(1)0 ⊗ p(2)

n ]∗

we come to the (still formal) three term recurrence relation

XiQn = An,iQn+1 + Bn,iQn + A∗n−1,iQn−1, (12)

1 This troublesome case untouched in [3], what is a contemporary lexicon on orthogonal polyno-mials, is left generously to the reader to be discovered and to be handled somehow.

An,1def

⎡⎢⎢⎣

a(1)n · · · 0 0...

. . ....

0 · · · a(1)0 0

⎤⎥⎥⎦ , Bn,1

⎡⎢⎢⎣b(1)n · · · 0...

. . ....

0 · · · b(1)0

⎤⎥⎥⎦ ,

An,2def

⎡⎢⎢⎣

a(2)0 · · · 0 0...

. . ....

0 · · · a(2)n 0

⎤⎥⎥⎦ , Bn,2

⎡⎢⎢⎣

b(2)0 · · · 0...

. . ....

0 · · · b(2)n

⎤⎥⎥⎦ .

Now the question is to determine an ideal, trivial or not, with respect to which therecurrence relation (12) holds. Due to (3), we can consider set ideals determinedby algebraic set which are Zariski closed. For d = 1 the only possibility for this iseither the set is finite or equal to R. Thus we have got four cases two of which giveus an ideal which is not trivial: both measures are finitely supported or one of themeasures is finitely supported, the other not; the latter is interesting enough to bediscussed below in some detail.

The example

To discuss the case just mentioned take two sequences of orthonormal polynomials:the Charlier polynomials Cn∞n=0 and the Krawtchouk polynomials Kn∞n=0.

The Charlier polynomials C(a)n ∞n=0, a > 0, are determined by

e−az(1 + z)x =

∞∑

C(a)n (x)

They are orthogonal with respect to a non-negative integer supported measureaccording to

∞∑

C(a)m (x)C(a)

n (x)e−aax

x!= δmnann!, m, n = 0, 1, . . . .

The orthonormalized Charlier polynomials C(a)n ∞n=0 are

C(a)n (x) = a−n

2 (n!)−12 C(a)

and they satisfy the recurrence relation

XC(a)n =

√(n + 1)a C

(a)n+1 + (n + a)C(a)

n +√

na C(a)n .

Both Charlier and Krawtchouk polynomials have probabilistic connections (Pois-son and binomial distribution), for more look at [4]. If we build product poly-nomials with these two we get polynomials in two variables having the followinginteresting features:

• the support of the orthogonality measure is a countable subset N×0, 1, . . . ,. . . , N of the integer lattice;• the support is not Zariski closed, its Zariski closure is R× 0, 1, . . . , N;• VL = ((−X2)(N)) and κV = +∞;

310 F.H. Szafraniec

• the three term recurrence relation (3tr) modulo the ideal VL is of full (i.e.,infinite) length;• the length of Qn stabilizes after n = N + 1.

Let us mention that all the details as well as extensive references to the literaturecan be found in [2].

Glossary

= 0, 1, . . . and Nd – d - fold Cartesian product of N by itself;

for i ∈ N i, jdef

= i, i + 1, . . . , j if i j < +∞ and i, jdef

= i, i + 1, . . . if j = +∞;R (resp. C) stands for the field of all real (resp. complex) numbers;linA – the linear span of a subset A of a linear space;basis means a Hamel basis;δi,j – the Kronecker symbol;

Xα def

= Xα11 · · ·Xαd

d for α = (α1, . . . , αd) ∈ Nd;Pd – the algebra of all polynomials in d commuting indeterminates X1, . . . , Xd

with complex coefficients;∗ – the unique involution in the algebra Pd such that X∗

i = Xi for all i = 1, . . . , d;∗-ideal – an ideal invariant under the involution ∗;deg p – the degree of p ∈ Pd;ℓ(Q) – the length (that is, the number of entries) of a column polynomial Q;

P〈k]d

= p ∈ Pd : deg p k, k ∈ N;(p1, . . . , pn) – the ideal generated by a set p1, . . . , pn ⊆ Pd;

L#s (D) – the set of all linear operators A on an inner product space D such that〈Ax, y〉 = 〈x, A〉, x, y ∈ D.

References

[1] T.S. Chihara, An introduction to orthogonal polynomials, in Mathematics and itsApplications, vol. 13, Gordon and Breach, New York, 1978.

[2] D. Cichon, J. Stochel and F.H. Szafraniec, Three term recurrence relation moduloan ideal and orthogonality of polynomials of several variables, J. Approx. Theory, toappear.

[3] R. Koekoek and R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonalpolynomials and its q-analogue, Delft University of Technology, Report of the Facultyof Technical Mathematics and Informatics no. 94-05.

[4] W. Schoutens, Stochastic processes and orthogonal polynomials, in Lecture Notes inStatistics, vol. 145, Springer-Verlag, New York – Berlin – Heidelberg, 2000.

Franciszek Hugon SzafraniecInstytut Matematyki, Uniwersytet Jagiellonskiul. Reymonta 4, PL-30059 Krakow, Polande-mail: fhszafra@im.uj.edu.pl

Operator Method for Solution of PDEs Based on Their Symmetries

Documents

Nöther Symmetries in Bianchi Universes

Chaotic Attractors with Discrete Planar Symmetries

Multilevel matrices with involutory symmetries and skew symmetries

Dynamical systems and sigma-symmetries

Approximate Noether-type symmetries and conservation laws via partial Lagrangians for PDEs with a small parameter

Dihedral Flavor Symmetries

An Operator-Fractal

A New Legendre Wavelets Decomposition Method for Solving PDEs

On the linear stability of hyperbolic PDEs and viscoelastic flows

BDD minimization using symmetries

Properties of ODEs and PDEs in Algebras

Symmetries leading to inflation

Classical Symmetries of Complex Manifolds

Symmetries in finite order variational sequences

A data-driven approach for multiscale elliptic PDEs with

An iterated Radau method for time-dependent PDEs

Operator Manual

Covariant Quantum Mechanics and Quantum Symmetries

Schrödinger invariance and spacetime symmetries

Solving PDEs with radial basis functions