Integral Equations and Operator Theory Vol.5 (1982)

0378-620X/82/050632-27501.50+0.20/0 © 1982 Birkh~user Verlag, Basel

Factorization Results Related to Shifts in an Indefinite Metric

Joseph A. Ball and J. William Helton

The celebrated theorem of Beurling [Brl] and Lax's vectorial generalization ([L,H-L]), as in the form given in Helson's book [Hls],characterize the form o f the simply invariant subspaces for the bilateral shift operator of arbitrary multipli- city. The proof of this result due to Halmos [H] hinges on the concept of wander- ing subspace for an isometry and leads naturally to the Wold decomposition for a general Hilbert space isometry. This structure theory for isometries comes up naturally in the factorization problem of nonnegative operator valued functions on the unit circle as presented by Sz.-Nagy and Foias [N-F] and Rosenblum and Rov- nyak [R-R], and leads to the Lowdenslager criterion for the factorability F(e it) -- A(ei!)*A(e it) (A outer) of the nonnegative operator-valued function F.

The authors [B-H1] recently extended Halmos' wandering subspace argument to obtain a Beurling-Lax theorem for shifts in an indefinite metric (also see [B] for an informal account); applications there principally involved a quick derivation and extensions of the linear fractional map parametrization of all solutions of a Nevanlinna-Pick, Caratheodory-Fejer interpolation problem. In this article we analyze the geometry of more general invariant subspaces of shifts in an indefinite metric and pursue further applications; one can think of most of the applications here as being complementary to the work of Sz.-Nagy and Foias and of Rosenblum and Rovnyak mentioned above. Specifically, we derive the factorization theorem of Nikolaichuk and Spitkovskii ([N-S], see also [C-G], Chapter V) for invertible self-adjoint valued functions on the unit circle. We then derive the symplectic inner-outer factorization due to Potapov [Po] via our methods, as well as a more general inner-outer factorization. Finally, we turn to Darlington's embedding of a rational function in H °° of norm less than one as the upper diagonal entry of a rational H~°(M2) inner function (see Arov [A] or Douglas-Helton [D-HI). Darlington's theorem plays a major role in a design procedure for passive circuits. We are able to give a simple geometric proof of this classical Darlington theorem for passive circuits and to quickly obtain the extension of it to active circuits given in [B-H2]. The proof given in [B-H2] relied on algebra plus the IN-S] factoriza- f i n n theorem which the authors_derived indet~endentlv for this aoolication.

Ball and Helton 633

§1. Krein space isometries and their invariant subspaces.

Following Bognar [Bog], a Krein space is an indefinite inner product space

(K, [ , ]) which can be written as the [ , ]-orthogonal direct sum K = K+ I'~ K_

(i.e., K+ MK_--{0} , K + [ ] K _ = K and [k+,k_]=0 for k + E K + and

k_ E K_) of K+ a postive subspace (i.e., [k+,k+] >~ 0 for k+ 6 K+) and K_ a

negative subspace ([k_,k_] ~< 0 for k_ E K_), such that the restrictions of [ , ]

to K+ and of - [ , ] to K_ make K+ and K_ Hilbert spaces. When this is the

case, the operators P: k+ + k_ -" k+ (k_+ E K+) and Q = I - P are bounded

[ , ]-self-adjoint projections on K with images K+ and K__ respectively, and the

Krein space K becomes a Hilbert space in the inner product

(k,klj = [Jk,k]

where J ~ P - Q (= 2P - I). An operator J arising in this way from a fundamen-

tal decomposition K = K÷ [ ] K_ of K as above is said to be a fundamental sym-

metry for K. The fundamental symmetry J is not unique, and therefore also the

Hilbert space inner product ( , >j is not uniquely determined; nevertheless the

norm and weak topologies determined by all such inner products are all equivalent

and therefore a Krein space has a well-defined Hilbert space norm and weak topol-

ogy.

The main focus of our interest is on [ , ]-isometries, that is operators U

defined on all of K which are isometric on the Krein space inner product

[Ukl,Uk2] = [kl,k2]

on kl,k2 E K . If the range of U is all of K, we say that U is a [ , ] -uni tary

operator. The spectral theorem for unitary operators and the general Wold decom-

position for Hilbert space isometries (see [N-F]) completely describes all such

isometries up to unitary equivalence if the inner product [ , ] is a Hilbert space

inner product. Moreover Beurling's theorem and general spectral theory can be

Ball and Helton 634

used to describe all the invariant subspaces of such a U in this case (see [His]).

For a general Krein space unitary operator, analogues of the spectral theorem have

been developed by the eastern European school (see references in [Bog]); for the

case of general non-unitary [ , ]-isometrics, recent work of McEnnis [McE] shows

that the analysis can be tricky.

More definitive results can be obtained if one restricts attention to fundamen-

tally decomposable isometrics on a Krein space K, and their restrictions to invari-

ant subspaces. Following the terminology in Bognar's book [Bog], we say that an

operator T on a Krein space K is fundamentally decomposable if there is some

fundamental symmetry J for K which commutes with T. For the case of an

[ ,]-isometry U, the fundamental decomposability of U implies that U is also

an isometry in some Hilbert space ( , >j-inner product. By a shift operator S we

mean an isometry on a Krein space K with the property that A S" --- {0}. We n>~0

say that an invariant subspace M for an isometry U is simply invariant if

tq UnM -- {0}; thus in this case the restricted operator S = UM is a shift. n>~0

For the case of Hilbert space, a shift operator S is classified up to Hilbert

space unitary equivalence by its multiplicity, namely, the codimension of ImS

(the image of S). In fact S is unitarily equivalent to the concrete shift operator

Mei t (multiplication by the coordinate function e it) o n the vector-valued Hardy

space HI~, where L is a Hilbert space with dim L = codim Im S. We say two

shifts $1 and $2 on Krein spaces K1 and K2 are unitarily equivalent (in the

Krein space sense) if there is a bounded ( [ , ]Iq,[ , ]K2)-unitary operator U from

K1 onto K2 such that S2U = US1. It develops that classifying shifts up to unitary

equivalence, while successful for Hilbert spaces, is somewhat over ambitious for

Krein spaces at this point, and hence we shall work with a notion which we shall

call quasi-unitary equivalence. Two shifts $1 on K1 and $2 on K2 are said to be

quasi-unitarily equivalent if there is a (possibly unbounded) isometry U with

Ball and Helton 635

domain D(U) dense in Kl which maps D(U) isometrically onto a dense linear

manifold in K2 and USllo(u)=S2U. That U is an isometry means

[Uk,Uk']K2 = [k,k']K~ for k,k' in D(U).

The Krein space analogue of the concrete shift discussed above for Hilbert

space is as follows. Let L be a Krein space with some compatible Hilbert space

n o r m tl IlL,

product

let HI~ be the usual Hardy space with values in L; then the inner

[f,g]H~ -- ~ [f(eit),g(eit)] L dt (1.1)

makes HI~ a Krein space, and the operator Meit o f multiplication by e it o n HI~

is a fundamentally decomposable shift. One of the main results of [B-H1] can be

rephrased in terms of quasi-unitary equivalence. Recall that a subspace M of a

Krein space K is said to be regular (or ortho-complemented in the terminol-

ogy of [Bog]) if M n M ' = {0} and M + M ' - - - K. (Here M ' - -

{x e K I Ix,y] -- 0 for all y in M}). In general for two subspace X and Y,

we write X + Y if [x,y]--0 for all x in X and y in Y, X n Y = {0}

and X + Y is dosed .

THEOREM 1.1. [B-H1] Suppose U is a fundamentally decomposable

isometry on a Krein space K and M is a regular simply invariant subspace.

Then S ---- U ~ is quasi-unitarily equivalent to Me~, on HI~ for an appropriate

Krein space L. Moreover, one may take L equal to M A (SM)' which we

denote as M [ ] SM.

For some applications it is necessary to consider isometries on certain degen-

erate inner product spaces. Given a degenerate inner product space K, by the

isotropic subspace K0 we mean the set of all vectors orthogonal to the whole space

Ball and Helton 636

(Ko= { x 6 K l [ x , y ] K = 0 for all y in K}). In the case where K0={0}, we

say that K is a nondegenerate inner product space; otherwise K is degenerate.

Given a degenerate inner product space, the inner product [ , ]K is well-defined

on the quotient space K/K0, and makes K/K0 a nondegenerate inner product

space. If in fact K/K0 is a Krein space, we will say that K is a quasi-Krein

space. If we assume that the isotropic space Ko is itself a Hilbert space (as will

always be the case in our examples), then any pseudo-Krein space K has a well-

defined separating weak and norm topology. If K is a Krein space and M is a

subspace of K, M is said to be pseudo-regular if M + M' is closed; in this case

the restriction of [', ]K to M makes M a pseudo-Krein space with isotropic

subspace M0 = M o M'. We next extend the notions of isometries, shifts and

quasi-unitary equivalence from Krein spaces to pseudo-Krein spaces; in the

definition of quasi-unitary equivalence one should add that the intertwining

isometry U be injective on its domain, since this no longer follows automatically.

To get a concrete example of a shift on a pseudo-Krein space, let L be any

pseudo-Krein space, let HI~ be the associated L-valued Hardy space with degen-

erate inner product given by formula (1.1) and again let S be the operator Me,

on HI~. Then HI~ is a pseudo-Krein space with isotropic subspace equal to HI~ 0

(under the natural identifications), where Lo is the isotropic subspace for L.

Note that HI~ 0 is invariant under the shift operator Meit. A more general result

from [B-H1] can be rephrased as follows.

THEOREM 1.2. [B-H1] Suppose U is a fundamentally decomposable

isometry on the Krein space K and M is a simply invariant pseudo-regular sub-

space such that the isotropic subspace M0 = M n M' is also invariant. Then

there is a pseudo-Krein space L such that S -- UMo is quasi-unitarily equivalent

to Meit on HI~.

Ball and Helton 637

Finally we consider certain shifts for which the isotropic subspace Mo is not

invariant; these were not considered in [B-H1]. An example, and as we shall ulti-

mately see, a model (for the finite dimensional case) of this situation is con-

structed as follows.

Let D(e it) (0 ~< t ~< 2~r) be an N × N off-diagonal self-adjoint matrix

function of the form

(1.2) D(e It) =

-iklt e

e-ik2t "

-ikrt e

ikrt

iklt

eik2 t e

where kl >~ k2 >t "" kr > 0 are positive integers, c~+ and c~_ are nonnegative

integers, I,~+ and I,~_ are respectively the c~+ × c~+ and c~_ × c~_ identity matrices

and N -- 2r + c~+ + c~_. Obviously D is self-adjoint. A matrix function D of

the form (1.2) will be called a winding matrixand the integers

- k l , • • • , - kr; 0 , . . . , 0 ; 0 , . . . , 0 ; kr, . . . , k 1 Or-- O;+

will be called the indices of D. It is easy to see that these agree with the partial

indices of D used by Gohberg and Krein [G-K]. The Hardy space H A of ~ -

valued functions becomes a Krein space, denoted H2(D), via the inner product

27"r 1

[f'g]H2(D) = 2n fo ( D ( e i t ) f ( e it) , g (e i t ) >L.N d t

Ball and Helton 638

The Krein space H2(D) has an isotropic space (H2(D))0 whose dimension is

kl + k2 + "" + kr. Of course Meit is a [ , ] isometry on H2(D). Indeed we have

an example of a pseudo-regular Krein space with an isometry acting on it which

does not leave the isotropic subspace invariant. The next theorem shows that all

such situations are equivalent to this one.

T H E O R E M 1.3. Suppose U is a fundamentally decomposable isometry of

finite multiplicity N on a Krein space K, and suppose that M is a simply

invariant pseudo-regular subspace such that Mo =-- M N M' is finite dimensional.

Then there is a unique N 1 x Nt winding matrix D such that S ---- ~ is quasi-

unitarily equivalent to Meit on H2(D), where N1 x< N.

It is easy to read off the indices of D from intrinsic properties of U acting on

M0. We shall do this and prove Theorem 1.3 later in the appendix.

As a corollary we obtain a non-Euclidean Beurling-Lax theorem more general

than that obtained in [B-H1]. Let IL m," be the usual Euclidean space

L ~ - ~ ~ ~ (N = m + n) but considered as a Krein space with the indefinite

quadratic form

[x @ y,x @ y]~,, --<x,x>t v - <y,y}o ,

x ~ y 6 L m ~ .

L2, . is the usual Lebesgue space of L~-valued functions on the unit circle, but

with Krein space inner product

[f'g]L~,, = '2~" [f(eit)'g(eit)]~'ndt "

Note Mei t o n L~, . is a fundamentally decomposable isometry.

3all and Helton 639

COROLLARY 1.4. Suppose M is a pseudo-regular simply invariant sub-

space of L~,n such that Mo -- M tq M' is finite dimensional. Then there is an

N 1 × N l winding matrix D and a square-integrable (N x N1)-matrix function

E(e it) satisfying

(1.3) o o , o , a o t

such that

(1.4) M - (,E • H ~ ) - .

Also N1 is no bigger than m + n.

Proof. By Theorem 1.3, Me~i is quasi-unitarily equivalent to Meit on

H2(D) for some winding matrix D as in (1.2). If U is the intertwining unitary

operator, we define a matrix function _= (e it) by

~(eit)x ---- (Ux) (e it)

for any constant vector x in L r~. Since UMei, = MeitU , one sees that U is sim-

ply multiplication by E(e it) on polynomials. Since U is an isometry of H2(D)

into L2(~'a), equation (1.3) follows. As a general property of such isometrics, it

follows that N1 < N. Since the matrix entries of ,.~(e it) are in L 2, we see that

_= is square integrable in norm. Finally, since U has dense range in M, equation

(1.4) follows.

§2. Factorization of continuous self-adjoint matrix functions on

the unit circle.

In this section we apply the results of the previous section to give a

Ball and Helton 640

geometric proof of a factorization theorem for invertible self-adjoint valued func-

tions on the unit circle. For related results see Nikolaiauk and Spitkovski [N-S]

and Chapter V of Clancey and Gohberg [C-G]. We let MN denote the N x N

matrices and L°°(MN) and H~(MN) the associated spaces of measurable uni-

formly bounded MN-valued functions on the unit circle (respectively, also having

analytic continuation to the unit disk), while C(MN) denotes continuous MN-

valued functions on the unit circle. A function A in H2(MN) is said to be ou ter

if A H °° (£Y) is dense in H 2 (IL~).

THEOREM 2.1. Suppose F is a self-adjoint invertible function in C(MN).

Then F admits a factorization of the form

F(e it) _ A(eit)*D (eit)A(e it)

where A is an outer function in H2(MN) and D is a winding matrix (as in

(1.2)). Note D is determined uniquely.

Proof. Let L2(F) be the Lebesgue space L2(I~) of norm-square-integrable

~-valued functions on the unit circle, but with inner product

27r 1

[f'g]L2(F)- 2~- fo <F (eit)f (e it) ,g (e it) > d, git .

Since F is self-adjoint and invertible, this inner product makes L2(F) a Krein

space. Note also that Melt is an isometry in this inner product. Define a subspace

S+ of L2(F) by

S+ -- {f 6 L2(F) [f(e it) is in the positive

spectral space of F(e it) for a.e.t}

Ball and Helton 641

and an analogous subspace S_ for the negative spectral subspace of F(eit). Then

it is easy to see that L2(F)= S+ + S_ is a fundamental decomposition for

L2(F) commuting with Mei t and thus Mei t on L2(F) is a fundamentally decom-

posable isometry. We now consider the space H2(I~) as a subspace of L2(F),

and denote it as H2(F). Since the matrix function F is continuous and invertible

on the unit circle, it follows from well-known facts (see [Dou] Corollary 1.4) that

the Toeplitz operator TF: f---' PH2(o)(Ff) on H2(I~) is Fredholm. It is not

difficult to deduce from this that H2(F) is a pseudo-regular subspace of L2(F). It

follows from Theorem 1.3 that S ~ Mei~H2(F ) is quasi-unitarily equivalent to Mdt

on H2(D) where D is a winding matrix. It follows as in the proof of Corollary

1.4 that the unitary operator V: H2(F) --, H2(D) intertwining Me~ t on these two

spaces, to wit VMeit-- MeitV , must be multiplication by a norm square integrable

function A satisfying

A(eit)*D(eit)A(e it) = F(e it)

a.e. in t. Since Im V is dense in H2(D), the function A must be outer.

Theorem 2.1 is proved.

Many refinements of this theorem are possible. We shall derive a few of the

more important ones. However, the proofs are not exercises in our Beurling-Lax

approach and so this is a bit of a digression from the main theme of this paper.

Consequently, some readers may prefer to skip directly to §3.

We shall now study factorization of rational invertible self-adjoint functions

on the unit circle. In this and in other important instances quasi-unitary

equivalence can be shown to be a strict unitary equivalence, that is, the unitary

operator U is bounded.

Ball and Helton 642

COROLLARY 2.2. (a) (Fejer-Riesz Theorem) Suppose that F(e it) is a

self-adjoint invertible matrix polynomial

F(e it) - An*e -int + "'" + A0 + "'" + An eint

with coefficients Aj in Ms and A0 = A~. Then there is an analytic matrix poly- n+k 1

nomial P(e it) -- ~ Pie ijt with coefficients Pj in Ms and a winding matrix D j=0

such that

F(e it) = p(eit)*D(eit)p(eit ) .

(b) Suppose F(e it) is a rational bounded self-adjoint and invertible matrix

function on the unit circle. Then there is an outer bounded rational function

Q(e it) such that F(e it) -- Q(eit)*D(eit)Q(e it) for some winding matrix D.

Proof. We prove (a) in detail; then we shall only sketch the modifications

needed to prove (b). The main idea comes from the work of Rosenblum and

Rovnyak [ R - R ] (see also [R]).

Hence suppose F is a self-adjoint invertible matrix polynomial on the unit

circle. Then by the theorem we know that F has a factorization

F(e it) --A(eit)*(eit)A(e it) for an outer A in H2(Mr~). For the following argu-

ment it is convenient to assume that A is bounded. It is then a technical exer-

cise to put in subtleties concerning domains to see that it still goes through even

for a priori unbounded factorizations. For K any function in L°°(M~), let

TK: H2(I~) ---' H2(I~) denote the Toeplitz operator TK(f) = PH~t~(Kf). Since A

is analytic and A* is conjugate analytic, the factorization F(e it) --

A(eit)*D(eit)A(e it) implies the operator factorization

T F = TA,TDT A •

Ball and Helton 643

Recall that the analytic Toeplitz operators (T G with G in H°*(Mr~)) form pre-

cisely the commutant of the shift T x. Here X (eit) = eit. By assumption FX n is

analytic, and hence TF×nT x = TxTF×n. We next claim A*Dx n is analytic. Indeed

(TA.DxnT x -- TxTA.Dxn)T A

= TA.DTxaTxT A - TxTA.DTxnT A

= (TA.DTATxn)T x -- T x (TA*DTATxn)

= TF×nT x - T×TFxn = 0 .

Since A is outer, TA has dense range on H2(IL~), and hence the above compu-

tation shows that A*Dx n is analytic (i.e., is in H~°(MN)) as claimed. Solving for

A* we see that A(eit) * at worst has the form

A(eit) * _ A_n_kl e-i(n+kl)t .+.... + A_I e-it

+ analytic.

Since A itself is analytic, we must have

i(n+kl)t A(e it) = A0 + A1 eit + "'" + An+kle

so A must be an analytic trigonometric polynomial of degree at most n+kl.

To prove (b), note that the assumption that F is rational implies that there

is a scalar finite Blaschke product ~ such that Ftk is analytic. One now can fol-

low the proof of (a) above, with the function ~k playing the role of X n.

As a corollary we pick up the following result which was needed in [B-H2] .

Ball and Helton 644

COROLLARY 2.3. Suppose F(e it) is a rational bounded self-adjoint and

invertible matrix function on the unit circle. Then there is a (not necessarily

outer) bounded rational matrix function R(e it) such that F(e it) = R(eit)*JR(eit).

Here J is the signature matrix J -- [0 Is_] where ~+ is the (constant) number

of eigenvalues of F(e it) greater than zero and ~_ is the number less than zero.

Proof. By Corollary 2.2(b) we can factor F as F(e it) = Q(eit)*D(eit)Q(e it)

where Q is outer and D is a winding matrix. But any winding matrix in turn

can be factored

D(e it) = A(eit) * J A(e it)

where A is an analytic trigonometric polynomial of the form

A(e it)

1 1

0

6

ik2t e

iklt e

The corollary follows by setting R = AQ.

We close this section by describing connections with other work. The posi-

tive integers kl >/k2 >/ "" >/kr appearing in the factorization results above are the

positive, (right) canonical partial factorization indices appearing in the earlier work

of Gohberg and Krein [G-K] . There have been recent efforts to construct expli-

Ball and Helton 645

cit formulas to evaluate them for some special cases (see the book of Clancey and

Gohberg [C-G]) . For the self-adjoint case which we consider here, generically it

happens that r - 0, that is, the partial factorization indices as defined by

Gohberg and Krein are all zero. This is shown for the self-adjoint case in

[B-H2], for the unrestricted case it is in [G-K].

Gohberg, Lancaster and Rodman [ G - L - R ] have recently obtained a very

general analogue for the real line of Corollary 2.2 above for polynomials. They

show that a polynomial P(z) with determinant not vanishing identically has a fac-

torization P(z) = Q(z)*J Q(z) for some signature matrix J and polynomial P if

and only if P has self-adjoint values on the real line With constant signature.

Their approach also gives detailed information concerning the factorization, includ-

ing spectral information for Q. Note the analog of kl + k2 + "" + kr is

1 order P - order Q. The Beurling-Lax approach given here has been successful 2

for factorization theorems on the real line for nonnegative matrix or operator poly-

nomials (see [R-R]) ; the best result for factorization of matrix polynomials self-

adjoint on the line which seems to follow directly from our methods is the follow-

ing.

2n THEOREM 2.4. Suppose P(z) = ~ Ak zk is an even degree matrix polyno-

k=0

mial with invertible self-adjoint values on the real line and with highest order

coefficient A2n invertible. Suppose also that the matrix function

Q(e it) = (1 + x(eit)2) -n P(x(eit)),

1 + e it where x ( e it) - - i ~ has zero factorization indices with respect to the unit circle.

Then P admits a factorization

P(z) = A(Z)* J A(z)

Ball and Helton 646

where A(z) is a matrix polynomial of degree n and J is a constant signatun

matrix.

The interested reader can construct the proof by mapping Corollary 2.2 abov~

from the unit circle to the real line and using some ideas in [R-R] . The mair

point is this. If P (x) - -A(x)*AA(x) with A outer and A constant, ther

(A*A)-IP equals A which is analytic in the U.H.P. and since (A'A) -1 is ana-

lytic and bounded in the L.H.P., A must be analytic in the L.H.P. with pole at

of order ~< order P. The Liouville Theorem says that A is a polynomial ol

1 order ~< order P. The argument that order A ~< -~- order P involves the com-

parison of outer factorizations of two functions P1 and P2 which satisf)

Pl(x) >/P2(x) for all x and is more subtle. The authors wish to thank I

Gohberg for a helpful conversation on this topic.

To extend this approach to the case of nonzero partial indices requires on~

thing which has not been worked out yet. The transform

Iz+i I

of the winding matrix D (e it) to the line can be replaced in our theorems by othe~

winding matrices; for example by o l ] J

[ - ~ ' l for Im a > 0. Thus we have a func.

tion of z which is singular at ~ where Im ~ < 0. One might think of the

[ G - L - - R ] factorization as one with a 'winding matrix' which has singularity at

oo. So ~i gap between our theorem and the [ G - L - R ] is deriving a correspon-

dence between winding matrices with interior singularity and 'winding matrices'

with boundary singularity; in particular at co.

§3. Inner-outer factorizations

In this section we derive inner-outer factorizations for meromorphic matrix

Ball and Helton 647

functions which are "compatible" with an indefinite metric [ , ]e~,. on

(N = m + m). We first consider an inner-outer factorization of the type con-

sidered by Potapov [Po]. Let J be the (N x N)-signaturematrix [ Im __0i l . , We

shall also have occasion to consider the ( 2 N x 2N)-signature matrix J = [J00j].

For any (m x m)-signature matrix j, the (m x m)-matrix function 0(z) is said

to be j-inner if it is meromorphic on the open unit disk, det 0(z) does not vanish

identically, 0(z)*j0(z) < j for all z in the open unit disk except for the possible

poles, and on the boundary of the disk

0(eit)*j0(e it) = j for a.e.t .

The matrix function F(z) is said to be a (j-analytic) j-contraction on the unit disk if

it is meromorphic on the unit disk and satisfies

F(z)*j F(z) ~< j

there except for possible poles. An outer j-contraction is any such function

whose inverse is j- bounded on the unit disk, i.e.,

[j F(z)-Ix,F(z)-lx] ~< M[jx,x]

F

for some M > 0 in H2(ILM). The following result is due to Potapov [Po]; his

proof is a brute force calculation and gives additional information on the form of

J-inner and outer factors, while our proof here is geometric. Also, to avoid messy

technicalities, we assume that our functions are uniformly bounded on the circle,

and that F is a strict J-contraction, i.e., either J - 15'(eit)*JF(e it) ~ 0 or there is

an ¢ > 0 such that

eJ ~ F(eit)*JF(e it) ~ (1 - e ) J .

Ball and Helton 648

T H E O R E M 3.1. Suppose the matrix function F(z) in L°°(MN) is a strict

J-contraction meromorphic on the unit disk, that is F(z) is pseudo-meromorphic

(i.e., OF 6 H°°(MN) for some inner function 0--see [ D - H I ) , and that det

F(z) ~ 0. Then there is a J-inner function B(z) and an outer J-contraction

G(z) such that F(z) = B(z)G(z).

Proof. If J - F(eit)*JF(e it) -~ 0 for a.e.t, choose B = F and G = I (the

identity). Thus assume J - F(eit)*JF(e it) >f eJ. This implies that the subspace

[F] H 2 ( ~ ) i s a uniformly negative subspace of L2(J). Let M be the subspace

FH2(L ~) of L2(J). Therefore there is an M-maximal negative subspace of M

which is uniformly negative. This means that M is a regular subspace of L2(J).

Since det F =--0, this subspace is a full-range subspace. Since F is pseudo-

meromorphic, this subspace is simply invariant for Me~t. Therefore by Theorem

1.1 Meit~vl is quasi-unitarily equivalent to Meit on H2(J). As in the proof of the

theorem in §2, the intertwining unitary operator must be multiplication by a

L 2 (MN)-function B satisfying B (e it ) *JB (e it ) = J. Thus F" H 2 ( ~ ) =

(B" H°°(L~)) -, so G - B-1F has the property that G . H°°(~) is contained

densely in H 2(L~). Therefore G is outer and clearly F --- B. G.

It remains only to show that both

morphic continuations to the unit disk.

modulus principle" from [B-H2]:

B and G have J-contractive mero-

For this we need the "J-maximum

If the rational matrix function K is J-contractive on the unit circle and

has no J-poles inside the unit disk, then F is also J-contractive inside

the unit disk.

(A J-pole z0 roughly is either a pole in the usual sense with residue living on a

positive space, or a point of analyticity where Ker K(zo) contains a negative sub-

Ball and Helton 649

space.) We apply this result to G = B-IF to see that G is J-contractive inside

the disk (G has no J-poles since G has no poles or zeroes, G is J-contractive

on the boundary since F is and B is J-isometric.) Also 1 G_ 1 _- 1 F_IB is J- E E

contractive on the boundary of the disk (since B is J-isometric and 1 F(eit) is E

J-contractive by assumption) and 1 G_ 1 has no J-poles inside the disk (since

G -1 is both pole- and zero-free there); therefore 1 G_ 1 is J-contractive on the

whole unit disk, so G is J-outer. Finally B = FG -1 is J-contractive on the

boundary of the disk and has no J-poles inside, and so by the same principle, B is

in fact J-inner.

Another type of inner-outer factorization is the following.

T H E O R E M 3.2. Suppose K is a Continuous matrix function in H°°(M N)

such that the boundary values K(e it) are invertible for all t. Then K has a fac-

torization

K(e it) = ~ (eit)A(e it)

where both _= and A are in H2(MN), A is outer and E satisfies

=_ (e it) *J~, (e it) = D (e it )

f .

for a unique winding matrix D.

Proof. Since the boundary value function K(e it) is invertible and continu-

ous, the function F(e it) -- K(eit)*JK(e it) satisfies the hypotheses of Theorem 2.1.

Thus F has a factorization of the form

Ball and Helton 650

F(e it) = A(eit)*D(eit)A(e it)

with A outer. But then

A(eit)*-lK(eit)*JK(eit)A(eit) -1 = D(e it) ,

that is, the function ,~(e it) ~--K(eit)A(eit) -1 is a generalized phase function.

Moreover, since A is outer _= is analytic, and clearly K = _=A.

§4. Darlington's embedding

For Jl and J2 two

J2 = [J2-0J ] " 2 Given any matrix

ated a linear fractional map

MN by the formula

ioatur, matr,, os, 011 and

Gg defined generically on all of MN with range in

Gg(m) = (am + fl)(Km + ,),)-1

for Km + y invertible. A more geometric picture is had by considering the graph

of m[{~]lL-~ c ~N I rather than m itself. Then for those g's for which Gg is

defined, we have

[Gglm']~ = [(am + fl' (rm + 3"-1 ] 1 L-~

= [('~m + fl)~Km + v)-'](,,:m + v)~ -- [: ~] [T]~

- -

Ball and Helton 651

Thus when Gg is viewed as acting on the graph subspaces of matrices rather than

on the matrices themselves, the action is simply multiplication by the symbol g.

We will have occasion to let g = 1~ v B] and m be functions on the unit circle.

Then we have an action defined by Gg(m)(e it) --= Gg(eit ) (m(eit)).

Let J1 and J2 be two N x N signature matrices, and let the corresponding

script letters J1 and J2 be the 2N x 2N signature matrices

J, IJ1 o1,2:IJ2 1 Then if the symbol g(e it) is a (J1,J2)-isometry for all t, the computation above

implies that the two self-adjoint matrices

Gg(m) (eit)*J2Gg(m) (e it) - J2, m(eit)*Jlm(eit) - J1

have the same signature for each t (i.e., the number of eigenvalues larger than

0, equal to 0, and less than zero is the same). In particular, for such a g, Gg

takes Jl-contractive valued functions m(e it) into J2-contractive ones.

Note that when J ----- J1 -- J2, the set of (J ,J)-unitary matrices is a group;

denote this group by U(J,J). Let R U(J,J) denote the group of rational matrix

functions whose values on the unit circle are in U(J,J). Then g---" Gg defines a

transformation group action (at least generically) of RU(J,J) on RL°°(M N)

(rational functions in L°°(MN). A natural problem for any transformation group

is to classify the orbits. The following theorem is a slight extension of a result

from [B-H2]. For j,k,~, three nonnegative integers with j + k + Z = N, we

define the set

RMj(j,k,~,) = {f 6 RL~°(MN) I

J - f(eit)*jf(e ~t) has j eigenvalues < 1 ,

k eigenvalues = 1 , and ~ eigenvalues > 1 for all t}.

Ball and Helton 652

THEOREM 4.1. The set of orbits of {Gg[g 6 RU(J,J)} acting on

RL°°(MN) containing a constant function is precisely the set { Mj(j,k,Z)[j,k,Z

nonnegative integers with j + k + .Z = N}.

The key to the proof is the following very general version of Darlington

embedding (see [B-H2]). The proof we give here does not involve the usual

explicit reliance on factorization results, but instead is very geometric. The proof

of Theorem 4.1 follows from the following lemma as in the proof of Theorem I. 1

in [B-H2], and will be omitted here.

LEMMA 4.2. Suppose the matrix function S is in RMj(j,0, ) (j + = N).

Then there is a rational g such that the boundary values g(e it) are

[[J1 -j10]'[J0 -0J]] "is°metries f°r all t where J1--[~ O£]1, and Gg(0) = S.

[J 01 Proof. Let J be the 2N x 2 N signature matrix _ j , and consider

[i S] H2(~) as a subspace of L2(J). Since J - S*JS is rational and is invertible

everywhere on the unit circle, this is a regular subspace of L2(J). Since S is

rational, we see also that it is simply invariant. By Theorem 1.1 we can write

[i S] H2(f2 "q) -- 0H2(~) for a phase function 0. By a signature argument on the

values 0(eit), we see that

0,e't, O I

Another simply invariant subspace of L2(J) which is [, ]U(j)-orthogonal to

[IS] H20L "N) is [I,] H2(L'~). As in the argument above, we therefore have

SI,] H2(¢2 ~) = qj H2(£ ~)

Ball and Helton 653

where

0(eit)*J~b(eit) = J1 for all t .

[.1 k , ]

g = [0 0] satisfies

M -- g H E(t~r~) where

Since S and S* are rational and M is a full range subspace, there is a rational

inner function 3' such that y M c H2(I~ N) and H2(J) 1"21 y M is finite dimen-

sional. From this it follows as in an argument of [B-H1] that g is rational.

Furthermore

By a computation above, this implies that Gg(0) = S. The lemma follows.

We conclude with the remark that our association of a subspace to the func-

tion S to obtain the Darlington extension of S is in the spirit of DeWilde's con-

struction for S E RMj(j,0,0)), c.f. [B-D] . He then took this space and analyzed

it algebraically rather than geometrically.

§5. APPENDIX--Proof of Theorem 1.3.

Assume that U is a fundamentally decomposable isometry on a Krein space

K which is of finite multiplicity N. Also assume that M is a simply invariant

pseudo-regular subspace such that ~ - - M N M' is finite dimensional and the

Ball and Helton 654

shift operator S is U~. We define a canonical basis for Mo as follows. First

choose a chain of vectors in Mo of the form {el, S e l , . . . , Skl-lel} and of maxi-

mal possible length for chains of this form in Mo. Choose a subspace M~ com-

plementary in 1Vlo to the span of { e l , S e l , . . . , Skl-lel}, and then choose a maxi-

mal chain {e2,Se2,...,sk2-1e2} inside ~ (k2 x< kl). After r steps (r < co

since Mo is finite-dimensonal) we get a basis for Mo of the form

S -- {e l ,Se l , . . . ,sk~-lel; e2, • • • ,Sk2-1e2; "" ; e r ,Se r , . . . ,Skr-ler} with the

additional property that S%ej ~ Mo for j = 1,...,r. Here kl >I k2 1> "" >/kr >I 1.

Such a collection S we call a canonical basis for Mo. It is not difficult to see that

the positive integers kl f> k2 >/ "-- >i kr are uniquely determined by S and Mo,

that is, they are independent of the particular maximal chains and complementary

subspaces chosen at each step. Let us call the integers {kl, • • • , kn} the winding

indices of the degenerate shifts.

We shall ultimately establish that S on M is quasi-equivalent t o Mei t on

an H2(D) space and that Mo maps to the isotropic space (H2(D))o of H2(D).

Thus it behooves us to take an arbitrary winding matrix D and analyze the action

of M e i t o n (H2(D))0 . Suppose D is a winding operator with indices

- - k 1 . . . . , k r ; 0 , . . . , 0 ; 0 . . . . , 0 ; , k r , . . . , k 1. I f E l , E 2 , . . . , E N i s the standard basis

for ~ , it is not diffacult to see that

i ( k l - 1 ) t i ( k 2 - 1 ) t S = { E l , e i t E 1 , . . . , e ~1;E2, • • • , e ~2;'";

• i ( k r - 1 ) t 1 Er , . . . , e ~ r /

is a canonical basis for (H2(D))o with respect to the shift Me~t on H2(D); thus

M e l t is a degenerate shift having the same winding indices {kl, •. • , kr} as S on

Mo. Note that, for 1 ~< Zl, ~ ~< N,

Ball and Helton 655

iJlt iJ2t 1 e E~l,e E~;21H2(D )

10 if ~ = N+I - Z 1

ffi and Jl =,J-2 + kh .

otherwise

Now that we have a grip on M0 we must analyze the remainder of M and

its relationship to Mo. Let N be the closed span of {SkM01 k >/0} and set

M1 = M ¢q N'. Then M1 is a degenerate space having isotropic space equal to

N. It follows as in the proof of Theorem 2.2 in [B-H1] that M1 = V SkL k>~0

where L = M f3 (SM)'; in particular, M1 is also invariant under S. Since M

is pseudo-regular, so is L, and L can be decomposed as L = Lr + L0 where

Lr is a Krein subspace of K and L0 is the isotropic subspace of L. (Of course,

it may happen that Lr = {0}). Since U is of finite multiplicity, it will turn out

that L~ must be finite dimensional. Let {er+l,..-,er+a,er+~,+!, • • • , er+c,+a} be

an orthonormal basis for Lr such that

01 j ~tZ [ej,ezl = j = Z ~< r+a

- 1 , j = ~, > r+a , j, Z = r+l , . . . , r+a+f l

Set a + = a and a - f f I B . Finally, for each p, 1 ~< p ~< r, choose a vector

eN+l-p which is orthogonal to the span of {S%pln >/kp+l} t3

{Snem I m ;~ p,1 ,<< m < r, n ~< kin} for which [Skpep,eN+l_p] = 1, and

[eN+l-p,eN+l-p] = 0. Here N = 2r + ~ +/3. This is certainly possible because

Skpep is a null vector not in M N M' by our construction. We next must argue

that the span of {Snem In >/0, m = 1,...,N} is dense in M. To see this suppose

that there were some vector x in which was simultaneously [ , ]-orthogonal to

{Snem In /> 0, m -- 1,...,N} and ( , ) -or thogonal to Mo. Since M is assumed to

be simply invariant the construction in Theorem 2.2 of [B-H1] implies

V SnL -- M N N'. So in particular x is [ , ]-orthogonal to M N N', that is n>~0

Ball and Helton 656

x is in N. Then, since x is also [ , ]-orthogonal to {Snem[n >~ 0,

N+l-r~<m~<N}, we must have that x is in M0. Finally, if also x is ( , ) -

orthogonal to M0, it follows that x = 0, and that {Snem In >~0, 1 ~< m ~< N} is

dense in M as claimed.

Now let the matrix function D be a winding matrix with indices

-kr, • • • ,-kl;0,.. . ,0; 0,...,0; kl, • • . , kr determined as above, and define a linear ~ m . . _ J

~ - - o ~ +

mapping U from the polynomials in H2(D) onto a dense subset of M so that

U: Meinte j ~ Snej; n /> 0, 1 ~< j ~< N .

Then it is easily verified that

[Up,Uq]M = [P,q]H2(D)

for any EN-valued polynomials p and q, and tha t UMei t = SU. Thus S is

quasi-unitarily equivalent to Me~t on H2(D) as asserted.

Acknowedgement

The authors would like to acknowledge their appreciation for the work of

Neola Crimmins in preparing this manuscript for typeset.

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Department of Mathematics Virginia Tech Blacksburg, VA 24061

Department of Mathematics University of California, San Diego La Jolla, CA 92093