ANNALS OF MATHEMATICS STUDIES

NUMBER 4

AN INTRODUCTION TO LINEAR TRANSFORMATIONS

IN HILBERT SPACE BY

F. J. MURRAY

PRINCETON PRINCETON UNIVERSITY PRESS

LONDON: HUMPHREY MILFORD

OXFORD UNIVERSI rv PRESS

Copyright I 941 PRINCETON UNIVERSITY PRESS

PRINTED IN U.S.A..

Lithoprinted by Edwards Brothers, Inc., Lithoprinten Ann Arbor, Michigan, 1941

The theory of operators

~'~--PREFACE.'·--- _:~

'• · .. 'i' ., --·--- ,/; ---.. j • .-<·:'

in Hilbe~t~§'i'if~'has its roots in the theory of orthogonal functions and integral equations. Its growth spans nearly half a century and includes investigations by Fredholm, Hilbert, Weyl, Hellinger, Toeplitz, Riesz, Frechet, von Neumann and Stone. While this subject appeals to the imag­ination, it is also satisfying because due to its present abstract methods, questions of necessity and sufficiency are satisfac~orily handled. One can therefore be confident that its developement is far from complete and eagerly await its further growth.

These notes present a set of results which we may call the group germ of this theory. We concern ourselves with the struc­ture of a single normal operator and at the end present the reader with a reading guide which, we believe, will give him a clear and reasonably complete picture of the theory-.

Fundamentally the treatment given here is based on the two papers of Professor J. von Neumann referred to at the end of Chapter I. kn attempt however has been made to unify this treatment and also recast it in certain respects. (Cf. the introductory paragraphs of Chapter IX). The elementary portions of the subject were given as geometrical a form as possible and the integJ;>al representations of unitary, self-adjoint and normal operators were linked with the canonical resolution.

In presenting the course from which these notes were taken, the author had two purposes in mind. The first was to present the most elementary course possible on this subject. This seemed desirable since only' in this way could one hope to reach the students of physics and of statistics to whom the subject can of'fer so much. The second purpose was to emphasize those notions which seem to be proper to linear spaces and in partic­ular to Hilbert space and omitting other notions as far as pos­sible. The importance of the combination of various notions cannot be over-emphasized but there is a considerable gain in clarity in first treating them separately. These purposes are not antagonistic. We may point out that the theoretical por­tions of this work, except §4 of Chapter III, can be read with­out a knowledge of' Lebesgue integration.

PREFACE

On the other hand~ for these very reasons, the present work cannot clairo to have supplanted the well-known treatise of M. H. Stone or the lecture notes of J. van Neumann. It is s1111ply hoped that the student will .f'ind it advantageous to read the present treatment first and follow the reading guides given in Chapters XI and XII in consulting Stone's treatise and the more recent literature.

To those familiar with the subject, it will hardly be neces­sary to point out that the influence of Professor von Neumal'lil is effective throughout the present work. Professor Bochner of Princeton University has also taken a kind interest in this work and made a number of valuable suggestions. I &~ also deeply grateful to my brother, Mr. John E. Murray, whose valuable assis­tance in typing these lecture notes, was essential to their prep­aration.

Columbia University, New York, N. Y. May, 1940

F. J. Murray

TABLE OF CONTENTS

Pref'ace

Table of' Contents

Chapter I. INTRODUCTION

Chapter II. IIlIBERT SPACE §1. The Postulates •••

§2. Linear Normed Spaces • §3. Additivity and Continuity

§4. Linear Functionals

§5. Linear Ma.nif'olds §6. Orthonormal Sets

Chapter III. REALIZATIONS OF IIlIBERI' SPACE §1. Prel:l.Jninary Considerations •••••

§2. 12 • • • • • • •

§3. n1 $ $ nn and n, $ n2 $ •••

§4. s::2 • • • • • • •

Chapter IV. ADDITIVE AND CLOSED TRANSFORMATIONS • §1. The Graph of' a Transf'ormation §2. Adjoints and Closure ••• §3. S-ymmetric and Self'-adjoint Operators

§4. C.a.d.d. Transf'ormations • • .•••••

Chapter V. WEAK CONVERGENCE §1. Weak Completeness

§2. Weak Compactness §3. Closed Transf'ormations with Domain n.

Chapter VI. PROJECTIONS AND ISOMEI'RY §1. Projections . • • • • . • • • • • §2. Unitary and Isometric Transf'ormations

Page

4 4

6

7 11

14

16

22

22

23 26 27

45

45

47 48

51

51

56

TABLE OF CONTENTS

Chapter VI. (Continued) §3. Partially Isometric Transformations §4. C.a.d.d. Operators ••••••••

Chapter VII. RESOLUTIONS OF THE IDENTITY §1. Self-adjoint Transformations with H Finite

Page

60

62

64

Dimensional • • • • • • • • 64 §2. Resolutions of the Identity and Integration 67 §3. Improper Integrals • • • • • • • • • • • 73 §4. Commutativity and Normal Operators 77

Chapter VIII. BOUNDED SELF-ADJOINT AND UNITARY TRANSFORMATIONS.. • 81

§1. Functions of a Bounded H 81 §2. (Hf,g) • • • • • • • • • 85 §3. Integral Representation of a bounded H 88 §4. Integral Representation of a Unitary Operator 89

Chapter IX. CANONICAL RESOLUrION AND INTEGRAL REPRESENTATIONS • •

§ 1 • The Canonical Resolution §2. Self-adjoint Operators §3. Normal Operators •••••

Chapter X. SYMMEI'RIC OPERATORS § 1 • The Cayley Transform • • §2. Structure and Existence of Ma.x:llna.l Symmetric

Operators

Chapter XI. REFERENCES TO FURTHER DEVELOPMENTS §1. Spectrum • • • • • • • • ·• • ••• §2. Operational Calculus •••••••• §3. Commutativity and Normal Operators §4, Symmetric Transformations §5. Infinite Matrices §6, Operators of Finite Norms §7. Stone's Theorem §8. Rings of Operators

95 95

1 00

102

110 110

116

122

122

123

124 125

125 126 125

127

TABLE: OF CONTENTS

Chapter XII. REFERENCES TO APPLICATIONS §1. Integral and Other Types or Operators §2. Dif'rerential Operators §3. Quantum Mechanics §4. Classical Mechanics

Page

130

130

130

131

131

CHAPI'ER I

The expressions:

or

T2f = p(x)i:ixf(x)+q(x)f(x)

or in the case of a function of two variables,

T f = a2fi2f 3 ax2 ay2

are linear operators. Thus the first two, when applicable, take a function defined on the un:1.t interval into another function on the same interval.

Now if we confine our attention to functions f(x) continu­ous on the closed unit interval and with a continuous derivative, we know that such a function can be expressed in the form,

f ( x) = r:=-oo xO!exp ( 2'1tiOOC)

where xoc=J1f(x)exp(-2'1tiO!x)dx. If Tif is of the same sort, 0

Tif =r:=-= yocexp(2'1ticxx:)

where Yot = j 1Tif exp(-2niocx)dx = 0 ~

I:;.-.!~ J0 T1 (exp( 2n~x) )exp(-2Td!xx)dx =I:f;-ao x~a~,a· Now for T3 a somewhat similar argument holds, although it

is customary to use a double summation. The important thing to notice is that the operator equation

Tf = g

can be, in these cases, replaced by an infinite system of linear equations in an infinite number of unknowns. We shall prove that this can be done in far more general circumstances.

One might attempt to solve such an infinite system of equa­tions by substituting a finite system and then passing to the limit, for example one might take the first n equations and ignore all but the first n unknowns. But this process is' in-

2 I. INTRODUCTION

effective in general and introduces certain particular difficul­ties of its own.

Other methods must be sought. The choice of the functions e:xp(2niOQ() corresponds to a choice of a system of coordinate a.xes in the case of a finite number of unknowns. In the finite case for a synnnetrical operator, the coordinate system can be chosen, so that,

6a,B = o if a + B,

6CX,<X = 1. (Cf. Chap. VII, § 1 , Lemma 4) Correspondingly in the infinite case we would seek a complete set of functions ~ex such that

When this has been done, inverting the equation becomes a simple process. For example, consider, T = 1:ix , ~cx(x) = e:xp(-2nicxx:). While this is in general impossible, nevertheless an effective method of generalizing the result in the finite dimensional case exists and we shall discuss it in the present work.

We shall want to give our discussion its moat general form and for that reason we consider not the set of functions whose square is summable, but rather an abstract space which has just those properties of this set which are needed in our develope­ment. This space, ~ , is called Hilbert space and we shall show in Chapter II the existence of something equivalent to orthogonal sets of functions.

In Chapter III, we discuss L2 and other realizations of abstract Hilbert space. In Chapters rv, V and VI, linear trans­formations are studied and certain preliminary properties estab­lished. Also a notion "weak convergence," which is of consider­able interest in the theory of abstract spaces, is introduced to establish Theorem V of Chapter v.

In Chapter VII, we shall develope the needed generalization of the notion of an operator in diagonal form. In Chapter IX, we shall show that a self-adjoint operator even if it is discon­tinuous, is expressable in the diagonal form.

Synnnetry is not sufficient in the discontinuous case as we shall see. The distinction between synnnetric and self-adjoint transformations is brought out in Chapter X. In Chapter XI, a

INTRODUCTION 3

brief outline of further developements in the theory is given. Except for Chapter XI, our discussion is based on the follow­

ing:

(1) F. Riesz and E. R. Lorch. Trans. of the .Amer. Math. Soc. Vol. 39, pp. 331-340 (1936).

(2) M. H. Stone. Colloquium Publications of Amer. Math. Soc. Vol. XV ( 1932).

(3) J. von Neumann. Math. Annalen Bd. 102 pp. 49-131 (1929). (4) J. von Neumann. Annals of Ma.thematics, 2nd series, Vol.

33, pp. 294-310 (1932).

CHAPI'ER II

The axiomatic treatment of Hilbert space was first given by J. von Neumann in. (4) pp. 64-69. He proposed the definition given below. We follow here the discussion given by Stone (2). (Numerals in parentheses refer to the references cited at the end of Chapter I.)

DEFINITION 1 • 1 • A class n of elements f, g, ••. is called a Hilbert space if it satisfies the following postulates:

POSTULATE A. n is a linear space; that is, (1) there exists a commutative and associative oper­

ation denoted by.+, applicable to every pair f, g of elements of n , with the property that f +g is also an element of n .

(2) there exists a distributive and associative operat~on, denoted by • , applicable to every pair (a,f), where a is a complex number and f is an ele­ment of n ;

(3) in n there exists a null element denoted by e with the properties

f+e = f, a·0 = e, O•f = 0.

POSTULATE B. There exists a numel"ically-valued f'unction (f,g) defined for every pair f, g of ele­ments of n , with the properties:

(1) (af,g) = a(f,g). (2) (f1+f2,g) = (f1,g)+(f2,g). (3) (g,f)=1T,gj. (4) (f,f) ~ o. (5) (f,f) = o if and only if f = e. The not-negative real number (f,f) 1/ 2 will be de­

noted for convenience by If I •

POSTULATE c. For every n, n = 1, 2, 3, ••• , there exists a set of n linearly independent elements

4

§1. THE POSTULATES

of n ; that is, elements f1, •.• , fn such that the

equation a1f1 + . • . +a.rln = e is true only when a1 •.• = ~ = o.

POSTULATE D. n is separable; that is, there exists a denumerably infinite set of elements of n , f 1, f 2 ,

.•. , such that for every g in n and every positive 6 there exists an n = n(g,£) for which lfn-gl < e.

POSTULATE E. n is complete; that is, if a sequence {fnl of elements of n satisfies the condition

then there exists an element f of n such that

In this statement of the postulates, certain notations and conventions were introduced. These are (a) -f = (-1)f, (b) f-g = f+ (-1 )g, (c) a denotes the complex conjugate of a.

5

We shall also use (d) af = a·f, (e) R(a) is the real part of' the complex number a , J(a) is the imaginary part of a , (f) lal is the absolute value of a •

The properties B(1) - B(5) imply

B(6) (f,ag) = a(f,g) B(7) (f,g1+g2 ) = (f,g1 )+(f,g2 )

B(8) la.fl = lal · lfl

We shall next prove that these imply

B(9) I (f,g) I ~ (fl· lgl, the equality sign holding if and only if f and g are linearly dependent.

For if Tl• 71. and µ are real and 71.2 +µ 2 + o then

is e for some choice of Tl, 71. and µ if and only if f and g are linearly ~dependent. Thus by B( 4) and B( 5 )

6 II. HILBERT SPACE

(ld'+exp(1r1)µg,i\.f'+exp(iri)µg) ~ o

and equality can only occur when f' and g are linearly depen­dent.

Expanding by means of' B( 2), B( 1 ) , B( 7) and B( 6) and using, with B(3) the f'act that .f'or any complex a, a+a = 2R(a), we obtain

with equality possible only if' f' and g are linearly depen­dent.

Now we can choose ri so that exp~iri)(f',g) =-I (f',g)I. Then the equation becomes in the linearly independent case

A21f'l 2-2Aµl(f',g)l+µ 2 lgl 2 ) O.

Now f'or linear independence, lgl + o and thus if' we let ii. = I g I , µ= I< f', g l I I I g I , we get

lf'l 2·1gl 2-l(f',g)l 2 > o.

On the other hand if' f' and g are linearly dependent it is easily seen that the equality holds.

B(10) lf'+gl ~ lf'l+lgl, with equality possible only if' f' and g are linearly dependent.

Proof':

lf'+gl 2 = (f'+g,f'+g) lf'l 2+2R(f',g)+lgl 2 ~

If' 12 +2 If' I • I g I+ I g 12 = < Ir I+ I g I > 2

§2

A weaker restriction than B is the postulate:

POSTULATE B 1 • There exists a real valued f'unction If' I of' elements of' ~ with the properties B( 4), B( 5), B( 8), B( 1 o).

The f'unction f f'I is called the norm. If' a space satisf'ies

§3. ADDITIVITY AND CONTINUITY 7

postulates A, B', and E it is usually referred to as a Banach space.* If D is also satisfied, the space is called separable. Thus Hilbert space is a separable Banach space but there are, as we shall see, separable Banach spaces, which are not Hilbert spaces.

The relation between Hilbert space and a separable Banach space is clearer if we consider

This equation is an immediate consequence of the equation

Thus in Hilbert space, we have B' and B( 11) and it can be shown that B' and B(11) are sufficient to insure that a separable Banach space is a Hilbert space.**

The major purpose of this book is to give as simply as possi­ble certain results in the theory of Hilbert space and these specific results do not hold in general separable Banach space. However the Hilbert space theory can be more clearly understood if one appreciates the precise dependence of this theory upon certain specific properties of Hilbert space. For this reason, we shall endeavor to give the fundamentals of our subject, without restricting ourselves to Hilbert space, to the largest extent consistent with our purpose.

§3

If the linear space £ has a norm lfl, then d(f,g) = lf-gl is a metric for th~ space, i.et, satisfies the conditions

(i) d(f,g) ~ o, d(f,g) = o if and only if f = g. (ii) d(f,g) d(g,f). (iii) d(f,g) ~ d(f,h)+d(h,g).

These conditions are consequences of B( 4), B( 5), B( 8) and B( 1 o ).

* These spaces have been investigated in a famous treatise · "Theorie des Operations Lineaires." bys. Banach (Warsaw ( 1 932)). ** J. von Neumann and Jordan, Anna.ls of Ma.thematics, vol.36 (1935), pp. 719-724.

8 II. HILBERT SPACE

Thus we are invited to introduce the notion of continuity in such a space.

DEFINITION 1. Let F(f) be a f\mction defined on a subset of £. This subset is called the domain of F. Let f 0 be an element of the domain of F. If for every & ) o, it is possible to find a 6 such that if f is in the domain of F and lf-f0 1 < 6, then IF(f)-F(f0)1 ( &, we say that F i.s continuous at f 0• If F is continuous at every point of its domain, F is said to be continuous.

If F assumes only complex numbers as its valuAs, it is called a f\mctional. Thus If I itself is a continuous f lctional.

DEFINITION 2. A f\mctioii F(f) will be cal'/ additive if whenever f and g are in its doma: . af +bg is also in the domain for any two complex mbers a and b and F(af+bg) = aF(f}+bF(g).

Notice that these definitions apply not only to f\mctionals but even to f\mctions, which assume values in any linear space.

LEMMA 1. For an additive f\mction F(f), and any f 0 in its domain, the following statements are equiva­lent.

(a) F is continuous at f 0• (b) F is continuous at e. (c) There exists a C such that IF(f)I ~ Cffl

for every f in the domain of F.

We note firstly that if f 0 is in the domain of F, f 0-f 0 =

e is also in the domain of F and F(0) = F(f0)-F(f0) = 0 1 •

(0 1 is the null element for th~ space of the values of F.) The element f is in the domain of F if and only if

h = f-f 0 is in the domain. Also

lf-f01 = lhl = lh-el

and

§3. ADDITIVITY AND CONTINUITY

IF(f)-F(f0 ) I = IF(f-f0 ) I = IF(h)-e 1 I = IF(h)-F(e) I. These statements give precisely the equivalence of (a) and (b) by a substitution.

We will show that (b) and (c) are equivalent. Suppose (b).

9

Then if & is taken as 1, the continuity at 9 implies that there is a o such that when lhl < o, h in the domain of F, then, IF(h) I < 1. Now if f is any arbitrary element of the domain, af (= af+1 ·9) is in the domain for every a. Let a = o I 2 • If I . then h = ar is such that I h I = I ( o I 2 • If I l · f I = 0/2 < o. Hence

1 ) IF(h)I = l!3F(f)I = lallF(f)I = (o/2lfl )• IF(f)I

or lf1(2/o)) IF(f)I, and hence 2/0 is a constant for which (c) holds. Thus (b) implies (c).

Now let us suppose (c) and that an e Let o be such that Co < e. Then if IF(h)-F(e)I = IF(h)I ~ Clhl < Co ( e. at e and (c) implies (b).

) o has been given. lh-91 < o, we have

Thus F is continuous

THEOREM I. Im. additive function F(f) is continuous at every point if it is continuous at one point. Im.

additive function F(f) is continuous if and only if there exists a C such that for every f in its domain F, IF(f) I ~ Clfl.

A set S in n will be called additive if whenever f and g are in it, af+bg is in it for every pair of complex numbers a and b. A closed additive set, m, will be called a linear manifold.

It is easily verified that the closure of an additive set is also additive and hence is a linear manifold. This depends on the fact that the limit of a linear combination in a Banach space is the linear combination of the limits •. Let fn ~ f and 15:ri ~ g. Then since

laf+bg-(afn+b15:ti) I = la(f-fn)+b(g-15:ri) I ~ lal lf-fnl+lbl lg-15:ril - o

we have that if f and g are in the closure of a linear set

1 o II. HILBERT SPACE

a.f+bg is also.

THEOREM II. The domain of an additive function F(f) is additive. If F(f) is also continuous, with values in a complete space, there exists a continuous additive function, [F} with the properties

(a) The domain of [F] is the closure of the domain of F.

(b) If f is in the domain of F, [F](f) = F(f). This [F] is unique.

The first statement is obvious from the definitions. We will prove our statements concerning [F] by specifying its values uniquely. Now if f is in the domain of F, [F](f) = F(f). Let f be any point of the closure of the domain of F. If

!fnl is a sequence of elements of the domain of F, such that fn---+- f, then the F(fn) 's are also convergent since IF(fn)-F(fm) I = IF(fn-fm) I ~ Clfn-fml ---+ o as m and n ---+oo Since the values of F(fn) are in a complete space, they ma.st converge to an f*. !my two sequences {f~! and {f~l with the same limit f must have lim F(f~) = lim F(f;;_) since otherwise the sequence of' F(f) 's consisting of elements which are takf:n alternately from one and then the other sequence of F( f) 's wou,ld have no limit. Thus f* depends only on f. We may take [F](f) = f*. (No contradiction with the previous definition of [F] on the domain of F is possible, for if f is in the domain of F, we may take fn = f). Furthermore if [F] is continuous,. this must be the definition. Thus the con­ditions (a) and (b) determine [Fl precisely.

To complete our proof it is only necessary to show that [Fl is additive and continuous. The additivity is a consequence of the facts given in the paragraph preceding the theorem, that the closure of an additive set is a linear manifold and that the limit of a linear combination is the linear combination of the limits. The continuity is shown by noting that if C is such that IF(f)I ~ Clfl for every f in the domain of F, then I [F](f)I -~ Clfl for evecy: f in the closure of this domain. Such C 's exist by Theorem I and the same theorem shows that this implies continuity.

§4. LINEAR FUNCTION.BLS

§4

f€C. and

For a

11

An additive functional which is defined for all which is continuous, is called a linear functional. Banach space, C. , the set of linear functionals, C.*, is again a Banach space as one can see as follows.

Firstly, we notice that the set of linear functionals satis­fies Postulate A if we define the sum of two linear functionals F+G by the equation

(F+G)(f) = F(f)+G(g)

and scalar multiplication by the equation

(a.F)(f) = a.F(f).

To prove Postulate B', we define IFI .9.s the gr. 1. b. of the C 's for which IF(f)I ~ Clfl for all f € c.. IFI is readily seen to be the least such C. B(4) and B(5) are obvi­ous from this definition, B(8) and B(10) follow from the defini­tion of scalar multiplication and of addition given in the preceding paragraph.

To prove Postulate E, we consider any sequence fFn! of linear functionals, and such that 1Fn-Fm1 - o as n and m ---. CD • It is readily seen that for each element f of C. ,

IFm(f)-Fn(f) I = I (Fm-Fn)(f) I ~ IFn-Fml ·I.fl -- o

as n and m --1> CD and furthermore, this approach to zero is uniform on those f 's for which lfl = 1. Thus Fn(f) has a limit F(f) for every f in the space. It is easily seen that F(f) is additive and that there is a C such that IF(f) I ~ C· lfl for every f € C. •

Now given e , take N so large that for n and m > N,

1Fn-Fm1 < e. This means that we have

IFn(f)-Fm(f) I ~ el.fl.

Let us fix m, and let n ~ CD • We then obtain

IF(f)-Fm(f) I ~ e !fl.

Thus for m > N, IF-Fml < e. This implies that F is such that Fm---+- F as m - CD and hence that f.* is complete.

THEOREM III. The set C.* of linear functionals on

1 2 II. HIT.BERT SP.ACE

a Banach space £ is again a Banach space.*

Now one of the essential facts concerning Hilbert space is that n* is equivalent to n . The specific relation is given by the following theorems.

THEOREM N. If F

the Hilbert space n, that for every f € n,

is a linear functional defined on then there exists a g E n such F(f) = (f,g).

Proof: If F = o, we can let g = e. Suppose then that IFI ) o. If we are given a sequence of positive numbers {en! with ~--.,> o, we can find a sequence !~I of elements such that

IFI • l~I ~ IF(~) I ~ ( 1-enl IFI • l~I

and F(~) + o. If we multiply ~ by 1 /IF(~) I we obtain a sequence 15:ri with F(15:ril = 1 and

IFI • l15:ril ~ 1 ~ (1-enl· IFI · l15:ril.

Now consider I 15:ri +~I • We have

IFl·l15:ri+~I ~ IF(15:ri+~ll = 2 ~ IFl·(1-enl·l15:ril+IFl(1-eml·l~I

or

Thus

l15:ri-~1 2 = 2(l15:ril 2+1~1 2 l-115:ri+~l 2 ~ 2 ( I 15:ri 12+I0m1 2 l- ( ( 1 -en l · I 15:ri I + ( 1 -Em l • I~ I l 2

* A proof of the fact that Postulate C for £ implies C for £* can readily be given if the Hahn-Banach Ex:tension Theorem is shown (Cf. Banach loc. cit. pp. 27-29). This has the conse­quence that if F is additive and continuous on a linear subset G1 its definition can be extended throughout·the space without increasing the norm. A proof of this is not on the main line of our developement but if this is assumed, one would proceed as follows.

Let f 1 , • • • , f , be n linearly independent elements of and G the set ofllJ.inear combinations of these. It is easily seen that one can define n linearly independent linear func­tionals F1, ••• , Fn on G. These can then be extended to the whole space by the extension theorem and this does not effect their linear independence.

§4. LINEAR FUNCTIONALS

Since 1~1----+ 1/IFI and en----+ o we have l~-~1 2 -- o as n and m - cc • Hence the ~' s form a convergent sequence. Define g so that ~----+ g. Then lgl = 1/IFI, F(g) = 1. Now if h is such that F(h) = o, we have that

, = IF(g)I = IF(g+Ab.)I ~ IFl·lg+i'Jll = lg+Ab.l/lgl

or for eve'!'y A,

Squaring we must have

lgl 2 ~ lg+Ah.1 2 = lgl 2+2R(A(h,g))+IAl 2 ·1hl 2 •

13

Now we can choose A so that 2R( A(h,g)) = -2 IA I· I (h,g)I. Thus

lgl 2 ~ lgl 2-2ri·l(h,g)l+ri2 1hl 2 for eve'!'y ri) o. But this is possible only if I (h,g) I = o. Thus if F(h) = o, (h,g) = o.

If h is arbitra'!'y, h = F(h)g+h' where F(h') = F(h-F(h)g)= F(h)-F(h)·F(g) = o. Let g0 = (l/lgl 2 )g. Then

(h,g0 ) = (F(h)g+h',g0 ) = F(h)(g,g )+(h 1 ,g0 ) =

F(h)(1/lgl~)(g,g)+(1/lgl 2 )•(h 1 ,g) = F(h),

using the fact that (h' ,g) = o since F(h') =O. Thus g0 satisfies the condition of the theorem.

The converse of Theorem N is the following:

THEOREM V. The equation (f,g) = F(f), f €fl defines for each g, a linear function F with IFI = lgl.

Proof': F is obviously additive. Also

IF(f)I = l(f,g)I ~ lfl · lgl.

This implies that F is continuous and IFI ~ lgl. Since however IF(g)I = lgl 2 = lgl·lgl," IFI ~ lgl,- and thus we obtain the theorem.

Theorem V tells us that (f',g) is continuous in each variable, separately. But since

I (f+6f ,g+6g)-(f ,g) I I (61' ,g)+(r ,6g)+(6f ,6g) I

~ l(6f',g)l+l(f,6g)l+l(6f,6g)I

~ l6fl·lgl+lfl·l6gl+l6fl·l6gl,

it is easy to show that (f ,g) is continuous in both variables.

14 II. HIIBERT SPACE

§5

The relation between linear f'unctionals and the elements of n has the following consequences. Consider a set S in any Banach space E... We can consider s• 1 the set of linear f'unctionals F such that F(r) = o for every f in S. It can be shown without difficulty that s~ 1 is a linear manifold in E..*. (The additivity is obvious and the closure is shown, by

recalling that H F is a limit of the sequence Fn, Fn(g) -­F ( g) for every element in E.) • If E. = n, we have correspon­ding to S.L 1 , a set S.L, in n, for which F € S.1.1 and F(r)= (f,g) for all f € n, imply g € S.1.. Thus ordinarily in a

Banach space the orthogonal complement S.1.1 to a set S must be regarded in E.*, but in n we may take S.L in the space it sell.

Now s.1.. as we have defined it above consists of all the g € n n, for which (f,g) = 0 for all g € n. This too is readily seen to be a linear manifold. In the case in which S is itsell a linear manifold m , we have the essential theorem:

THEOREM VI. Let m be a linear manifold in n and let m.L be as above. Then if

f = f,+r2, where f, €'11, is tmique.

f is an arbitrary element of n, f 2 € m.1., and this resolution

We first note that any such resolution f = f 1+f 2 must be tmique since if we have f = f 1+f2 and f = f{+f2, then g = f,-f{ = q-r2 is in both m and m.L and hence fgf 2 = (g,g) = o and g = e.

Now if f is in 711, f+0 = f is the desired resolution. We can suppose then that f is not in m. Consider P the set of .. elements f-g, WhE1re g € 7n. Let r = gr.l.b. ff-gf, g € '1l • Now r + o, since otherwise we will have a sequence Sn. such

that If-Sn_ I - 0. and Sn. -- f. Sin-ce m is closed this would imply that f € m contrary to our hypothesis.

We can therefore find a sequence l1n_ in the form f-Sn_, Sn. €

m and such that f!1n_I __. r. Since ~(!1n_+~) = f-~(~+Sm_), we have

§5. LINEAR MANIFOLDS 15

Then by B(11);

ll\1-1\nl 2 = 2(11\il 2+11\nl 2 )-11\i+1\nl 2 ~ 2(Jh:n_l 2+11\nl 2 )-4r2 •

Since 11\il ----+ r, we see that 11\i-1\nl - o as n and m -+ex> and thus the 1\i' s converge to some element h with !hi = r. The ~ = f-1\i also converge to a g € m and thus h = f-g is a minimal element of P.

Now if g I . is any element of' m, h+ll.g I = f- ( g-71.g I ) is in the set P. Thus

lhl ~ lh+Ag' I for every value of 71. • As in the proof of' Theorem IV, this implies (g' ,h) = o. Since g' was any arbitrary element of m, h must be in'7l.L.. Thus the resolution f = g+h is the desired one with g € m, h € m"'".*

COROLLARY 1 • folds with m1

where f' 1 € m1

If' m1 c m2 ,

and f' 2

and m 2 are two linear mani­

then if' f' € m2, f = f1+f2

€ '1l2·77l~.

We must show that in the resolution f' = f' 1+f'2, f'1 £ m1 , f' 2 € '7!~, we have f' 2 € 77l 2 • Since f' and f' 1 are in m2, this is true.

As a consequence of' this, we have,

COROLLARY 2. If' '7!1 C '71 2 but m 1 + m 2, then m2 C

m~ c m:t but 'm1 + m~. THEOREM VII. If 77l is a linear manifold, ( '1l .1.) .L = 77l •

Proof': It is readily seen that '71 C('m ... ) .... We must demon-

* In some spaces other than n, it is possible, given an F € £*, to f'ind a g f'cir which F(g) = !Fl· lg!. Furthermore inequali­ties similar to B(11) hold in these spaces (Cf. J. A. Clarkson, Trans. of the Amer. Ma.th. Soc. Vol. 4o, pp. 396-414, (1936)). These inequalities imply that the correspondence is not additive.

Furthermore it is possible to show that in these spaces (not n), there is not even a generalized equivalent of' Theorem VI. For it can be shown that there exists linear manifolds '7l1 ,. f'or which no linear manifold 77l2 exists such that '7!1 •'7!2 = fel and f' = f'1+f'2 1 f'1 € 77l1, f'2 € '7!2.. f'or every f' € £. (Cf'. F. J. Murray, Trans. of' the Amer. Ma.th. Soc. Vol. 41, pp. 138-152, (1937)).

1 6 II. HILBERT SPACE

strate (m..1.).L Cm. Let f € ('1l.L).L. We have f = f 1+f2, f 1 E m, f 2 E m.L. Since f 2 = f-f 1 and f E ('1l.l.)..a., f 1 E '1l C ('1l...1.).l., we have f 2 also € ('1l.l.).l.. Hence f 2 €

('1l.l.)(m...l.).J. = {el and f = f, € m. Thus f € ('1l.1.).l. implies

f € m and this completes the proof.

§6

If S is an arbitrary set of elements, let U(S) denote the set of linear combinations of the elements of S, i.e., the set

of a 1f 1+ ••• +~fn, fi Es. For this notion, the following properties are easily obtainable: U( S) .L = s.L. If s 2 C s 1 ,

U(S2) c U(s,). Also if s2 c U(s, ), U(S2) c U(s, ). The closure of U( S), we denote by 71l( S). For this, again

we have, 71l(S).l. = U(S).l. = s.i., and if S2 C '1HS1 ), '1l(S2 ) C

7n ( S1 ) • It follows from these and Theorem VII that 71l( S) = ('1l(S).._).._ = (S.J.).l..

To develope these notions further, we prove the following lemma:

LEMMA 1. If £ is a separable metric space, and S is a non-empty subset of £, then there exists a finite or denumerably infinite subset s 1 of S which is dense in s.

Proof: Let f 1 , f 2, • . • be dense in £, We define for each a, a set of elements g r.t in s, which will be finite a,,. or denumerably infinite depending on a. Let r°' = gr.Lb. If -gl, g € S. If r_ = o, we can chose a sequence g n

°' - °'' such that g n ~ f • If r is not o, we can find a gES °'' °' °' such that lf..:gl < 2r. We let g 1 be such a g and let ,

°'' . 1 g 2, g 3 , •.• , remain undefined. In this last case -2 1f -g I ~ • °' °' < r < If -gl for every g € s. There is at most a denumerable

°' . number of the g n• a,,.

Now suppose a g € S and an E ) o are given. Choose ex

so that lfcx-gl < E/3. If ra= o, we can find a gcx,k such that If -g kl < 2E/3. If r_ + o, we have that If -g 1 1

. °' a, - °' °'' < 2lfcx-gl < 2e;3. Thus we can find a gcx,k such that lg-gcx,kl < E. Thus the set fgcx,~! is dense.

A set of elements S, will be said to be orthonormal if for

<1>-E S, l<1>I = 1 and if <I> and ljl E S and <I> + ljl, then (<!>,ljl)=O.

§6. ORTHONORMAL SEI'S

T:irn:OREVI VIII. Pn orthonormal set S in n can con­tain at most a denumerable set of elements.

17

Let f 1, f 2, be a dense set in n. For each <I> of S we choose an °' = cx<I> such that 1<1>-fcxl < ~· This correspon­dence <I> "' ex is one-to-one. For by our choice of ex, to each <I> there is only one cc • Furthermore to each ex, there is at most one <I>, since if <1> and ljl € S are such that ex = cx<I> =

cxljl' then

1<!>-ljll = l<<1>:-fcx)-(1j1-f0t:)I ~ f(<1>-fal+l1j1-fcxl < 1.

But if <I> + ljl, then

f<!>-ljl1 2 = l<1>1 2-2R((<1>,ljl))+l1jil 2 = 2.

This contradicts 1<1>-ljll < 1 so we must have <I> = ljl. Since the elements of S, are in a one-to-one correspondence

with a subset of the positive integers, there is at most a denum­erable nUl!lber of them.

We can therefore enumerate the elements of an orthonormal set S, <1>1, <1>2, ••• Then the orthonormal condition can be written (<l>cx,<l>B) = 6cx,B where 6cx,B is the Kronecker symbol and equals zero if cc + B and one for ex = B·

THEORE!v'I IX. Given any denumerable set S not all of whose elements are e, we can f'ind an orthonormal set s1 such that U(S1) = U(S).

Let g1, g2, ••• be the given set S. Let k1 be the least integer such that gk1 + e. Let .p1 = (1/lgk1 I )gk1• Then 1<!>1 I = 1. Let g:X = ~-(gcx1 <1> 1 ).p1• For cc < k 1, gcx = e and hence g:x = e.

gk1 = gk1 - < gk1 ,.p, >c1>, = gk1 - < gk1 ' ( 1 I I gk1 I )gk1 )( 1 I I gk1 I )gk1

Also = gk,-(gk1'gk1 >< 1/lgk1 l2 >gk1 = 9•

( g~, cl>1 ) = ( gQ( - ( g(X, <I>, ) <I>, , <1>1 ) = 0

since ( <1>1 ,<jl1 ) = 1 • Now suppose that we have by the repetition of' this process,

arrived at a def'inition of a sequence g~s), g~s), ••• 1 a number ks and an orthonormal set ci>1, ••• , cl>s, with the

18 II. HILBERT SPACE

following properties:

(1) ks-1 <ks and g~s) = e for 0( ~ks. ( 2) ( g~ s >,<l>i) = o for a = 1 , 2, • • • , i = 1 , • • • , s.

Now H g( s) = e for every cc, we do not define <l>s+p' p ~ 1. But H gfs) $ e, the method of the preceding paragraph can be applied t~ determine ks+l, a <l>s+l and a sequence fg~s+l) l, with (g~s+l) •<l>s+l) = o for every a. The sequence fg~s+l )l and ks+l have property (1 ), above.

We also have for i = 1, ••• , s, that b

(<l>s+1''h) = (g~s) '<l>i) = 0• s+1

Thus <1>1 , .• • • , <l>s+ 1 is an orthonormal set. By our construc-tion, we already have property (1) for s+1 and we have proper-ty (2) in the case of i = s+1. If i ~ s,

( s+ 1 ) ( s ) ( s ) (goc ,<l>i) = (goc -(ga ,<l>s+l )<l>s+1 •<l>i)

= (gis) '<l>i)-(g~s) •<l>s+l H<l>s+1'<1>i) = 0

since (gis),<j>i) = o by the hypothesis of the deduction and

(<l>s+l'<l>i) = o as above. Thus the process either stops with some s with _g~s) = e

for every ex or it continues indefinitely. In any case, we have defined an orthonormal set, cp1, cp2, • • • • We note that each <l>i is a linear combination of the g 's.

Furthermore each g°' is a linear combination of the <I> 1 s. for either Sex= e or there is a least s such that g~s)= e. Then

Let s1 be the orthonormal ·set <1>1 , • • • , <l>n' • • • Each goc is then in U(S1 ) hence s C U(S1 ) and heHce tltf::iJ c UtS1 ). Bimilarly ti ( s1 ) C U( S), since each <l>cx is a linear combination of the Sex's.

This process of "orthonormaliz1ng" the sequence g1, g2, ••• is usually referred to as the "Gram-Schmidt" process.

THEOREM X. Given a set S, having elements other

than e, :1n U(S)

§6. ORTHONORMAL SErS

it is possible to f:1nd an orthonormal set s1 such that 7n(S1 ) = m(S).

19

Proof: By a previous lemma, there is a denumerable set S' dense :1n S. Then U(S') is dense :1n U(S). We can find an

orthonormal set s1 such that U(S1 ) = U(S'). Thus U(S1 ) is

dense :1n U(S). Thus the closure of U(S1 ) equals the closure of U(S) or 7n(S1 ) = 7n(S).

COROLLARY. For evecy m, there is an orthonormal set

s, such that 711( s, ) = m.

Proof: In Theorem X, we can let S = 711.

LEMMA 2. Suppose S = f<1>1, <1> 2 , ••• I is an infinite orthonormal set. Then for a sequence of numbers a 1,

a2, • • • , CID n

r: oc=1 aoc<l>oc = lim .roc=1 aoc<l>oc fr+C>D

exists if and only if I:'; =1 faocl 2 <co. We also have

(I~=1aoc<l>ocf 2 = r:~=1 laocl2

when either limit exists.

Proof: We note that for m ~ n+ 1 , m m

(I: oc=n+1aoc<l>oc' r:f3=n+1aB<I>~)

m m - ( ) I: OFn+1 I: B=n+1aexaf3 <l>oc•<l>13

m m -= I: OFn+1 I:13=n+1 aexa136ex,(3

=I: !:n+1 laexl2.

Thus if fn = I:.~=,aex<l>ex• ffn-fmf 2 = r::=n+1 faexf2. Thus the convergence of the sequence fn is equivalent to that of the

partial sums I:~= 1 I ace( 2 • Furthermore putting n = o :1n our first equation we get

II:'=iaoc<l>exf 2 = lim I I!:1aex<l>exl 2 = lim r:m 1 fa (2 =I _ fa 12. ~... m+= CG= cc ex-1 ex

LEMMA 3. Suppose <1> 1, , cl>n is a (f'inite) ortho-normal set. For f € J; , we def'ille a ex =(f', cl>ex). Then

20 II. HILBERT SPACE

Proof': For ~ = 1 , • • . , n,

Thus

(f',<1>13)-r !:,a.c:X(<l>ex•<l>13l

( f', <1>13)-a.~= 0.

o ~ 1r-r~,a.cx<1>cxl2 = (f- I:~=1a.ex<1>cx'f-I:~=1a.13<1>13l =( (f-r ~1 aex<f>cx'f')-r ~=,aa(<l>cx'f-I: :=,a.~4113 )

(f'-I:~=1aex<l>oc'f) = (f,f)-I: ~=1aex(<l>ex•f) = lf'l 2-r n a a = 1r1 2-rn la 12• ex= 1 ex ex ex= 1 ex

This completes the proof' of the lemma.

COROLLARY. normal set and (f 1<1>a), then orthogonal to

I:~=~ laexl 2•

If <1> 1, <1> 2, • • • is an infinite ortho­if for a given f € ~. we define a.ex =

r==,aex<l>ex exists and f-I:~=1aex<l>ex is <l>ex for ex=1,2, ••• and lfl 2 ~

Since lfl 2 ~ I:~= 1 1a.exl 2 , we must have lfl 2 ~ I:;=1 1aexl 2 • This implies that r•~ex=l aexcj>a exists by Lemma 2. Since

= ( n f- I:OC=l aex<f>ex = 11.m f- r cx=l S.ex<l>oc) l}+CZ>

Lemma :; implies ( f- I:'::1 a ex<l>ce<l>l3) = o for 13 = 1 , 2, • • • •

THEOREM XI. If !171 is a closed linear manifold, we can find an orthonormal set s1, <1>1 , <1>2, • • • (finite or infinite) such that 7Jl(S1 ) = 7Jl. For every f €~ when we define a.ex= (f,<f>ex), we have that lfl 2 ~

2 I:ex la.exl and I:exaex<f>ex = f 1., exists and is in 7Jl. If we define f 2 as f-f1, then f 2 € m..1. and f = f +f2 is the resolution of Theorem VI. lfl 2 lf1 I2+1f 2 1 ~, f € m if and only if If 12 = I ex I a.ex I 2•

The first sentence is the corollary to Theorem X restated. The second sentence is a consequence of Lemma :; and it·s corollary.

§6. ORTHONORMAL SEI'S

Since St = m ... , f' 2 € '71.J. also f'ollows f'rom Lemma 3 and its corollary. f' = f' 1+f'2 is the resolution of' Theorem VI since that resolution is unique. Since f' 1 and f'2 are orthogonal, lf'l 2 = lf' 1 12+1f'2 12• By Lemma 2 (and its proof'), we can show

2 2 2 I 12 that lf' 1 f = I:cx laoi:I • Thus I.ti = Icx a°' if' and only if' lf'21 2 = 0 or f' = r, € m.

If' Theorem XI is applied to Ji as a closed linear manif'old we obtain:

THEOREM XII. There exists an orthonormal set S, <1> 1 , <1>2, • • • such that '1l(S1 ) = f;. To every f' € f;

we can f'ind a sequence a 1 , a 2, ••• , a°'= (f' •<l>cx) with

If' 12 = rcocx=l I acxl 2 and f' = I:~=l acx<l>cx· If' f' "' la1 ,

a 2, • . • ! and g "' lb1 , b 2, • • • ! then

(f' ,g) = I~=lacxboi:.

In connection with the last sentence, we note that

(f',g) = ( r::'=,aoi:cj>cx' I:~ 1 b~ci>~)

= ~:ZO ... ( r~=lacx~oc' I:';:Jb~<I>~) = lim r:~,aOl:(~cx· r~,b~cj>~)

n-+<» n . m

= llm.I:a=la<X lim (<l>oi:• r~=l b~cj>~) n-+co m+ ...

= lim r:;=,a~OI: = r:=,a~a· n-+ ...

The coITespondence f'"' la1 , a 2, ••• l has certain other obvious properties. For instance

f'+g"' la1+b1 , a 2+b2,

af' "' laa1 , aa2 , • ·.:

e "' 10, o, •.•

<l>a "' I 6a 1 , 6oi: 2, l . . ' An orthonormal set s1 such that '1!( s1 ) = f; is called

complete.

CHAPI'ER III

REALIZATIONS OF HILBERT SPACE

§1

In this chapter, we will give certain examples of J;, Our

method of procedure will be t.o specify a set of elements, f, define + and a· so that Postulate A is satisfied, then define (f,g) so as to yield Postulate B. Postulate C will be 1n general almost trivial and it will then be necessary to estab­lish Postulates D and E.

Thus we will deal with the sets >;1 which are lmown to satis­

fy Postulates A and B. As we remarked 1n Chapter II §3, J;' is a linear metric space. Thus we can apply to >; 1 , certain lmown theorems on metric spaces directly and this will 1n general sim­plify the proofs of separability. Thus if s1 is dense in' s2, the closure of s1 is the closure of s2 . Furthermore, the no­t ions ti ( S ) and 71l ( S ) can be defined and we have the lemma .

LEMMA 1. If >; 1 satisfies Postulates A and Band if S is a denumerable set 1n J;' , then 71!( S) is separable.

Proof. Since 71l( S) is the closure of ti( S), if ti ( s) is separable, 71l( S) is separable. • But U( S) consists of elements

1n the form r ~=1acla• fa E s. Now let E > 0 be given. Let ra be a number 1n the form p1 +ip2 where p 1 and p 2 are ra­

tional numbers and such that lfal·laa-ral < E/n, Then

I r !:1 aafa - r ~:i1i_rafa I is easily seen to be < E. Thus the set of elements I: a=1 rafa (which we will denote by tlr(S)) is dense 1n U ( S) •

But Ur( S) has only a denumerable number of elements. For let us enumerate the elements of S; f 1, f 2 , .. • • We then see

that the set of elements r 1f 1+ ••• +rnfn for n fixed must be denumerable since an n 'tuple sequence can 'berearranged 1n a sin­gle sequence. Now Ur(S) is the set of all of these, i.e., for every n, and hence is a denumerable sum of denumerable sets.

Thus Ur(S) is denumerable. But Ur ( S) is dense 1n U ( S) whose closure is 71l( S) and

thus '1l ( S) must be separable. 22

III. REALIZATIONS OF HILBERI' SPACE 23

To show the 1n.fin:1te dimensionality i.e., Postulate C, the f'ollowing lemma is useful.

LEMMA 2. Let fl' satisfy Postulates A and B. Let <1> 1 , ••• , ~n be the property that

~,, .•. , <l>n are

n non-zero elements of' fl', with (cj>i,<j>j) = o, if i + j. Then

linearly independent

For if' a 1<1>1+ ••• +Bn_<l>n = a, we have o = (0,<j>i) =

( I:~=laa<l>a'<l>i) = ai(<l>i'<l>j). Since (<l>i'<l>i) + o, we have ai=O. Thus a 1<1> 1 + .•. +Bn_<l>n = 0 implies a 1 = •.. = 8n = o a.rid the <l>i 's are linearly independent.

§2

DEFINrrION 1 • Let 12 of' complex ;numbers !a1 ,a2, We def'ine

denote the set of' sequence ... 2< such that :t a=l I aocl oo.

l+lb,,b2, ala1,a2 ,

l = !a1+b1,a2+b2, l = !aa1 ,aa2 , •• •

0 = 10,0, ••• l (!a1 ,a2, ••. l,!b1 ,b2, ••• l)= I.~ 1 aocboc.

We note that la+bl 2 ~ 2( lal 2+lbl 2 ) and thus if' ·Ja1,a2, •· •• l and {b1 ,b2, ••• l are in 12, then !a1+b1 ,a2+b2, ••• is also. !aa1 ,aa2, • • • l obviously is in 12 if' !a1 ,a2, .•. l is. Now it can be shown in precisely the same way as we we established B(9) in §1, Chapter II, that

(I:~, 1aabJ)2 = (I:!:, laal · lbal )2 ~ ( r~, 1aa1 2 )( r:~, lb131 2 )

"'° 2 < "'° 12 < Thus if' I: a=l laal oo and r: 13=1 lbf3 oo, then the sum z~1 aJia is absolut·ely convergent and thus the inner product is def'ined f'or every pair of' elements of' 12 .*

* If' we consider lp, p ) 1 , the set of sequences f'or which r:-=1 laalP < oo, thee operation + and -a· will be also univer­sa'fly applicable. But to f'orm an inner product, we would take !a1 ,a21 ••• l in l:tl and !b1 ,b2, ••• l in 1-01 where 1/p+1/p 1 = 1. (Cf'. s. Banach. loc. cit. pp. 67--68) This corres­ponds to the f'act that l~ = lp 1 f'or these spaces. (Cf'. Chapter rI, §4 above) However th~ completeness, separability and infin­ite dimensionality of' lp can be demonstrated in a manner quite analo.gous to the corresponding proof's f'or 12 •

24 §2.

THEOREM I. 12 is a Hilbert space.

Proof'. From the previous def':1n1tions, it is readily seen that Postulates A and B are satisf'ied.

To show Postulate C we notice that s1 , the sequence of'

elements ~1 , ~2 , • • • where ~°' = ! c5cx, 1, 6°'' 2, • • • l ls a de­numerable orthonormal set. Thus given n, we take the f'irst n of' the ~a 's, and Lemma 2 of' the preceding section tells us that these are linearly independent.

To show Postulate D, we note that U(S1) ls the set of' se­quences !a1,a2, ••• l ror which there ls an N such that lf' n ~ N, ~ = o. Now U(S1) is dense in 12 since lf' f' ..., la1, a 2, ••• l and E) o ls given, we can choose an N so that 1f'

n ~ N, I: ~1 f aal 2 ) If' f 2-e2• Now 1f' f'n 1.s the sequence fa1, •.• , ~· o, .•. l then f'n € U(S1 ) and ff'-f'nf < E. Thus 12 = mes,) and Lemma 1 now implies the separability postulate.

To show the completeness, we must consider a sequence of' elements (f'nl such that ff'n-fml -- o. We must show the ex­istence of' a sequence g = lb1, b 2 , • • • l such that I:':.:1 f baf 2< co and f f'n-gf -+ o.

Now if' f'n = l~, 1 ,~, 2 , ••• we def'ine f'n,p = l~, 1 , ••• ,

~,p' 0, ••• j. Then f'or 11 ) o. there is an N = N(11) such that if n and m ) N, then

q2 > If' -f' 12 = r- I a a 12 n m a=l ~,a--:m,a

~ I:~=1 l~,a-~,af 2 = 1f'n,p-f'm,pl 2 (a')

or (a)

f'or every p. Thus lf' we f'ix p, since f f'n-f'mf ....:...._ o, f f'n,p-f'm,pl ---+- o and the sequence f'n,p must be convergent. This means that f'or q ~ p, ~,q-- bq. We remark that since p can be taken indef':1n1tely large that bq ls def'ined f'or every q. Let Sp= lb1, ••• ,bp,o, ••• j, Obviously· f'n,p-+ Sp as p-+co.

Furtherniore f'or N = N(TJ), we have that

p--+- co, For ff'N""f'N,pf2 = r:.p+1 f81J,af2. we can f'ind a P(11) such that 1f' p ~ P(11),

ff'N-f'N,pf < 11

f'N,p -+ f'N as Thus given 11 ) o,

(~)

III. REALIZATIONS OF HILBERT SPACE

Now let n and m be ~ N(ri), p ~ P(ri). Then by (ex), ( ~), and (ex) again, we have

lfn-fm,pl = lfn-fN+fN-fN,p+fN,p-fm,pl

25

~ lfn-fNl+lfN-fN,pl+lfN,p-fm,pl (-y)

~ 3Tl·

If we let m --+cc, then fm,p --+ 8P and we obtain that for n ~ N(ri), p ~ P(ri),

( "Y. 1 )

IT in particular we let n = N, we have lfN-Spl ~ 3Tl ·for every p ~ P(ri). This implies ISpl ~ (fN(+3ri for p ~ P(ri). Thus

r~1 fbexf 2 < ( ffN(+3ri) 2 .

We may let p -+ cc and obtain

r ;'.,1 fbexf2 ~ ( ffN(+3ri) 2 < =· Thus we may let g = !b1 , b2, • • . I .

We observe that Sp---+ g. This and (-y.1) imply

ffn-gf ~ 3ri ("Y. 2)

for n ~ N(ri). This implies that fn--+ g. The existence of a g with this property indicateB the completeness of 12 and we have demonstrated Theorem I.

When we recall Theorem XII of Chapter I, we obtain,

THEO:RllM II. Every Hilbert space is equivalent to 12, in the sense that to every f of n, there is an element ! 8n I of 12, f "' ! 8n I and this correspondence is one­to-one, and preserves the operations + , a. , e and (f,g).

A set of postulates is said to be categorical if for any two realizations, there exists a one-to-one correspondence which preserves the relations of the postulates. Since a:n:y two reali­zations are in such a relations with 12, they must be related 1n this way to each other. Thus the axioms of Hilbert space are categorical.

Now if a linear manifold, 7n is infinite dimensional, 1.e.,

§ 3. DIRECT SUMS

satisfies postulate C, it satisfies all the postulates, with the original def'initions of + , a· , e and ( , ) • Thus we obtain:

COROLLARY. Every infinite dimensional manifold in a Hilbert space is equivalent to the space itself. It is also equivalent to 12•

§3

DEFilUTION 1. Let n = 2, 3, • • • be a given integer and let us suppose that we have n Hilbert spaces, n 1 ,

. . . , nn. Now consider the n 'tuples of' elements ff,, ••• ,fn! with fi € ni for i = 1, ••• ,n. Define

!f1 , ... ,fn!+lg,.

a!f 1 ,

( !f1 , ••• ,fn!, !g1 ,

We call this set of

.Bn_!

,fn!

9=

,Bn_I·> =

n 'tuples,

ff 1+g1 , ••• ,fn+Bn_!

!af1 , ... ,af'n!

fe 1 , ••• ,en! (f, ,g, )+ .•. +(f'n•Bn.)

n,$ ... $ nn•*

THEOREM III. n1 $ • • • $ nn is a Hilbert space.

The proof of this theorem is most elementary.

DEFINITION 2.. Let n1 , n2 , • • • be a sequence of Hilbert spaces. We consider the sequences ff 1 ,f2, •••

such that fee€ nee and I::=1 lfccl 2< =· We define

ff 1,f2, ... !+!g1 ,g2, !f1+g1 ,f2+g2,

far,;ar2 , ...

e = f e, , e 2, • • • !

(!f',,!'2, ... J,!g,,g2, ... I)= (f,,g,)+(f2,g2)+

* If n 1 , • • • , fin are Banach and not Hilbert spaces, we can make similar def'initions of' the sum. and scalar multiplications of n 'tuples. For the norm, it is however sometimes more con­venient to use the def'inition

l!f1 , ... ,fn!I = (lf1 1P+ ... +lfnlP> 1/P.

III. REALIZATIONS OF HILBERI' SPACE 27

The proof that the operations + , a. , and ( , ) are Ulliver­sally defined is quite analogous to the discussion given in §2 for 12• We proceed to the theorem:

THEOREM N. H1 e f>2e • • • is a Hilbert space.

A proof of this theorem quite analogous to the proof of Theorem I of this chapter is possible. .Another proof can be obtained if one considers for each a, a complete orthonormal set l<l>a nl for f>a. Thus if we have a sequence !f1 ,f2, ••• with r~=, 1fa1 2 <co, we can find for each fa a sequence aa 1,aa 2, • • • of numbers with aa n. = (f ,cj>ty n.> such that

' 2 ' 2 11• a 2''• lfal = I:~=11aa, I . Then r:;=1 r~, laa,(31 < co. Now it can easily be shown that any method of corresponding a double sequence to a single sequence, determines an equivalence between H1e H2e ••• and 12 •

§4

DEFilUTION. Let E be a measurable set of finite non­zero measure in an n dimensional space. Let C2 consist of the Lebesgue measurable functions f(P) defined on E and such that fElf(P) I 2dP < co, However two functions are to be regarded as identical if they differ only on a set of measure zero. Sum and a· are defined in t):l.e ·usual way for functions, e is the function which is zero. (except possibly on a set of measure zero). Finally

(f,g) = f ~(P)g(P)dP. If n and E are not specified, it will be understood

that n = 1 and E is the set of x such that o ~ x ~ 1 •

The 'Operation a· is obviously Ulliversally defined in 1::2 , Since la+bl 2 ~ 2(1al 2+lbl 2 ) for a and b complex is is readily seen that + is also universally defined. To obtain the corresponding result for ( , ) , we introduce the notion fA(P) for positive A 1s. fA(P) =A if f(P)) A, fA(P) = f(P) if -A~ f(P) ~A and fA(P) =-A if f(P) <-A. For f A and gA, we obtain without difficulty as in the proof of B( 9) in § 1 , Chapter I that

28 §4. c

efElf' Al· lgAldP) 2 ~ ef Elf' Al 2dP)(/ElgAI 2dP)

~ <fElf'l 2dP)(fElgl 2dP).

Due to a well lmown theorem on a monotonically increasing se­quence of' positive measurable f'unctions, this implies

VElf'l • lgldP) ~ VElf'f 2dP){[Elgl 2dP).

Hence f'eP)·geP) is a sumable f'unction of' P and thus e , ) is universally defined.

THEOREM V. S::2 is a Hilbert space.

It is possible to show Postulates A and B without diff'iculty. To show Postulate C we observe that since E is of finite

non-zero measure it is possible to find n mutually exclusive measurable sets, E1, .•• ~ included in E and each of non­zero measure. Let xi eP) be the characteristic function for Ei' i.e., the f'unction which assumes the valus 1 on Ei and

o elsewhere. Then x1, ••• , Xn are a set of mutually orthog­onal non-zero f'unctions to which Lemma 2 of' §1 may be applied. Thus Postulate C is satisfied.

The proof of Postulate D depends on Lemma 1 of § 1 • Let s 1 denote the set of cha.racteristic f'unctions of the measurable sets F C E. Since every f'unction of S::2 can be approximated by step f'unctions, tt follows that mes,) = s::2. Let s2 denote the subset of these, in which the F is an intersection of' an open set G with E. It is well lmown that s 2 is dense in S1 • Thus mes2) = mes1 ) = S::2• Let s 3 denote the subset of s 2, in which G is the interior of' an n -dimensional cube whose faces have the equation xi= pi' where Pi is a rational number. Now a:ny open set is a denumerable sum of such n -dimen­sional cubes regarded however as closed point sets, but whose interiors are mutuall~ exclusive. The faces of the cubes in such a sum form a set of measure zero and thus it is possible to show that mes3 ) contains S2 and thus mes3 ) = m(S2 ) = S:: 2• Lemma. 1 of § 1 of this chapter now implies Postulate D since s 3 is denumerable.

It remains to prove Postulate E for s;. Let f 1 , f 2 • ••• be a sequence of elements such that f.fn-fml __.,. d as n and

III. REALIZATIONS OF HILBERT SPACE 29

m ~ =· Let E1, E2, • • • be a sequence of positive numbers

such that r:=1 Ea< = and Ea > o. We take an increasing sequence of positive integers n 1, n 2, such that if n

and m are ) na' lfn-fml < Ea" It fbllows that lfn -fn I < EOC" Thus a+1 a

We let k = Let

We have seen that

f Ehn_(P)dP = t'{;=1fElfn (P)-fn (P) jdP a+1 a

S. rn_ (m(E)) 1/ 2 1f -f IS. m(E) 1/ 2 .k. - a-1 na+1 na -

Thus the 1\i_(P) are a monotomically increasing sequence of posi­

tive f\m.ctions whose integrals are bounded. It follows that for

almost every P, 1\i_(P) ~ h(P) < =· We will next show that h(P) is in £2 • For consider hA(P)

(Cf. above). Obviously hxi,A(P)-+ hA(P) for almost every P, Since these f\m.ctions are uniformly bounded, we have that

fil~(P)dP = lim /~ A(P)dP n+- ,

~ lim./~(P)dP = lim ll\i_l 2 n-+- n-+-

~ lim < r~1 lfn -fn I )2 = k 2 • n-+- a+1 a

Since this holds for every A, we have again an increasing se­quence of positive f\m.ctions h~(P) whose integral is bounded.

Thus for the limit we have h h2(P)dP ~ k 2 < =· From the existence of h(P), we can conclude that the series

fn1(P)+(fn2(P)-fn1(P))+(fn3(P)-fn (P))+ ••• is absolutely conver­gent fbr almo'St every P. Call tfie sum g( P). Since I g( P) I ~ lfn (P)l+h(P) for almost every P, it is easily seen that g(P) is k £2 •

It is readily seen that for almost every P

lg(P)-fn (P) 12 = I I~oc(fn__ (P)-fn (P)) 12 a 1~1 P

~ ( I:?-alf~1 (P)-fllp(P)I )2 ~ h 2 (P),

30 §4. c

also f'lloc(P)---+ g(P). Thus by the majorant theorem of' Lebes­gue hElf'n (P)-g(P) 12dP-+ o. Thus f'n -+ g.

a . a Suppose now that e ) o is given. From the above we can

f'ind an a such that if' ~ ~ a, lg-f'~I < e/2. We can f'ind an N = N(e/2) such that if' n and m ~ N, lf'n-f'ml < e/2. Let P(e) =max (N(e/2),n«). If' n 2 P(e) and n~ ~ P(e) we

have lfn-gl = lfn-f'n~+f'nB-gl ~ 1.f'n-fnBl+lf~-gl < e. Thus fn - gES and the completeness is shown.

If E is a measurable set of' infinite measure, we can com­bine Theorem IV of §3 and Theorem V to get the result:

COROLLARY. The restriction that E be of' f'inite measure 1n Theorem V may be omitted and the result will still be valid.

CHAPI'ER N

ADDITIVE AND CLOSED TRANSFORMATIONS

§1

The purpose of this section is to introduce a number of notions.

DEFINITION 1 • A transformation T from n1 to n2

is a single-valued function of the elements of n,, which assumes values Tf in n2 . The set of f 's for which Tf is defined is called the domain of T, the set of Tf 's is called the range. The set ! of pairs {f;Tfl in n1$ n2 (Cf. Chapter III, §3, Theorem III) is called the graph of T. If T' is a transformation f'rom n2 to n1, its graph is the set of pairs IT'g,gJ.

LEMMA 1 • A set S in n1 $ n2 is the graph of a transformation from n1 to n2 (from n2 to n1 ), if for a given f E n,, ( g E n2 ) there is at most one pair of S having f (g) as its first (second) element.

T is obviously the transformation for which Tf is undefined if there is no pair of S with f as its first element and for which Tf = g if {f,gl E s.

DEFINITION 2. Let T be a transformation from n1 to n2 and let ! be the graph of T. Now if ! is also the graph of a transformation from n2 to n1, this second transformation T- 1 is called the inverse of T.

The range, of T,

inverse of T when it exists, has for its domain and the range and domain of T. Also if f is in the domain T- 1 (Tf) = f and for g in the range of T, T(T- 1g) = g.

31

IV. ADDITIVE AND CLOSED TRANSFORMATIONS

DEFINITION :;. If' T1 and T2 are two transf'ormations f'rom n1 to n2 such that ! 1 C ~, then T1 is called a contraction of' T2 and T2 an extension of' T1• We write this S'YJ!lbolically T1 C T2 •

LEMMA 2. T2 is an extension of' T1 if' and only if' f'or evecy f' in the domain of' T, T 2f' is def'ined and T2f' = T1f'.

llMilA 3. A transf'ormation T is additive. (Cf'. Chapter II, §3, Def'inition 2.) if' and only if' its graph ! is an additive set.

LEMMA 4. kn additive set of' a transf'ormation f'rom n, !91 ,hl € ! implies h = e2•

! c n,$ n2 is the graph to n2 if' and only if'

By Lemma 1, the condition is necessary. It is also suf'f'icient. For suppose !f',g1 l and !f',g2 J are in !. Then since ! is additive, !f',g1 l-!f',g2 l = !91,g1-g21 €I. Our condition implies g1 = g2 and thus there is at most one pair If' ,gl €'I with f' in the f'irst place.

DEFINITION 4. Let T be a transf'ormation with graph !. If' U(I) is the graph of' a transf'ormation, Ta, this latter transformation is called the additive exten­sion of' T.

DEFINITION 5. A transf'ormation T f'rom 1i1 to fi2 will be said to be closed, if' its graph is a closed set in Ji1$ 1i2 • If' [I] is the graph of' a transformation [T], [T ] is called the closure of' T •

We note that '1l(I) is the graph of' [Ta] when this latter transf'ormation exiSts. In general, given T, Ta will not exist. (A necessary and suf'f'icient condition that Ta exist can be obtained by applying Lemma 4 to U(I)). However $ven if' Ta exists, [Ta] need not exist. We give an example of' this.

Let 1P1 , 1P2 , • • • be an orthonormal set. We def'ine T~i= 1P1 •

§1. THE GRAPH OF A TRANSFORMATION

n Then it is easily seen that Tex exists. For if Iex=iaex<l>ex = 0, then aex = Ci and hence In 1a = o. Thus if {0,fl is in ex= ex U('.t), h = 0. Furthermore Ta( r!.1aexcj>ex) = ( r!:1aex)cj>1• But [Tex] does not exist since {0,cj>1 I is in [U(I)]. To see this

we notice that Ta ( r ~=l ( 1 /n)cj>ex) = <1>1 and thus ! r~1 ( 1 /n)cj>ex' 4>1 l is in U(I). Now Ir !_1 ( 1 /n)<1>exl = ( 1/n) 1 / 2- o as

n -- CD. Hence ! r::=1 ( 1 /n)cj>cx'cj>1 l -- {0,<1>, j. Thus 10,cj>, I is in [U( I)] and the latter is not the graph of a transforma­

tion. However Theorem II of Chapter II, §3, tells us that if Ta is

a continuous transformation [Ta] exists and has domain, the closure of the domain of. Ta. Thus if a continuous additive T has domain, a linear manifold, T is closed.

DEFINITION 6. IT T 1 is a transformation from n1 to n2 and T2 is a transformation from n2 to n3 then T2T1 is the transformation whose domain consist

of those f 's for which T2(T1f') is defined and has the value T2 (T1f), i.e., (T2T1 )f = T1(T2f).

Since a continuous f'unction of a continuous f'unction is continuous, it follows that if T1 and T2 are continuous, T2T1 is also continuous. IT in addition T1 and T2 are additive with bounds c 1 and C2 (Cf. Chapter II, §3, Theorem I.), then

T2T1 has bound C2C1•

DEFINITION 7. IT T 1 and T 2 are two transformations from n1 to n2 , T1+T2 is the transformation, the domain of which is the set of those elements for which T 1 f and T2f are defined and for which (T 1+T 2 )f' '=

T 1f+T 2f. aT 1 is the transformation, whose domain is that of' T1 and f'or which (aT1 )f = a(T 1f').

The sum of two continuous transformations is again continuous

and if in particular T1 and T2 are additive with bounds c 1 and C2 , then the bound of' the sum is ~ C1+c2 •

N. ADDITIVE AND CLOSED TRANSFORMATIONS

§2

THEOREVJ I. If T is a transformation from n1 to n2 , with graph I and domain D then :r• is the graph of a transformation from fi2 to n, , if and only if 7n(D) = n1 •

Proof: I• is· the graph of if and only if fh1 ,02 1 EI• is equivalent to ·

a transformation from fi2 to n, , implies h = 01 • But !h,02 1 EI•

o = (!h1 ,02 1,ff,Tfl) = (h,f)

for every f in the domain of I. fh,02 1 EI• is equivalent to h E D • • Thus I• is the graph of a transformation if and only if h ED• implies h = 0 1 •

But h ED• implies h = 0 1 , if and only if D•= {01 1. Thus I• is the graph of a transformation if and only if D• = {01 I. But D• = 0 1 is equivalent to (D•)• = {01 I' = n1 by Theorem VII of Chapter II, §5. But since (D•) .. = ( '1l(D).•)•=

'1l(D), we see that D• = {0 1 I is equivalent to '1l(D) = ~·

From a preceding statement we see that I• is the graph of a transformation if and only if 711( D) = n.

If in particular T is additive, D is additive and U(D) D. Thus '1l(D) = [U(D)] = [D] and the statement 711(D) =Ji1 is equivalent to D is dense. We have then:

COROLLARY. If T in Theorem I is also additive, then I• is the graph of a transformation, if and only 1f D is dense.

DEFINITION 1. If T is a transformation from n1 to Ji2 and if '!• is the graph of a transformation from Ji2 to Ji1 , we.will denote the latter transformation by T' and -T' by T*.t

THEOREM II. Let T be such that T• exists. Then (a) A pair fg1 ,g2 1 E Ji1$ n2 is such that T'g2 = g1 if

t If T is a transformation between two Banach spaces £.1 and £e, T• can be regarded as a transformation from £.~ to er (the conjugate spaces).

§2. ANOINTS AND CLOSURE

and only if' f'or evecy f in the domain of T,

(f,g1 )+(Tf,g2 ) = o;

(b) A pair {g1 ,g2 1 € n1 ~ n2 is such that T*g2 = g 1 if and only if for evecy f in the domain of T,

(f,g1 ) = (Tf,g2 ).

Since T* = -T', (a) and (b) are equivalent. Inasmuch as

({f,Tf!,{g1,g2 !) = (f,g1 )+(Tf,g2 ),

the condition in (a) is equivalent to {g1 ,g2 1 € '.t'.

COROLLARY. Let T be such that T • exists. Then (a) A transformation T' is C T' if and only if. for

evecy f in the domain of T and evecy g in the domain of T''

(f,T 1g)+(Tf,g) = o.

(b) A transformation T' is C T*, if and only if for evecy f in the domain of T and evecy g in the domain of' T • ,

(f,T'g) = (Tf,g).

THEOREM III. ·Let T be a transformation from n1 to n2 for which T' exists, 1.e., 7Jl(D) = n. Then [Ta] exists if and only if T' (or T*) has domain dense.

35

By Definitions 4 and 5, we see that [Ta] exists if and only if 7Jl('I) is the graph of a transformation. But 7n('I) = (!•)•.

(Cf. Chapter II, §6) . Furthermore since 'I• is a linear mani­

fold, T' is additive. Thus the corollacy to Theorem I of this

section states that "Domain of' T' dense" is equivalent to "('I•)• is the graph of the transfonna:tion." Since ('I')• =

m ('I), the first sentence in this paragraph shows that this latter statement is equivalent to " [Ta] exists."

COROLLARY 1. [Ta J exists if and only if (T • )• (= (T* )*) exists. When they exist [Ta]= (T')• (= (T*)*).

IV. ADDITIVE AND CLOSED TRANSFORMATIONS

COROLLARY 2. If T tion with domain dense, equals T.

is a closed additive transforma­(T') • (= (T*)*) exists and

Thus a closed additive transformation with a dense domain is symmetrically related to its perpendicular and to its adjoint. We will abbreviate "closed additive with a dense domain" to "c.a.d.d."

We call a continuous additive transformation whose domain is the full space a "linear" transformation. AfJ we remarked before Definition 6, in §1, a linear transformation is closed.

THEOREM IV. If T is a continuous additive transfor­mation, whose domain is dense and with bound C, (Cf. Chapter II, §;, Theorem I), then T• (and T*) is a linear transformation with the same bound as T.

PROOF: [T] exists by Theorem II of Chapter II,§;. Since [T P = T •, we may suppose that T = [T] and has domain the full space. By Theorem III of this section, T • has domain dense. It is also c.a. since ~· is linear manifold. Thus Theorem II of Chapter II, §;, implies that T• is linear 1f it is contin­uous.

By Theorem I of Chapter II, §3, we have for. every f ITfl ~ CI f I (we have assumed that [T] = T). Hence for every f and

S:. I (Tf ,s>I ~ ITfl ·Isl ~ C· lfl ·ls.I. If s is in the domain of T•, we have by (a) of Theorem II of this section that

(T•g,f )+(s,Tf) = o.

Hence

l(T•s,f)I = l(g,Tf)I ~ C·lfl·lgl.

If we let f = T•g, we get IT•sl 2 ~ C·IT•st·lsl which implies IT•gl ~ C·lsl. Theorem I of Chapter II implies that T• is continuous, with a bound c• ~ c. Since however (T• )4 = T,

we also have C ~ C • and thus the bounds must be equal.

THEOREM V. If T 1 and T2 are additive transformations with dense domains, then 1f (T2T1 )* exists (or (T1+T2 )* ),

we have that TlT~ is a contraction of (T2T1)*, (Tl+T~ is

§2. ADJOINTS AND CLOSURE

a contraction of' (T1+T2)*), (aT 1 )* = aT* if' a+ o. If' T1 and T2 are linear, we have that Tr T~ = (T2T1 )*. (Similarly Tt+T~ = (T1+T2)*).

37

PROOF: If' f' is in the domain of' T 2T 1 and g in the do­main of' TtT~, (b) of' Theorem II of' this section implies

(T2T1f',g) = (T 1 f',T~g) = (f',TtT~g). Now (b) of' the Corolla~ of' Theorem II of' this section implies TtT~ C (T2T1 )*. In the case of' T1+T2 the argument is similar.

To show that (aT1 )* = aTt, we note that if' a+ 0 (T1f',g1 ) =

(f',g2 ) is equivalent to (aT1f',g1 ) = (f',ag2). If' T 1 and T 2 are linear Tt and T~ are also by Theorem

IV above. Thus TtT~ is everywhere def'ined and has no proper extensions and we must have TtT~ = (T2T1)*. This argument also applies to the sum.

COROLIARY. If' T1 is c.a.d.d. and T2 is linear, then (T2T1 )* = TtT~.

PROOF: We lmow that (T2T1 )* ::> TrT~. On the other hand let f' be in the domain of' (T2T1 )* and let g be in the domain of' T 1 and hence in that of' T 2T 1 • Then

(g,(T2T1)*f') = (T2T1g,f') = (T 1 g,T~f').

Since this holds f'or evecy g _ in the domain of' T 1 , we have that Tt(T~f') exists and equals (T2T1)*f'. This implies that TtT~ ::> (T2T1 )*.

LEMMA 1. Let T be c.a.d.d. Let 71* be the set of' f' 's f'or which T*f' = o. Let 7t denote the range of' T. Then 7t • ='1 *.

Since T* is c.a., 7l* is closed. Since

(fe1,g!,ff',Tf'!) = (g,Tf'),

we see thil.t 10,gl is in 'P if' and only if' g € 7t•. Thus 7t• is the set of' zeros of' T • = -T*.

It is evident geometrically that if' and T*-l exist, then (T- 1 )* = T*- 1•

T is c.a.d.d. and T- 1

Lemna. 4 of' § 1 and the

:;8 IV. ADDTIIVE AND CLOSED TRANSFORMATIONS

preceding Lemma. shows that T*- 1 exists if' and only if' [!R] = f:i and that T- 1 exists if' and only if' [7t*] = f:i.

THH:OREVI VI. Let '1 denote the zeros of' T, 1l* denote the zeros of' T*, 7t the range of' T, !R* the range of' T*.

Then '1* =7t • , '1 = CR*)•. T- 1 exists if' and only if '1* = (!R*}' = !0!. T*-l exists if and only if '1* = (7t)' = 19!. If T- 1 and T*- 1 both exist, (T- 1 )* = T*- 1 •

§:;

We now introduce certain notions which are fundamental in

our discussion.

DEFINTIION 1. An additive transformation H within n, will be called s:ymmetric if (a) the domain of H is dense and (b) :f'or every f and g in the domain of' H,

(Hf ,g) (f ,Hg).

From §2, Theorem I, we see that H* exists. By (b) of' the

corollary to Theorem II of' §2, we see that H C H*. Thus we obtain the following Lemma.

I.EMMA 1. An additive transformaion H is s:ymmetric, if (a) it has domain dense and,(b) H CH*.

I.EMMA 2. If H is s:ymmetric, [H] exists and is s:ymmetric.

PROOF: H* is a closed transf'ormation. Since H C H*, we must have the graph of H in a closed set which is the graph

of a transf'ormation. Thus Lemma 1 of §1 of this Chapter, shows that the closure of the graph of' H must be the graph of' a transformation. Thus [H] exists. From the graphs, it follows

that [HJ' = H• and hence [H]* = H*. Lemma. 2 permits us in general to consider only closed s:ymme­

tric transformations •.

DEFINTIION 2. If H* = H, H is called self-adjoint.

§3. SYMMEIT'RIC AND SELF-ADJOINT OPERATORS

LEMMA 3. A self'-adjoint transf'ormation is symmetric. If' H is closed symmetric and H* is symmetric, then H is self'-adjoint. If' the domain of' a symmetric trans­f'ormation H is the f'ull space, H is self'-adjoint. A synnnetric linear transf'ormation is self'-adjoint.

The f'irst sentence is a consequence of' Lemma 1 • If' H is closed symmetric and H* is synnnetric, we obtain by Lemma 1 and Corollary 2 of' Theorem III of' the preceding section that H C H* C (H*)* = H. The third statement f'ollows f'rom Lemma 1 of' this section since a transf'ormation with domain the f'ull space can have no proper extension. The f'ourth statement f'ollows f'rom the third.

39

LEMMA 4. If' H1 and H2 are symmetric and the domain of' H1+H2 is dense, the latter transf'ormation is symmetric. If' a is real, a.H1 is synnnetric and if' H1 is self'­adjoint, a real, then a.H1 is self'-adjoint.

This is a consequence of' Theorem V of' the preceding section. For if' the domain of' H1+H2 is dense, (H1+H2)* exists. Then too, H1+H2 C Ht+H~ C (H1+H2)* by this theorem. The second sentence is an illlmediate consequence.

LEMMA 5. If' H1 is sell-adjoint and H2 linear sym­metric (and hence self'-adjoint by Lemma 3 above) then H1+H2 is self'-adjoint.

PROOF. The domain of' H1+H2 is the same as that of' H1 and thus is dense. Hence Lemma 4, tells us that H1 +H2 is symmetric and that -H2 is self'-adjoint. Furthermore ( H1 +H2 ) + ( -:-H2) has domain the domain of' H1 • Hence H1 = (H1+H2)+(-H2) C (H1+H2)*+(-H2)* C ((H1+H2)+(-H2))* =Ht= H1• This implies that the domain of' (H1+H2)* which is the same as that of' (H1+H2)*+(-H2)* is included in the domain of' H1 which is also the domain of' H1+H2• Since H1+H2 is symmetric, H1+H2 C (H1+H2)*. Since the domain of' (H1+H2)* is included in that of' H1+H2, we must have (H1+H2)* = H1+H2•

40 N. ADDITIVE AND CLOSED TRANSFORMATIONS

LEMMA 6. If H is symmetric, and if H- 1 exists, H- 1 is symmetric if H*- 1 exists, i.e., if [!R] = ~.

This is a consequence of Theorem VI of the preceding section. We note that if H*_, exists, since H CH*, we must have H- 1 C H*-l.

LEMMA 7. If H is self-adjoint and H- 1 exists, then H- 1 is self-adjoint.

This is a consequence of the last sentence of Theorem VI of the preceding section.

Another consequence of Theorem VI of the preceding section and H C H* is Le= 8.

LEMMA 8. If H is closed symmAtric, then ~ C '1* = !R •.

Suppose H1 and H2 are closed symmetric and H1 C H2• ~en we have H1 c H2 c H~ c Hr. Now if H1 is self-adjoint since H1 =Hr, we see that H2 must equal H1 and H1 has no proper symmetric extension. On the other hand, it is also conceivable that H1 is symmetric with graph A1 and ArAi is one dimensional. If H2 is then a symmetric closed exten­sion of H1, we have H1 C H2 C Hr. This last inclusion and Chapter II §5, Corollary1to Theorem VI imply that either H2 = H1 or H2 = Hr. But Hr is not symmetric because (H1 )** = H1 is c Hr, but H1 =f Hr. Under these circumstances then H1 would have no proper symmetric extension and yet not be self-adjoint. We shall show later the existence of an H1

having these properties and give a complete discussion of this phenomena. But for the present, we simply introduce the defini­tions.

DEFINITION 3. If H1 and H2 are closed symmetric transformations such that H1 C H2, then H2 is called a symmetric extension of H1• If in addition H1 =f H2, H2 is called a proper symmetric extension of H1 • If H1 is closed symmetric and has no proper symmetric ex­tensions, H1 is called ma.xilllal symmetric.

§:; • SYMMEI'RIC AND SELF-AD.JOINT OPERATORS 41

LEMMA 9. A self-adjoint transformation is maximal summetric.

IT H is symmetric and f is in the domain of H, then (Hf',r) = (f,Hf') = (Hf',f). Thus (Hf',f) is real and we may make

the following definitions.

DEFINITION 4. Suppose H is symmetric. If there is a real number C such that for every f (=f. e) in the

domain of H, C(f,f) ~ (Hf',f), we let c_ be the least upper bound of such C 's. Obviously c_ is such a C.

If no C exists, let C_ =-ex>. If there is a real num­ber C such that (Hf',f) ~ C(f,f) for every f (=f. e)·

in the domain of H, we let C + be the greatest lower bound of such C 1 s. Othetwise we write C + = ex> •

If' C_ ~ o, we say that H is definite.

LEMMA 10. If C =max (IC+l,IC_I), is <ex> then H is bounded with bound C.

We notice that for every f in the domain of H,

l(Hf',f)i ~ C·l:f'i 2 •

If f and g are in the domain of H, then

(H(f±g),f±g) = (Hf',f)+(Hg,g)±2R((Hf',g))

since (Hg,f) = (g,Hf') = (Hf',g). Hence

R( (Hf',g)) = *( (H(f+g),f+g)-(H(f-g),f-g)),

This equation and the preceding inequality on C yields

IR(Hf',g) I ~ ic( lf+gl 2+l:E'-gl 2 )

= ~( lfl 2+1gl 2 )

using B(11) of Chapter II, §1.

Now if (Hf',g) = <;·i(Hf',g)I where is in the domain of H. Furthermore R( (Hf'' ,g)) = R(<;- 1 (Hf',g)) = i(Hf',g) I.

of the preceding paragraph becomes

I <;I = 1 , then f' = i;- 1 f

if' 12 = lfl 2 and Thus the last inequality

i(Hf',g)I ~ ~C( lf'l 2+fgf 2 ).

42 "IV. ADDITIVE AND CLOSED TRANSFORMATIONS

This holds f'or every f' and g in the domain of' H. But since the domain of' H is dense, we see by continuity that this in­equality holds f'or every g.

Furthermore, if' 11. is real and not zero, we may let f'' = (1/11.)f', g' = i\g and the inequality becomes

l<Hf',g)I ~ -ic«1/11.2 )· lf'l 2+1-· lgl 2 >

ror every real non-zero 11.. If' f' =f e and g =f e, we may let 11.2 = lf'l/lgl and obtain

l(Hf',g)I ~ C·l.fl"lgl.

If' either f' = 9 or g = e; this last inequality is obvious. This inequality holds f'or every f' in the domain of' H and f'or every g. If' we let g =Hf', we get

1Hf'l 2 ~ C·lf'l·IHf'I

which implies IHf'I ~ C0 lf'I. Theorem I of' Chapter II, §3, now gives the result.

§4

THEOREM VII. If' T is c.a.d.d., then (1+T*T)- 1

exists, is self'-adjoint, has domain the f'ull space and is def'inite and bounded with a bound ~ 1.

PROOF. If' {h,kl is a:rry pair of' ~,$ ~2 , it can be ex­pressed as the sum of' an element of' ! and an element of' !• by Theorem VI of' Chapter II, §5. Thus given h and k there is a unique f' in the domain of' T and a g in the domain of' T' such that

{h,kJ {f',Tf'J+{T•g,gJ = {f',Tf'J+{-T*g,g!

or such that h = f'-T*g k = Tf'+g.

In particular if' k = e, this means that to every H, there is an f' in the domain of' T such that

h = (1+T*T)f'.

This f' is unique, since if' there were two distinct f' 's we would have two resolutions of' fh,91.

§4. C.A.D.D. TRANSFORMATIONS

Thus for every h, (1+T*T)- 1h exists. (1+T*T)- 1 is sym­metric. The domain is dense and (b) of' Definition 1 of the preceding section can be shown as follows. Let h and k be any two elements in n. Let f' = (1+T*T)- 1h, g = (1+T*T)- 1k. f and g are in the domain of T*T. Hence

(h,(1+T*T)- 1k) = ((1+T*T)f',g)

=(f,g)+(T*Tf ,g) = (f,g)+(Tf,Tg)

= (f,(1+T*T)g) = ((1+T*T)- 1h,k).

LeilllllB. 3 of the preceding section now shows that ( 1 +T':T )- 1 is self-adjoint.

It is also definite and bounded with a bound ~ 1. For

((1+T*T)- 1h,h) = (f,(1+T*T)f')

(f,f)+(f',T*Tf) = (f ,f)+(Tf',Tf) = lfl 2+1Tf'l 2 ~ o.

Thus A= (1+T*T)- 1 is definite and f'urthermore

(.Ah,h) ~ (f ,f') = (.Ah,.Ah) = I.Ahl 2•

Now for every h, we must have

l.Ahl·lhl ~ l(.Ah,h)I ~ l.Ahl 2 •

This implies lhl ~I.Ahl.

THEOREM VIII. If T is c.a.d.d, T*T is self-adjoint. If T' denotes contraction of T, with domain the domain of T*T, then [T' J = ·r.

PROOF. By Theorem VII, (1+T*T)- 1 is self'-adjoint. By LeilllllB. 7 of the preceding section, 1+T*T is self-adjoint. If' in LeilllllB. 5, we let H1 = 1 +T*T, H2 = • 1 , we obtain that T*T is self-adjoint.

It remains to prove our statement· concerning T 1 • Since T 1 C T, we must have [T ' ] C T. If then [T 1 ] f T, there must be a non-zero pair !g,Tgl of ~ which is orthogonal to all (f',Tf1 for which T*Tf can be defined. (Cf. Coroll­ary 1 , to Theorem VI of Chapter II, §5). Thus f'or every f' in the domain of' T*T,

0 = ( !g,Tgl, If ,Tfl) (g,.f)+(Tg,Tf')

44 "IV". ADDITIVE AND CLOSED TRANSFORMATIONS

= (g,f)+(g,T*Tf) = (g,(l+T*T)f).

But for every h in f> , we can find an f in the domain of T*T such that h = (l+T*T)f, by Theorem VII of this section. Thus for every h in f>, we have ( g,h) = o, and thus g = e1 , Tg = e2, contrary to our assumption that {g,T.g! is a non-zero pair. This contradiction shows that [T'] = T.

COROILARY. Theorems VII and VIII hold if T* is written in place of T, T in place of T*.

This is a consequence of corollary 2 of Theorem III of §2 of this Chapter, since this result permits us to substitute T* for T.

CHAPI'ER V

WF.AK CONVERGENCE

§1

In this section, we shall discuss the weak convergence of elements in Hilbert space. This notion applies in more gener­al spaces as we shall indicate.

DEFINITION. A sequence of elements !fnl of n, will be said to be weakly convergent, if to every g € n, the

~-(fn,g) exists.*

We shall establish for every weakly convergent sequence !fnl the existence of an f E n, such that for every g, (fn,g)- (f ,g).

LEMMA 1. Let Tn be any sequence of continuous additive functions' whose domain is the full space n and whose values are in a linear space. Then if there is a sphere A and a constant C such that for f E A, ~·rnfl ~ C()o), then the Tn 1 s are uniformly bounded.

We lmow .from Theorem I, of Chapter II, §3, that to every Tn we have a en such that I Tnf I ~ en If I for every f. We must show that the en 's are bounded. Suppose that they are not. Then if r is the radius of A, it must be possible to find a Cn such that Cn) 6C/r. Then given E, it is possible to find an f (= fE) such that ITnfl) (1-E)Cn·lfl and we may take lfl = r/2. Thus if f 0 is the center of A, f 0+f is in A and we must have I Tn ( f 0 +f) I ~ C. Hence I Tnf 0 +Tn! I ( C, which implies 1 ·rnf I ~ 2c since I Tnf 0 I < C. But I Tn! I ~ (1-E)Cnlfl =~(1-E)Cn·r ~ (1-E)3C. Now (1-E)3C cannot re-main less than 2C for every E) o and thus we have a con-

* In general Banach space, a sequence of elements is said to be weakly convergent if for every linear functional, F,F(fn) is convergent.

46 V. WEAK CONVERGENCE

tradiction. Hence the Cn 's are uniformly bounded.

THEOREVI I. Let Tn be a sequence of continuous additive functions, whose dornain is the full space n and whose values lie in a linear space. Suppose that for every f in n, Tif is convergent. Then the bounds of

Tn 1 s are bounded.

PROOF: Let us suppose that the Theorem does not hold fo~ a specific sequence {Tnl· Then Lemma. 1 above implies that the I Tnf I s.re unbounded in every sphere.

Now suppose that for i = 1, ••• , k we have specified a

function Tn , a sphere ~i' with a center f i and radius ri

~d such t~t if f E ~· l'rnifl ~ i. Suppose also that ri ~ 2ri-l ~ 1/2 and ~i+l C ~·

We know that the Tnf 's are not bounded in the sphere with

center fk and radius ~rk. We can therefore find a Tilk+l and

an fk+l within this sphere, with ITnk+/k+l I ~ 2(k+1). Since Tnk is continuous, we can find a closed sphere ~k+l with

ceni~r fk+l and radius rk+l ~ ~rk ~ 1 /2k+l for which

ITilk+/ I ~ k+1 for f E ~k+l. If g is in ~k+l,

lfk-gl ~ lfk-fk+1+fk+1-gl ~ lfk-fk+1 l+lfk+1-gl ~ rk

or g in ~k. Hence ~k+ 1 C ~k and we see that we may define a sequence of Tni' ~i' fi' ri, which have the properties given in the preceding paragraph for every i.

Since each ~i contains all that follow and rn---+ o as n-+ oo, the fn 's form a convergent sequence, whose limit f

is in every sphere ~i. Consequently ITnifl---+ co as i --+ex>

and the Tni f 's cannot converge. This contradiction shows that the Tn 's must be uniformly bounded.

COROIJ.ARY 1 • If in Theorem I, for each f, ITnf I is bounded, then the result still holds.

COROIJ.ARY 2. A weakly convergent sequence of elements

{fnl must have the norms ! lfnl l bounded.

Let fn be a weakly convergent sequence. We have a C such

§2. WEAK COMPACTNESS

that lf'nl ~ C. Now f'or evecy g, (g,f'n) --+ F(g), where F(g) denotes the value of' the limit. S1nce I (g,f'n) I ~ lf'nl • lgl ~ C· lgl f'or evecy g, we have IF(g) I ~ C• lgl. S1nce F is obviously additive, Theorem I of' Chapter II, §3, :iJIIplies that F is a 11near f'unctional. Thus there is a f' € n such that (g,f') = F(g) f'or evecy g, by Theorem IV of' Chapter II, §4. Thus we have established:

THEOREM II. If' {f'nl is a weakly convergent sequence of' elements of n, there exists an f' such that f'or evecy g 1n n, (g,f'n) -- (g,f').

§2

Thus if' a sequence {f'nl is weakly convergent it has a weak limit f', 1.e., (f'n,g) ---+ (f' ,g) f'or evecy g. Thus n is complete f'or weak convergence too.

S1nce (f' ,g) is cont1nuous 1n f', we have

LEMMA 1. If' a sequence {f'nl is strongly convergent to f', it is weakly convergent to the same limit.

The converse of' this lemma. does not hold. For let <1>1 , <1>2 , •••

be an 1nf'1nite orthonormal set. For evecy g we have I::=1 1aa1 2 < ~, where acx = (g,<l>cx). Thus f'or evecy g, ( g,<1>al ---+ o and the <l>cx 's f'orm ~ weakl~ convergent series. S1nce however: l<1>a-<1>~I = .../2 1f' ex+ ~, they are not strongly convergent.

This example also shows that there are bounded 1nf'1n1te sets of' elements, which have no limit po1nts. Thus Hilbert space is not locally compact. However f'or weak convergence, we have a k1nd of' compactness.

THEORllM III. If' {f'nl is a bounded sequence of' elements, there exists a weakly convergent subsequence.

PROOF: Let g1 , g2 , • • • be a denumerable set, dense 1n n. The numbers (g1 ,f'cx) are bounded and thus we can f'1nd a subse­quence {f'~! f'or which (g 1 ·,f'~) is convergent. Similarly, we can chose a subsequence {f'~l of' {f'~! such that (g2 ,f';) is

48 V. WEAK CONVERGENCE

convergent. By this process, we can continue to choose sub­sequences so that (gi,f~n)) is convergent for i ~ n. The "diagonal sequence" ff(a)} then bas the property that for each

a n (when the first n elements are ignored) it is a subsequence of ff~n) J. Hence (gi,f(a)) is convergent for every 1.

The norms of the !f~a<lJ l are bounded and thus the linear functionals (g,f~a)) are uniformly continuous on every bounded region. Since these functionals also converge on a dense set, they must converge for every value of g.

§3

Our purpose in this section is to prove Theorems IV and V below. For this, we prove the following lem:nas.

LEMMA 1 • Let T be a linear transformation from n1 to n2 • The domain of T is the full space and we let

~ denote the set in n2 of those elements in the form

Tf, lfl ~ n, ITfl ~ 1. The set ~ is closed.

·since T is linear, T* is also linear. (Cf. Chapter IV, §2, Theorem IV) Now let g be a limit point of An· We can

find a sequence !gil with gi--+ g and such that gi = Tfi for an f i with If i I ~ n. Since the f 1 s are uniformly bmm­ded we can find a subsequence ff~!, which converges weakly to an f with lfl ~ n (Cf. TheoranIII, §2 above). Let gf = Tfi. Then for every h,

(f,T*h) = lim (f~1T*h) a+CD

= lim (Tf 1 ,h) Ql:+CD QI:

= lim (gc'.x,h) Cl:+CD

= (g,h).

Since (T*)* = T, (b) of Theorem II of Chapter IV, §2 implies

Tf = g. Since lfl ~ n, lgl ~ 1, g is in An• Thus ~ is closed.

LEMMA 2. Let · T be a linear transformation fromm n1 to n2 for which T- 1 exists. Let Fin be as in Lemma. 1 • Then if for some n = n0 , An contains a sphere Iii, then

§ 3, CLOSED TRANSFORMATIONS WITH DOMA.IN

T- 1 is bounded.

Let R have radius r and center g 1• For g E R C Rn' we have that g = Tf for an f with If' I ~ n. In particular this is true for g1 = Tf1, Now if h is such tha.t fhl < r, we have that g = g1+h is in R and thus h = g-g1 = T(f-f1 ).

Since ff-f1 I ~ lfl+lf1 I ~ 2n, h is in R2n. Thus R2n con­tains the sphere with center 02 and radius r.

This implies that T- 1 is defined everywhere and has a. bound

~ 4n/r. For if g E n2 and g + 02, let h = (r/2!gl)g. Since h E R2n, (Cf. ,above) we have that IT- 1hl {. 2n, which is equivalent to IT- 1gl ~ (4n/r)·lgl. -

We introduce certain set-theoretic definitions which have played a. very important r<lle in the general theory of linear spaces.

DEFINITION 1 • A set S is sa.id to be nowhere dense if every sphere contains a sphere of the complement of S. A

set will be sa.id to be of the first category if it is a. denumerable sum of nowhere dense sets.

The following Lemma is important.

LEMJIA. 3. A set S of the first category does not con­tain a:rry sphere •

Let S = s1 +S2+ • • • where Si is nowhere dense. Let R be any sphere. We shall show that S does not contain R, Within R, we can find a. R1 belonging to the complement of s1 a.nd we ca.n suppose that the radius r 1 of R1 is ~ 1 /2. Within R1 , we ca.n find a. sphere R2 of the complement of s2 with radius r 2 ~ 1 /22• Conti:Dlllllg, we ca.n find a. sequence of spheres {Ri I ea.ch containing a.11 subsequent spheres, ~ in the complement of Si a.nd whose radii a.pproa.ch zero. The centers lfil of the spheres {Ril form a. conver­gent series, with a limit f, which is in every sphere includ­

ing R. Since f is in every sphere, Ri' it is in the comple­ment of every Si a.nd hence in the complement of S. Since f is in ~. , S does not contain R.

50 V. WEAK CONVERGENCE

LEMMA 4. Let T be a linear tra.."'1.sformation from n1 to n2 for which T- 1 exists. Let ~ be as in Lemmas 1 and 2. If T- 1 is not bounded,. each ~ is nowhere dense.

PROOF: Let fq be any sphere of R2 • An does not contain fq by Lemma 2. Thus A contains a point f of the complement of An. Since the complement of ~ is open by Lemma 1 and its intersection with A is not empty, ~ and the complement of ~ must have a sphere in connnon. Thus ~ is nowhere dense.

THEORJlM IV. Let T be a linear transformation from ~ to n2 whose inverse T- 1 exists. Then if T has n_ as its range T- 1 is bounded.

'"' Under these circumstances, the sum of the ~n· of Lemmas 1,

2, and 4 contains the unit sphere of n2 • If T- 1 were not bounded then this sum would be of the first category by Lemma 4 and hence could not contain a sphere by Lemma 3 . Thus T- 1 is bounded.

THEORJlM V. If T is a closed transformation whose domain is a closed linear manifold, then T is bounded.

PROOF: We consider A the transformation from '! (=n,) to D ( = n2 ), .defined by the equation A! f, Tf l = f. A has dome.in ! and bound 1. Thus A is linear. The inverse A- 1 exists and we note A- 1f = ff,Tfl. The range of A is 1>2 • Thus we may apply Theorem IV and obtain that there is a C such that for every f € D,

C· lfl ~ I ff,Tfl I

• CHAPI'ER VI

PROJECTIONS AND ISOMEI'RY

In this chapter we will consider four special kinds of trans­formations of particular interest in the theory that follows.

§1

DEFINITION 1. Let '11 be a linear manifold of n and for every f, let f "= f 1+f2, f 1 € '111, f 2 € '11 •. (Cf. Chapter II,' §5, Theorem VI) The transformation E which is defined by the equation Ef = f 1 is called a projection.

Lemma 1 • E is a linear self-adjoint transformation with C 2 o, C ~ 1. (Cf. Def. 4 of Chapter IV, §3) - - + -Furthermore E2 = E.

PROOF: The uniqueness of the resolution of Theorem VI of Chapter II, is easily seen to imply that E2 = E and that E is additive. We also see from the same Theorem that the domain

c 2 2 2 ' of E is "· Since !fl = lf1 I +lf21 , !fl ,£,. IEfl .9Ild thus E is bounded •. Hence E is linear.

Now for f and g € n, we resolve f = f 1+f2, g = g1+g2 and then since f 1 and g1 are orthogonal to f 2 and g2 we obtain,

(Ef,g) = (f1,g1+S2> = (f1,g1) = (f1+f2,g1) = (f,Eg).

Thus E is SYllIDletric. Since it is also linear, we know by Lemma 3 of §3 of Chapter IV that E is self-adjoint. Since we also have

2 2 (Ef,f) = (f1 ,f) = (f1 ,f1 > = lf1 I ~ 1r1

we see that c_ ~ o, C + ~ 1 • 1!0}, we easily verify that

Conversely, we have

If both '11 and '11• c+ = 1 and c_ = o.

are not

LEMMA 2. If E2 = E and E is closed symmetric then E is a projection.

Since E2 = E, n the set of zeros of E, includes all ele-51

52 VI. PROJECTIONS AND ISOMEI'RY

ments g f'or which g = ( 1-E)f', f' in the domain of' E. For since f' is in the domain of' E, and thus Ef' = E2f' = E(Ef'),

Ef' must also be in the domain of' E. Hence g is in the do­main of' E and Eg = E( 1-E)f' = (E-E2 )f' = 9.

By Chapter IV, §3, Lemma 8, we lalow that ~C ~* = [!R]L.

Thus 7l and !R are orthogonal and for f in the domain of E,

f = Ef'+(~-E)f

lf'l 2 = 1Ef'l 2+1(1-E)fl 2 •

Thus fEf'I ~ lfl and E is bounded. Since E is closed con­tinuous and has domain dense, E is linear. (Cf. the defini­

tion of "linear" in Chapter IV, §2 preceding Theorem IV) Thus E has domain the full space.

Let m = [!R]. For every f, E gives a !'esolution f =

f 1+f2, Ef' € m, (1~Jlt € '7! 1 • Since only one such resolution is possible, it follows from the definition of projection, that E is the projection on m.

COROLLARY. If E is c.a.d.d. and E2 = E and :R is orthogonal to ~ then E is a projection.

LEMMA 3. If' E is a projection with range m, then 1-E is a projection with range m •.

PROOF: 1-E is self-adjoint and (1-E)(l-E) = 1-2E+E2 = 1-E. Lermna 2 implies that 1-E is a projection.

Now Ef' = 9 1f and only 1f f € '11 1 • Aliio ( 1-E)f = f if and only 1f f is in the range of 1-E, since 1-E is a pro­jection. But (1-E)f = f is equivalent to Ef = 9 and thus m 1 is the range of 1-E.

LEMMA 4. If' E1 is a projection with range m 1 and E2 is a projection with range '71 2, then E1E2 is a

projection 1f and only 1f E1 ·E2 = E2 °E1 • If E1E2 is a

projection, its. range is m1.m2.

The condition E1E2 = E2E1 is by Theorem V of Chapter IV,

§2, a necessary and sufficient condition that E1E2 be self'­adjoint. Thus the condition of our Lemma is necessary and when

§ 1 • PROJECTIONS 53

it holds we lmow that E1E2 ls self'-adjoint. When it holds, we 2 also have ( E1 E2 ) = E1 E2E1 E2 = E1 E1 E2E2 = E1 E2 • Lemma 2 shows

then that E1E2 ls a projection.

If' E1E2 = E2E1, then the range of' E1E2 ls in both that of'

E1 and that of' E2, 1.e., in "11 • "1 2 • Also 1f' f' E m 1 • "1 2,

E1E2f' = E1(E2f') = E1f' = f', 1.e., f' ls in the range of' E1E2 • Thus the range of' E1E2 ls "11 ·"1 2 , when E1E2 ls a projec­tion.

LEMMA 5. If' E1, ••• , Fn_ are. n projections with ranges "11, • • • , mn respectively, then E1 + • • • +Fn_ is a projection, if' and only if' EiE. = o if' 1 + j. If'

E1 + ••• +~ is a projection, then bi is orthogonal to

m j f'or i + j and the range of' E1 + . • • +Fn_ is 2l( m, u ••• u mn) ( u indicating logical sum).

Let us suppose that E1+ ••• + Fn_ is a projection.

EiEj + o f'or i + j. Hence there is an f" such that

9~ Let g = Ef. Then g E mj and Eig + 9. Hence

lgl 2 ~ ((E1+ ••• +Fn_)g,g) = I~=1 <E~,g) = r~=, IE~l 2

~ 1Eigl 2+1Ejgl 2 = 1Eigl 2+1gl 2 > lgl 2 •

This contradiction indicates that EiEj =O.

~ the other hand if' EiEj =O f'or i + j, then (E1 + ••• + Fn_) = E1+ ••• +~. Since E1+ ••• +~ ls self'-adjoint, it is a projection by Lemma. 2 above.

If' EiEj = o, Ej = Ej-EiEj = (1-Ei)Ej. It f'ollows f'rom Le:mma.s 3 and 4 above that mj is in m1 mj or mj in m1. Thus m j ls orthogonal to ml.

Obviously the range of' E1 + • • • +~ must be in 2l( m 1 u • • • u

71tn )'. On the other hand 1f' f'E 2l ( m 1 u • • • u mn), 1t is readily

established that f' = f' 1 + • • • +f'n, where f' 1 E m 1 , • • • , f'n E

mn. Now (E1+ ••• + ~)(f' 1 + ••• +f'n) = f' 1+ ••• +f'n since

E1f'j = E1Ejf'j = 9, if' 1 + j. Thus f' 1+ ••• +f'n is in the range of' E1 + • • • +Fn_ and U( m 1 u • • • u mn) is included in this range.. This and the previous result prove the la13t statement of'

the lemma.

DEFINITION 2. Two projections E1 and E2 are called

orthogonal if' EiEj = o, or what is equivalent, 1f' "1 1 ls orthogonal to m2.

54 VI. PROJECTIONS AND ISOMEI'RY

LEMMA 6. If E1 and E2 are projections with ranges m 1 and m2 , then E1 -E2 is a projection if' and only

if' E2 = E1E2 • If' E1-E2 is a projection then 7n 2 C 7n1,

and the range of E1 - E2 is 7n1 • m 2.

It follows f'rom Lemma 3, that E1-E2 is a projection if a.nd

only if' 1-(E1-E2 ) = 1-E1+E2 is a projection. By Lemma 5, (1-E1 )+E2 is a projection if' and only if' E2 (1-E1 ) = o or E2 = E2E1• These two results ilIIply the f'irst statment of' the Lemma.

If E2 = E2E1, Lemma 4 ilIIplies 7n2 = 7n1 7n2 or m2 C 7n1 • Since E1-E2 = E1-E2E1 = (1-E2 )E1 Lemmas 3 and 4 imply that the

range of' E1 - E2 is '1t2 m1 •

LEMMA 7. Let E1, E2, . • • be a sequence of' mutually orthogonal projections, with range m1, m2, . . . respec­

tively. Let E = r ;'=1E°' (i.e., Ef =ii_~ r~= 1 Ecf whenever this limit exists). Then E is a projection

with range me m, u m2u ••• ) (where u denotes the logical sum).

PROOF: We note that if' n ~ m and f' E fi then by Lemma 5

lfl 2 2 I< rn E )fl 2 = I rn E f'- r:m E fl 2 - <X=m+ 1 <X <X=1 oc <X=1 <X

= I r:=m+1Ecfl2 = r:~=m+1 IEcfl2.

If we let m = o, we obtain lfl 2 ~ r:~= 1 1Eaf'1 2 • Hence lf'l 2 ~ r :=1 I Ecf 12 • This ilIIplies that r: ~m+ 1 I E<Xf' 12-- 0 as m and n - oo. Our f'irst inequality then shows that I r:~=1 Ecf-r: ::=l E°'f I -- o as m and n .--.. oo. This shows that Ef exists f'or every f'.

Now E is S'YlJI!lletric, since for every f' and g of' fi,

(Ef,g) = <ti_~ r:~=1 Ecf ,gl =ti_~ (( r~= 1 Ea)f,g) = ii_~(f,( r:~ 1 Ea)g) = (f,(ii_~ r:~ 1 Ea)g) = (f',Eg).

Furthermore

( r:;=,E<X)( r;=1Ef3) = ft~ ( r::=1Ea)( r:;;=1E[3)

=ii_~ ( r:n E )(11!a Im E ) - lim lim ( r:n )( m CIC> a=1 °' m CIC> [3=1 f3 - n'*OO m~ a=l E°' I f3=l Ef3)

= ft~i!i~ ( I.~=1E<X) =ft~ r::=lEa = r.;=1Ecx.

§ 1 • PROJECTIONS 55

Hence Lellllll8. 2 above shows that E is a projection. We now prove the last statement of' the Lellllll8.. Let '1 denote

the set of' elements f' of' n in the f'orm f' = I::,, f'cx, where

f'cx € ma. Since the ma 's are orthogonal, lfl 2 = I:~= 1 1f'a1 2 • Now a proof' similar to the completeness argument of' Theorem I of'

Chapter III, will show that n is a closed set. Furthermore

U(m 1 um2 u ••• ) C n and every· f' € n is the limit of' elements of' 21(m 1 um2u ••• ). Thus 1l is the closure of' U(m 1um2u ••• )

that is m< m,u m2u ••• ) . One can easily verify that n is the range of' E and this completes the proof' of' the Lemma.

DEFINITION 3. We will write E2 ~ E1 , if' m 2 C m 1 •

LEMMA 8. The f'ollowing statements are equivalent:

(a) E2 ~ E1;

(b) E~1 = E2 (c) For every

PROOF: (a) implies (b). For if' f' € n, we lmow that f' =

f' 1+f'2 where f' 1 € m 1 , f' 2 € mt By Corollary 1 to Theorem VI of' Chapter II, §5, we have t; = f' 1 1+f'1 2 , where f' 1 1 € m 2, , , , f' 1 , 2 € m2•m 1 • Thus f' = f' 1 , 1+(f'1 , 2+f'2 ). Since m; C m2 by Corollary 2 to Theorem VI of' Chapter II, we have f'2 € m~.

Thus f' 1 , 2+f'2 € 7n2, f' 1 , 1 € m 2 and these imply E2f' = f' 111 •

But by the method of' def'inition of' f' 1 , 1 , E2E1f' = f' 1 , 1 • Hence E2E1f' = E2f' and (a) implies (b).

(b) implies (c) since IE2f'l 2 = IE2E1f'l 2 ~ IE1f'! 2 • (c) implies (a). For if' f' € m 2, lf'l 2 = IE2f'l 2 ~ IE1f'l 2 ~

lf'l 2 • Thus lf'l 2 = IE1f'l 2 Since l(1-E1 )f'l 2 lf'l 2-IE1rl 2 = o, we must have (1-E1 )f' = 0 or f' = E1f' € m 1 • Thus m 2 C m 1 •

LEMM. 9. If' E1 , E2, • • • denote.a a sequence of' pro­jections with ranges m,, m2, . • • respectively and such

that Ea~ Ecx+l' then E = liln:n+oo~ is a projection such that Ea~ E. The range of' E is m( m 1 u m 2u ••• ) •

PROOF: Since Ea~ Ecx+l' Lemma 8 implies that Ea·Ecx+1= Ea• Lemma 6 implies that Ea 1-E is a projection. Since F. =

n-1 + . ex -.n E1+:rcx=l(Ea+1-Ecx) is a proJection, Lemma 5 implies that E1,

VI. PROJECTIONS AND ISOMEIT'RY

E2-E1, E3-E2, • • • are mutually orthogonal. Furthermore E = l1m -w. ~ E1+ I:°" 1(E 1-E ). Thus Lemma 7 states E is a

--:tl.+co -:n CC= CC+ CC projection.

The expression f'or Fn in the previous paragraph and the resulting orthogonality relations, shows that if' 13 2 n,

E'n(E~ 1 -El3) = o. It f'ollows that FnE = E1 +I:~= 1 (Ecc:l-Ecc) = ~ and thus by Lemma 8 that Fn ~ E.

It f'ollows f'rom Lemmas 7 and 6 that the range of' E is '1?('1? 1 u'1?fm 2um~m 3u ••• ). Corollary 1 to Theorem VI of' Chap­

ter II, §5 shows that U('1?cc,'1?tx·'1?cc+l) = mcc+l" These two results imply that the range of' E is '1? ( m 1 u '1?2u ••• ) •

LEMMA 10. Let E1, E2, ••• be a sequence of' projec­tions with ranges ml, m2, • • • respectively and such

that E cc~ Ecc+ 1 • Then E = 11llh•co Fn is a projection such that E ~ Ecc f'or every cc, and the range of' E is

m, · m2 • • • • •

PROOF: Let f'~=1-E~. Then E 2E+l' implies F ~F 1, ~ ~ a - cc a - cc+

since F cc •F cc+ 1 = ( 1-Eac) ( 1-Eac+l ) = 1-Eac-Eac+l +EacEc&l = 1-Eac = F cc' when we use Lemma 8 (b). Lemma 9 tells us that lim Fa is a projection. Hence 1-lim F°' = 11m(1-F<X) = lim Eac= E is also a projection. Since lim Fcx ~ Fn, we must have E ~ Fn·

If' f' is in 7n, • '1?2 • • • • then Eacf' = f' f'or every ex and hence Ef' = f'. Thus f' is in the range of' E and '1.n1 • m 2. • • • is included in this range. On the other hand, if' f' is in the range of' E, we have f' = Ef' and since we also have ~E = E we have E'nf' = E'nEf' = Ef' = f'. Thus f' € mn. Since this holds f'or every n, f' € m 1 • m 2. • • • and this set in­cludes the range of' E. These results together imply the last statement of' our lemma.

§2

DEFINITION 1 • A tra.nsf'orma.tion U f'rom fl1 to fl2 with domain fl1 and range fl2 and such that (Uf',Ug) (f' ,g) f'or every pair of' elements in fl1 is called a

mli tary tra.nsf'orma.tion f'rom fl1 to fl2 •

§2. UNITARY AND ISOMEI'RIC TRANSFBRMATIONS

LEMMA 1. A unitary transf'orma.tion U is linear, u-l

exists and U* = u- 1 •

PROOF: Given f' 1 and f'2 , we have f'or every g,

57

(U(af' 1+bf'2 )-aUf'1-bUf' 2 ,Ug) = (U (af' 1+bf'2 ) ,Ug)-a(Uf' ,Ug)-b(Uf' 2 ,Ug)

= (af'1+bf'2 ,g)-a(f'1,g)-b(f'2'g) = o.

Since the set of' u g Is f'ill out n2, we have u ( af' 1 +bf' 2) = aUf' 1 +bUf' 2 and U is additive. Since If' 12 = ( f', f') = (Uf' ,Uf') = !Uf'l 2 we see that U has bound 1. Thus U is linear. Since 1-fl 2 = IUf' 12 , Uf' = e2 implies f' = e, • Thus u- 1 exists. It is readily seen to be unitary.

Since U is linear, U* exists. For every f' E n1 , g E n2

we have (Uf' ,g) = (f' ,u-1 g).

Thus u- 1 CU* and since the f'ormer'has domain n2 , U* = u-1•

LEMMA 2. If' U is a unitary transf'ormation f'rom n1 to n2 and ~1' ~2' ~·· is a complete orthonormal set in n,, then u~,, u~2· . . . is a complete orthonormal

set in ~·

PROOF: It is easily seen that the u~,. U~2 • • • • is an or­thonormal set. If' g is in n2 , g = Uf' f'or an f' E n1 • f' =

I';=,aoc~oc by Theorem XII of' Chapter II. Thus g = Uf' = I';=1 aocU~oc and 711( IU~oc! ) = n2 • Thus the set U~1 , U~2 , •••

is complete.

LEMMA 3. If' ~, , ~2 , • • • is a complete orthonormal

set in n, and "111 ~' • • • is a complete orthonormal set in n2 then the tra.nsf'ormation def'ined by the equa­

tion U( I:';=i acc~oc) = r;=, acclilcc is unitary.

This is an innnediate consequence of' Theorem XII of' Chapter II.

DEFINITION 2. Im additive transf'orma.tion V with the property that (Vf',Vg) = (f',g) for every f' and g in its domain is called isometric.

58 VI. PROJECTIONS AND ISOJV!Err'RY

LEMMA 4. kn isometric tranaf'ormation V is bounded and has an isometric inverse. [VJ exists and is isometric with domain the closure of' the domain of' V, and range, the closure of' the range of' V.

PROOF: The f'irat statement is proved in a manner analogous to the proof of the corresponding statements in Lemma 1 • Theorem II of §3, of Chapter II implies that [VJ exists and has domain the closure of the domain of' V. The continuity of' the inner product shows that [VJ is isometric. A similar argument will show that [V- 1] exists and has domain the closure of the range of' V. From graphical considerations we see that [VJ- 1 = [V- 1J. Hence the range of' [VJ is the closure of' the range of V.

LEMMA 5. kn additive transformation V, with the property that f'or every f in its domain IVfl = lf'I, is isometric.

Thia is a consequence of' the identity 1 '2 21 .. 2 2

(f,g) = 4( lf'+gl -lf-gl )+4i( lf+igl -lf-igl ).

LEMMA 6. Let V be a closed isometric transformation with domain m and range 11. m and '1 are closed. Let s1 be an orthonormal set ci>1 , ci>2 , • • • such that m ( s1 ) = m . Then v cl>, , v 4>2, is an orthonormal set s2 such that "1( s2 ) = 11.

m and 71 are closed since V is closed and bounded. The proof of' the remainder of the Lemma is analogous to the proof of LelllillB. 2 above.

Theorem XI of Chapter II, §6 has the consequence:

LEMMA 7. Let s1 denote the orthonormal set ci>1 , ci>2 , •••

S2 denote the·orthonormal set ~1 , ~2 , ••• and suppose that s1 and s2 have the same number of elements. Then the transformation V defined by the equation V( I:cxacx<i>cx> = r:cxacx~cx is a closed isometric transformation with domain m ( s1 ) and range m < s2 ) •

§2. UNITARY AND ISOMEI'RIC TRANSFORMATIONS

DEFINITION 3. Let m 1, m2 , • • • be a sequence of' mutually orthogonal man:i.f'olds and V 1 , V 2, • • • be a sequence of' isometric transf'ormations such that Va has domain ma and range 7la. Let us suppose f'urther that the 7la 1 s are mutually orthogonal. It f'ollows f'rom the last paragraph of' the proof' of' LellllIIB. 1, of' the preceding section that m( m1 u '7!2u ••• ) is the set of' elements in

the f'orm I:af'a, f'a E ma.' We def'ine

V1&V2& ( raf'a) = I:aVaf'a.

LEMMA 8. V1&V2& • . • is isometric with domain

m ( m1 um2u ••• ) and range m(7!1u~u ••• ) •

59

If' V1 and V2 are closed isometric and v2 is a proper ex­tension of' V 1 , the domain and range of' V 2 include properly the domain and range of' v1 respectively. Thus D• and ~· f'or v 1 are not {el. On the other hand; if' D• and ~· are not e let <1>' with 14>'1=1, be ED• and IJI' with 1"1'1=1, be E ~·. Let V 0 be def'ined by the equation V 0(acj>1 ) = alj/1 • Then by Lemma 8, V 1 &V 0 is a proper isometric extension of' V 1 • Hence

LEMMA 9. A closed isometric transf'ormation v1 has a proper isometric extension if' and only 1f' both D• and ~·

are not f 9!.

The dimensionality of' a closed linear ma.nif'old m is number of' elements in an orthonormal set s1 such that = m. In terms of' this def'inition, we may state

the

m(s1 >

LEMMA 10. A closed isometric transf'ormation v1 has a unitary extension U 1f' and only if' the dimensionality of' D• is the same as that of' ~·.

To" show that this is necessary, suppose that V has a unitary

extension U. W:e take a complete orthonormal set <1>1 ,<1>2 , • • • ,

Cl>{, Cl>2, • • • with <1>1, <1>2, ••• E D, <I>{, Cl>2, ••• E D•. This can be done by using Theorem VI of' §5 of' Chapter II and Theorem XI of' §6 of' Chapter II, because these two results imply that an

60 VI. PROJECTIONS AND ISOMm'RY

orthonormal set which consists of' a complete orthonormal f'or D and another f'or D• is complete. Since U is an extension of' V, Ucj>oc = V<1>a € !R and since U is unitary U<1>~ € !R•. Since the set U<1>1 , U<1>2 , ••• ; U<1>{, ll<l>2• • • • is complete by Lemma 2

above and IU<1>1 , U<1>2 ,... l = IV<1>{, V<1>2 , • • • l determines !R by Lemma 6 above, we must have that 14>{, Uci>2, • • • determines !R•. Thus n• and !ff• must have the same cllJl!ensionality.

On the other hand, if' D• and !R• have the same cllJl!ensional­ity, we can f'ind. a partially isometric, V0 such that V0D• = !R• by Lemma 7 above. Lemma 8 can be used.to show that v1ev0 is a unitary extension of' v1 •

DEFINITION 1 • An additive transf'ormation W f'rom n1 to n2 which is isometric on a linear manii'old m and zero on '7l • is called a partially isometric transf'orma­tion. '7l is called the initial set of' WJ! '1 the range of W is called the final set of' W.

LEMMA 1. A partial isometric W is linear. The f'inal set of' W is a linear man1f'ol:d. Let V be the contrac­tion of W with domain 7Jl, let E be the projection of' n1 on 7Jl. Let F be the pl'Ojection of ~ on 11. Then

W =VE= FVE, W* = v- 1F = EV- 1F, W*W = E, WW* = F.

PROOF: Since f' = f' 1 +f 2, f' 1 € 7Jl, f 2 € 7Jl ', Wf is def'ined and equals Wf 1 • Hence W has n1 as its domain. Since 1Wf1 I = lf1 I ~ lfl, W is botmded. Thus W is linear.

Since V is botmded and has a closed linear ma:nif'old as its domain, V is closed. Thus the range of W which is the range

of V is closed. Since Wf = Wf 1 = Vi' 1 = VEf' for every f, we have W = VE. Since the range of V is 71, W = FVE. Inasmuch as v- 1 is isometric on 71, we have

(w _, -1 f,g) = (FVEf',g) = (VEf',Fg) = (Ef'.,V Fg) = (f,EV Fg)

for eve'I"Y f and g. Hence EV- 1 F C W*. Since the fo:nner is everywhere defined, W* - EV- 1F. Now

§3. PARTIALLY ISOMEl'RIC TRANSFORMATIONS

Similarly WW* = F.

I.EMMA 2. A c.a.d.d. W such that W*W = E ls a projection is partially isometric with initial set, the range or E.

61

PROOF: Since W*W ls everywhere derined, W ls everywhere derlned. If' r 1 and g1 are in the range or E, Er1 = r 1 and we have (f1 ,g1 ) = (Er1 ,g1 ) = (W*Wr,g1 ) = (Wr1 ,Wg1 ).

Thus W ls isometric on the range or E. If' r ls orthogo­nal to this range, Er = o and we have

o = (Er,f) = (W*Wr,r) = cwr,wr) = 1wr1 2

and thus Wr = e. Hence W ls partially isometric.

I..EMMA 3. The rollowlng statements are equivalent ror c.a .. d.d. transrorma.tlons: (a) W ls isometric, (b) W* ls isometric, ( c) W*W is a projection, ( d) WW* ls a pro jectlon, ( e) WW-llW = W, ( r) W*Wtil* = W*.

Lemmas 1 and 2 imply that (a) and (c) are equivalent. Simil­arly (b) and (d) are equivalent. But Lemma 1 shows that (b) implies (a) and also that (a) implies (d). These results show that the r!rst rour statements are equivalent.

If' W ls 1so111etrlc, we know that WE = W and that E = W*W. Thus WW*W = W and (a) implies (e). On the other hand, 1r

Wtl*W = W, we have W*Wtil*W = W*W or (W*W) 2 = W*W. Since W*W ls sell-adjoint and (W*W) 2 = W*W, W*W ls a projection by Lemma. 2 or § 1 or this chapter. Thus ( e) implies ( c) and hence (e) also must be equivalent to axry one or the rlrst rour. Taking adjolnts shows that (e) 1s equivalent to (r).

I..EMMA 4. Suppose Wcx, ex = 1, 2, • • • ls a partially isometric transro:mia.tlon with domain mcx and range nex" We shall suppose that the m a 1 s are mutually orthogonal and that the na 's are mutually orthogonal too. Then W = r:awcx ls partially isometric· with the initial set m ( m,u '1l2u •••. ) and the rlnal set '11('\u\u ••• ).

62 VI. PROJECTIONS AND ISOMETRY

Lemma 8 of' the preceding section shows that .roe Woe is isomet­

ric on '1l( m 1 u m 2u ••• ) • Now if' Eoe is the projection on moe'

E = l:oeEoe is the projection on '1l('1l1um2u ••• ) (Cf'. Lemma 7

of' §1 Gf' this Chapter). Then WE= ( roeWoe)E = I:oeWoeE = l:oeWoeEaE

= roe W~o: = I:oeWoe = W. Since WE = W, W is zero on the set m( m, u m2u ... )1 and w is isometric.

§4

We return brief'ly to the considerations of' §4 of' Chapter IV.

THEORBM I. If' A and B are c.a.d.d. operators f'rom

~ to n2 such that A*A = B*B, then there exists a partially isometric W with initial set, the closure of' the range of' A and f'inal set the closure of' the range of'

B such that B = WA, W*B = A, B* = A*W*, B*W = A*.

If' f' is in the domain of' A*A, Af' = 9 implies

O = (A*Af',f') = (B*Bf',f') = (Bf',Bf') = 1Bf'l 2

or 1Bf'l 2 = o. Thus the set in n2E& n2 of' pairs IAf',Bf'J, f' in.the domain of' A*A, is the graph of' a transf'ormation V. The domain of' V is the range of' A' the contraction of' A with domain the domain of' A*A.

Now V is isometric since f'or f' and g · in the domain of' A*A,

(Af',Ag) = (A*Af',g) = (B*Bf',g) = (Bf',Bg).

Thus if' q>. = Af', ljl = Ag, Vq> = Bf', Vljl = Bg and

(.q>,ljl) = (Vq>,Vljl)

f'or every

where B' B*B.

<P and ljl in the domain of' V. Furthermore VA 1 = B' is the contraction of' B with domain the domain of'

Let f' be an element in the domain of' [A 1 ] • We can f'ind a

sequence f f'n,A'f'nl such that lf'nA 'f'n! ~ ff', [A 1 ]f'j. Hence the sequence IA'f'nl converges and owing to the isometry rela­

tion, V, the sequence !B'f'nl must converge to a g*. Thus

lf'n,B'f'nl converges .so lf',g*}. Then by the def'inition of' [B' ], [B' Jf' exists and equals g*. We also have IA'f'n,B'f'nl ~ I [A' Jf', [B' Jf' j • Hence this latter pair is in the graph of'

§4. C.A.D.D. OPERATORS

[V] and [V] [A' ]f = [B' ]f. This last equation holds for every f in the domain of [A'] and hence [V] [A'] C [B']. If we take f in the domain of [B 1 ] , a precisely similar argument

shows the reverse inclusion and hence [V] [A 1 ] = [B'].

Theorem VIII of §4 of Chapter N states that [A 1 ] = A, [B 1 ]

B. Thus [V]A = B. It is obvious from the graphical considera­tions made in the above that A = [Vf1B, that the domain of [V] is the closure of the range of A and that the range of [V] is

the closure of the range of B. Let the closure of the range of

A have a projection E and that of B have a projection F. Let W = [V]E. Then W = F[V]E, W* = E[V]- 1F, W is partially

isometric with initial set the range of E, and final set the range of F. B = [V]A = [V]EA =WA. A= [V]- 1B = E[V]- 1FB =

W*B. Now if B = WA, B* = A*W* by the Corollary to Theorem V of

§2, Chapter N. Similarly A*= B*W. It should be remarked that the partially isometric W is

introduced because in general [V]* does not exist.

CHAPI'ER VII

RESOL1JrIONS OF THE IDENTITY

§1

In this section we will discuss certain properties of self­adjoint transformations whose range is finite dllnensional. While these results will be applied later, they should also be regarded as indicating what results are desired in the general case.

LEMMA 1 • Let H be a self-adjoint transformation with a finite dimensional range m which is determined by the orthonormal set 411 , • • • , 4>n. Then H is zero in m• and there is ~ n 1th order matrix (acr:,~) er:,~= 1, ••• , n

with acr:,~ = a~.cr: such that H4>cr: = I~acr:,~4>w H is botmded.

Since H = H*, H is zero on m• by Lemma 8 of §3 of Chap­ter IV. Let E be the projection on 7n. m• C DH implies thatfor f€Dil, (1-E)f€Dil and Ef€DH. Since~ is dense, this implies that 111 is dense in m. Since 7n is finite dimensional and ~ is additive, ~ must contain m. Since H4> € 7n, H4> = I ~1 acr:,~4>~ by Theorem ~ of Chapter II, §6. acr:,~ = (H4>a•4>~) = <4>a,H4>~) = (H4>w4>cr:) = af\cr:"

The botmd of H is the botmd of its contraction defined on m and for this we have

IHicr:Xcr:4>cr:1 2 = I Icr:Xcx114>cr:1 2

2 2 I I~ ( Iaxa!l-«,~)4>~1 = I~ I Iaxa!l-cr:,~1

~ ( I:cr:,~lacr:,~1 2 )(Icr:lxcr:l 2 ). The converse of Lemma 1 holds.

I.m.MA 2. A finite orthonormal set S, 4>1, • • • , <l>n• and a S'YJilllletric ma.triX (acr:,~) er:,~= 1, ••• , n (1.e., acr:,~ = a;~.a.) determine a self-adjoint transformation by means of the conditions: If f € m(s)•, Hf = e; H4>cr: = I:~ acr:,~4>W

64

§ 1 • SELF-ADJOINT TRANSFORMATIONS 65

H is readily seen to be s;ymmetric and def'ined everywhere. The essential result of' this section is that <1>1, ••• J <l>n

can be chosen so that aa,~ = i'l.a6oc,~ f'or real "'a· Thus we obtain the "diagonal f'orm" f'or the matrix.

LEMMA :; • If' H is a non- zero self'-adjoint transf'or­ma.tion whose range is a f'inite dimensional '11, then we can f'ind a <1> in '1! and a non- zero l\. such that if' '11 1 is the set of' lji in '1! which are orthogonal to cj>, then f'or every f' in Ji,

Hf'= i'l.(f',cj>)cj>+H,f'

where H1 is a self'-adjoint transf'ormation whose range

is m 1 •

PROOF: Since H is not zero, either C+ ) o or C_ < o. (Cf'. Def'ini tion 4 of' §:; of' Chapter IV) • We shall suppose C +>o. (Otherwise our argument would apply to -H) Let E' be the pro­

jection on m. Then EHE = H since H is zero on '7? •. For every f' we have (Hf' ,f') = (EHEf ,f') = (HEf' ,Ef').

Now let f'1, f'2, • • • be a sequence of' elements with lf'nl=l

and such that (Hf'n,f'n) - C+. It f'ollows that lEf'nl ~ 1, (HEf'n,Ef'n) ---+ C+. All the Ef'n 1s are in 7Jt whose unit sphere is compact. Thus a subsequence of' the Ef'n 1 s llllll!t converge to a g in '1! such that (Hg,g) = c+, lgl ~ 1. Furthermore lgl = 1, since if' lgl < 1, (Hg,g)/lgl 2 ) C+ a contra.diction.

We let <I> = g. If' lji € 7Jt1, then (lji,<I>) = o and f'or every value of' et, I COSCtcj>+sinaljil 2 = 1 • Thus if f' = cos~+sinalji,

(Hcj>,cj>) = C+ ~ (Hf',f') = cos2a(H41>,cj>)+2sinetcosaR(Hcj>,lji)+sin2a(Hlji,lji).

This is only possible if' R(Hcj>1 1ji) = o. Multiplying lji by a constant does not ef'f'ect lji € IC!>!• and thus we have by a f'a.mil­iar process I (HIC>,lji) I = o.

Let <I>, "11 , • • • , "'n- l be an orthonozml set complete in m and thus

~ = l\.cj>+b1"11+ •• • +bn-1"1n-1 •

But "1a is in 7111 and thus bet = (H41>,1jiet) = o. Hence Hcj> = i'l.<1>,

where i\ = (H<1>,<1>) = C+ is real and not zero. If' E1 is the projection with range 711,, we have

66 VII. RESOLUTIONS OF THE IDENTITY

(HE1f,q) (E1f,Hcj>) = o ·for every f. Thus the range of HE1 is orthogonal to <j>. The range of HE1 is included 1n 71! and this means that it is included in 77! 1• Hence HE1 = E1HE1 • E1HE1 is self-adjoint.

Now the projection on facj>l is given by the equation Eqf = (f,<j>)cj>. Hence Ef = (f,<j>)cj>+E1f and

Hf= HEf = (f,<j>)Hqi+HE1f = A(f,<j>)cj>+H1f

where H1 = HE1 = E1HE1 is self adjoint with range included in 77! 1 • Dimensionality considerations show that the range of H1 must be m, .

Call <j>, <1>1 and apply Lemma 3 to H1• This gives us a <1>2 in m, such that

Hf = A, ( f, <I>, )cp1 +i\2 ( f ,<j>2 )+H2f

where H2 has range in 77!1 orthogonal to cp2 and of course to lf>, • Repeating, we obtain the· following lemma.

I..Jl)f.fA 4. If H is a self-adjoint transfonna.tion with

a finite dimensional range 71!; we can find an orthononna.l set S, cp1, ••• , cpn and real non-zero constants A,, ••• , "'-n. such that

Hf= A,(f,<1>,>ci>1+"'-2(f,1f>2)ci>2+ ~ •• ~(f,<j>n)lf>n.

The "'-a satisfy the inequality c_ ~ "'-a~ c+.

For such a transfonmtion, H2, H;;, • • • can be defined. Let p(x) = anx:11:!8'n._ 1x11- 1+ ••• +a0 be a polynomial. We define p'(H) = ~H11+an__ 1 H11- 1 + ••• +a0E where E is the projection on 71!. Then it is easily verified that

p'(H) = p(i\1 )(f,<1>1 >ci>,+p(i\2)(f,<1>2><1>2+ ••• +P("'-n_)(f,cpn)lf>n.

In general we have

LEMJIA. 5. The co.i.•respondence p(x) ... p 1 (H) preserves the operation of addition, multiplication and multiplica­tion by a constant.

p' (H) = ~Jill~_,H11-1 + • • • +aoE

= p(A.1 )(f,4>1 >4>,+ • • • + p(A.n)(f',cpn)cpn

§2. RESOLUrIONS OF THE IDENTITY AND INTEGRATION 67

when p(x) = ~x11+~_ 1 x11- 1 + ••• +a0• When the a 's are real and c 1 and C2 are two constants such that f'or

c_ ~ x ~ c+, we have c, ~ p(x) ~ c2 then f'or every f',

C1(Ef',Ef') ~ (p'(H)f',f') ~ C2(Ef',Ef'),

To show t:he last statement we note that p 1 (H) = Ep 1 (H). Thus

(pr (H)f' ,f') = (pr (H)f' ,Ef')

(P(A, )(f',<1>1 ><1>1+ ••• +P("ii)(f',cj>n)<l>n1 (f',cj>1 )cf>,+ +(f',cj>n)cj>n)

= p(i\,)l(f',cf>,)12+ ••• + p(i\n)f(f',cj>n)l2.

This will imply the desired inequality.

We will obtain an inf'inite analogue of' Lennna.s 4 and 5. How­ever it must be remembered that in deali...--ig with an inf'inite di­

mensional space, one must consider not sums but limits of' sums.

Thus r-;=1 (or I:~-= ) represents that special limiting pro­

cess in which one (or both) limits of' summation are permitted to

approach CD. The integral ~ in the Rieman-Stieljes sense, is a more general process of' taking the limit of' a sum, which in­cludes the preceding method. Thus the generalization of' the ex­pression f'or Hf' in Lemma 4 need not be a.n inf'inite sum

r;,._= i\x(f' ,cj>a)<l>a

but a more general method of' taking the limit of' a sum.

§2

DEFINITION 1. A f'a.mily of' projections E( i\) def'ined

f'or -CD < i\ < +CD is called a resolution of' the identity if'

1 • E(i\) ~ E(µ) f'or i\ ) µ.

2. E( A+O) = E( i\) • *

3• 11~-+ -CDE( i\) = o, limi\~ CDE(A) = 1.

A resolution of the identity will be said to be finite, if' there is a A1 such that E( A1 ) = o and a i\2 such that

E(A2 ) 1. **

2 2 * E(A+O) limE-+ OE(i\+E ), E(i\-0) = limE-+ OE(i\-E ) • **The following are examples of a resolution of the identity.

(a) Let ••• <1>_ 1 , <1>0 , cp1 , • • • be a complete orthonormal set

68 VII. RESOLUTIONS OF THE IDENTITY

It follows f'rom Lemma 6 and 8 of' Chapter VI, §1, that if' i\1 ) i\2, E(i\1 ) - E(i\2 ) is a projection. It f'ollows f'rom

Lemma 5 of' Chapter VI §1, that if' i\1 ) i\2 ~ µ1 ) µ2, then (E(A, )-E(i\2 ))(E(µ 1 )-E(µ 2 )) = o since E(i\1 )-E(µ 2 ) = E(i\1 )-E(i\2 )+

E(i\2 )-E(µ1 )+E(µ 1 )-E(µ 2 ). It is also a consequence of' Lemma 8

that E(i\1 )E(i\2 ) = E(m1n(A1,i\2 )) = E(i\2 )E(i\1 ).

DEFINrrION 2. If' b ) a, we shall def'ine a "partition" Tr of' the interval (a, b) as a set of' points x 0 = a, x 1 ,

x2, ••• , ~ = b, with xa < xa+l which subdivides the interval (a, b) into n smaller intervals (xa- l ,xa). The interval (xa- 1 ,xa) will be said to be marked 1f' a

point x& with xa_ 1 ~ x~ ~ xa is chosen in it. If' each smaller interval of' n is marked, we will say that the partition n is marked and denote the marked partition

n•. If' n 0 is a subdivision y 0,y1, ••• , Ym such that

every xa = y~a for some ~a then n 0 is called a finer subdivision of' TI. We indicate tl:llls n 0 -< n. The mesh of'

a subdivision m(n) = ma.x(xa-xa_ 1 ). If' E(i\) is a resol­ution·of the identity, cj>(i\) a complex valued f'unction

defined on the interval a ~ i\ ~ b and n• a marked sub­division of' this interval we def'ine

I:n,cj>AE(i\) = I:~= 1 cj>(x~)(E(xa)-E(xa- 1 )).

LElVMA 1. 1TJ.c!>t.E( i\) is a bounded tra.nsf'omation with

bound C = ma.xl11><x:X> 1. We also have

(a) ~'1>16E(i\)+ I:n,'1>2AE(i\) = I:n,<4>,+'1>2)AE(i\).

Cb> (I:n1+1AE( i\) >CI: n 1+2t.E(i\ > > = I: n 1'1>1+2t.E< i\) •

(c) (I:n 1cj>AE(i\)f,f') = I:~= 1cj>(x~)((E(xa)-E(xa- 1))f',f). (d) II:n,lj>t.E(i\)f( 2 = I:na= 1 1<1>Cx:X>l 2 ·1(E(xcx)-E(xcx-l))f'l 2 •

whose indices range over the integers f'rom -oo to oo. If m ( i\) is the ma.nif'old determined by the <l>cx f'or which a s. i\, and E( i\) is the projection on m(i\), then one can easily­verif'y that E( i\) is a resolution of the identity.

(b) Let us realize Ji as C2 (Cf'. Chapter III, §4, Def'ini­tion 1). If' i\~O,wedef'in~ Xi\(X)=01 If' O(i\~1, x (x) = 1 when x S. i\ 1 x (x) = o if' i\ ( x. If' i\) 1, xi\ (x) := 1 • It is readilf verif'ied that the transformations ~ineq by the equation E(i\)f(x) = xi\(x)f(x) f'orm a resolution of the ideD.tity.

§2. RESOLUrIONS OF THE IDENTITY AND INTEGRATION 69

These results are immediate consequences or

Thus

1In<!>AE(i\)rl 2 = r!= 1 1<1><x:Xll 2 • l(E(xa)-E(xa-- 1 ))rl 2

~ C2 ( I~= 1 I (E(xa)-E(xa-- 1 ))r1 2 )

= C2 1(E(b)-E(a))rl 2 ~ c2 1r1 2 •

DEFINITION 3. We shall say that S~<I>( i\)dE( i\)r exists ror a given r ir there is an element r* 1n fl such that ror every sequence or partitions n 1 ,n2 , • • • such

that nee < na_ 1 and m(T\.) --+ o as n - oo, then

1r*-In1<l>AE(i\Jr1 --+ o. n

We derine S~<l>(i\)dE(i\)r = r*.

LEMMA 2. Let cj>(i\) be continuous on the interval (a,b), and let n• and nb denote two marked subdivisions with T'fo a riner subdivision or n. Given an E > o, there is a number µ = µ(E) such that H the mesh or n, m(TI) is ~ µ(E ), then

I I n•<l>AE(i\)r- I n•<l>AE(i\)rf ~ E fr f. 0 -

PROOF: Since <I> is unHormly continuous, we may derine µ(•l as the 6(E) ) o, such that when (x1-x2 ( < 6(E) then (cj>(x1 )-cj>(x2 ) I < e. Let us suppose m(TI) ~ µ. We derine

~· (x) by the equation •n• (x) = cj>(x:Xl H xat-1 ~ x < xa• Since m(TI) ~ µ, l<l>n• (x)-<l>(x) I • Furthermore ror n0( TI

Lno4>n1AE( i\) = I:n<l>AE( i\).

Hence the tra.nsrorma.tion

I: n•<l>AE( i\)-I: n ,cj>AE( i\) 0

= I: n ·~AE(i\)-I: n1<l>AE(i\) 0 0

= I TI.! ( <l>'-<l>)AE( i\) 0

has a bound ~ c by Lemma 1 above.

The existence of S~<l>(i\)dE(i\)f follows from Lemma. 2 1n a lllEl.Illler entirely analoguous to the proof of the existence of the

70 VII. RESOLUTIONS OF THE IDENEITY

Rieman-Stieljes integral in the ordinary sense. Thus Lemma 2 implies that every sequence of' partitions n, ,n2, ••• with

ncx < n~, and m(f'\i) - 0 as n --+ CX) will have I: n~<l>l!.E( 71.)f' convergent. The sequences n1 , 1 ,n2, 1 , • • • and ~, 2,n2, 2, • • • will have the corresponding r. convergent to the same limit as one easily sees if one considers a subdivision

'1i_ which is a finer subdivision of both nn, 1 and nn, 2 •

THEOREM I. If' cj>(x) is a continuous f'unction on the

interval a~ x ~ b, then f'or every f,

Tf' = S~<l>(7'.)dE(7'.)f'

exists. Tf' is linear with bound ~ maxa S.. x ~ bcj>(x). cj>(x) is real, T is also self-adjoint. -

If

The continuity and the bound of' T are consequences of' Lemma 1. Suppose cj>(x) is real. If' we let f'or the moment Tn = rn cj>l!.E(?..) we see that each Tn is self-adjoint. Thus for every f' and g.

(Tf',g) = lim(Tnf',g) = lim(f',Tng) = (f',Tg).

Hence T is s-ymmetric and since it is linear it is self'-adjoint.

LEMlllIA 3. If' H = S~<l>(7'.)dE(7'.) for <!>(?..) real and con­tinuous then

(Hf',f') = S~<1>(7'.)d(E(7'.)f',f')

where the integral is the Riema.n-Stieljes one in the ordinary

sense. C+ f'or H is ~ maxa S...x S.. b<1>(x) = M, when M ~ o. c_ f'or H is ~ mina_s: x $.. b4>('X) =-m when rp ~ o,

1Hf'l 2 =-S~l;(?..)f2dfE(7'.)fl 2 •

This Lemma f'ollows readily f'rom Lemma 1 above. We ma;y rema:r'k that the conditions M ~ o, m ~ o in Lemma 3

are necessary because there may be non-zero f' 's such that (E(b)-E(a)f' = 0. If' no such f' occurs, these conditions may

be omitted. (Cf. Lemma 1 above and its proof.)

LEM4A 4. If a.~ a' < b 1 ~ b

§2. RESOLUTIONS OF THE IDENTITY AND INTEGRATION 71

<S~<1>(ll.)dE( 11.) )(E(b 1 )-E(a 1 ))

(E(b 1 )-E(a 1 »S~<l>(i\)dE(i\) = S~:<l>(i\)dE(i\).

This is easily seen if' one considers the orthogonality re­

lations for the dif'f'erences E(xcx)-E(xcx- 1 ) •

We can now point out the precise circumstances under which,

we will have C+ < maxa ~ x {. b<j>(x) = M, M) o in Lemma 3. Let 11.0 be a point of' tlie lii.terval (a,b) for which <j>(i\ 0 ) = M.

Then there are two numbers "1 -and 11.2 in the interval (a,b)

with "; ~ i\0 ~ 11.2 , ~ + 11.2 and such that for 11.1 ~ i\ ~ 11.2 ,

<j>(i\) ~ C++6 for a c5) o.

For any such pair, we must have E(i\2 )-E(i\1 ) = o. For let

f' be in the range of' E(i\2 )-E(i\1 ). Then Tumma 8 of' Chapter VI,

§1 and the preceding results imply

<S~<1>(ll.)dE(11.)f',f') = <S~<1>(ll.)dE(i\)(E(11.2>-E<11.1 ))f',f')

= <S~<1>(11.)dE(11.)f',f') = S~2<t>(ll.)d(E0i\)f',f') i\1 "1"

~ rll.2<1>(ll.)dlE(i\)f'l 2 2 (C +cS)S~~dlE(i\)f'l 2 J i\1 - + ·1

(C++cS))~d(E(i\)f',f') = (C++cS)({E(i\2 )-E(i\1 ))f',f')

= < c + + 6) If' 12 •

The definition of' C implies that this is only possible if' 2 +

lf'I = o. A further consequence of' this situation is

S~<1><11.>dE<11.> = S~1 <1><11.>aE<11.>+S~<1><11.>aE<11.>.

LEMM 5. For 411 ( i\) and <1>2 (11.) continuous, we have

<a> S~<1>1dE<11.>+S~<1>2dE<11.> = S~<<1>1+<1>2>dE<11.> (b) <S~<1>1 (i\)dE(i\) HS~<1>2(i\)dE(i\)) = S~<1>1<1>2dE(i\).

PROOF: The statement (a) is obvious. To show (b) we note

first that since <1>1 is continuous on a closed interval

111> 1 (i\) I is bounded by some C. As in Lemma 2, we can find a

function µ(e) defined for e ) o such that if' lx1-x2 1 < µ(e)

1~2 (x 1 )-<1>2 (x2 )1 < e and also such that if' m(n) < µ(e) and

T'b < n, then

72 VII. RESOLUTIONS OF THE IDENTITY

Letting m(n6) - o, we have

J I:n,cp2AE(i\)f-S~<t>2 (i\)dE(i\)f'I ( elfl.

Hence by Lerm:na 3, since 1<1>1 I ( C,

I S~<I>, ( i\)dE( i\) I: n'<l>2AE( i\)f'- (s ~<I>, ( i\)dE( l..) ) s~ <l>2dE( i\)f' I ~ Ce If' I •

Now let <1>2,n, be def'ined as in Lemma 2, 1.e. <1>2,n1 (x) =

~2 (x') if' x 1 S. x ( x • Then Lemma 4 implies a a- - a x <S~<1>1 dE(A) )I: n1<1>2AE(A)r = r~= 1 <1>~<xc'.x>Jx a <1>1 (i\)dE(i\)

ac-1 = S~<1>2,n• <"><1>, (i\ldE< i\) •

The preceding inequality now implies

IS~<1>2 ,n1<1>1 dE(i\)-(J~<!>2dE(i\))(J~<!>1 dE(i\))I ~ Cef'.

However l<1>2,n1 (x)-<!>2(x) I ( e. Thus

rb rb 2 Sb 2 IJa~.n1<1>1dE(i\)f'-Ja<Pi<l>2dE(i\)f'I =I a<l>1<<l>2,n1-<l>2)dE(i\)f'I

= J~l<1>1 1 2 (<1> 2 ,n,-<1>2 l 2d!E(-i\)f'l 2 ~ c2e2 1f'l 2 •

This and the last inequality of' the preceding paragraph imply

I (5~<1>2dE(ll.) HS~<1>1 dE(i\) )r-S~<l>1 <!>2dE(i\)f'I ~ 2Celf J. Since e is arbitrary, we must have f'or every f',

(J~<1>2dE(i\) HJ~<1>1 dE(i\)) = J~<1> 1 <1>2dE(i\), which implies the last statement of' our Lemma..

LEMMA 6. If' <I> (x) is continuous and ) o f'or a (

x ~ b then the zeros of' T = J~<l>(i\)dE(i\) f'orni the range

of' 1-E(b)+E(a).

Let 7?0 denote the range of' 1-E( b) +E( a) • Since every

E(xa)-E(xa- 1 ) in I:n<l>AE(ll.) is orthogonal to 1-E(b)+E(a) it

f'ollows that '10 C 71. '\j is the range of E(b)-E(a) and we

shall show that if' f' + e is in 71~, then Tf + e. Since E(a+o) = E(a) (Cf'. Definition 1 of' this section) and

since (E(b)-E(a) )f' = f' + e, we can f'ind an a 1 such that b )

a 1 ) a and (E(b)-E(a' )f' + e. Since <!>(x) ) o and continuous

f'or b ~ x ~ a', it f'ollows that there is a 6) o, such that

<j>(x) ~ 6 in this interval.

(Tf',f') = J~cp(x)d(E(i\)f',f') ~ J~ 1<j>(x)d(E(i\)f,f') ~ cS((E(b)-E(a'))f,f') = cSJ(E(b)-E(a 1 ))fl 2 ) o.

§3. IMPROPER INTEGRALS 73

Hence Tf' =f e and 7'1Q·'1 = {8l. Since 7b C '1 Corollary 1 to Theorem VI of' Chapter II, §5, will show that 'n0 = 'n.

If' 11>(b) = o, the same sort of' argument will yield that the zeros of' T f'orm the range of' 1-E(b-o)+E(a). If' <1>(c) = o

f'or a c between a and b and ~(x) =F o otherwise, the zeros

of' T f'orm the range of' 1-E(b)+E(c)-E(c-)+E(a). This result is

easily generalized to the case of' n zeros.

We deal in this section with "improper" integrals. There are a number of' possibilities f'or improper integrals. For instance,

we may have that <I> (x) has a singularity at x = a. Then we

def'ine

s~~i\.)dE(i\)f' = lim Sba 211l(i\)E(i\)f' E+O +e

when this limit exists. This will be called an improper integral

of' the f'irst kind. The second kind is def'ined by the equation,

s.:. cp(i\)dE(i\)f' = a+-l.~++.,)~cp(i\)dE(i\)f' when this limit exists. The discussion that f'ollows is analog­

ous in each case. We will theref'ore at times prove our results only f'or the seoond kind and assume them f'or both kinds.

LEMMA 1. If' -= < a < b < =, then

(E(b )-E( a)) S:"..., <!>( i\)dE( i\) C <S:"ao <!>( i\)dE( i\)) (E(b )-E( a))

= s~~< i\)dE(h).

PROOF: If' f' :Is such that S~ao cp(i\)dE(i\)f' is def'ined, we have by Lemma 4 of' the preceding section, that

(E(b )-E(a))S:0 cp( i\)dE(i\ )f' = (E(b )-E(a)) lim S ~ :cp(i\)dE(i\)f'

= lim (E(b)-E(a))s~:<l>(i\)dE(i\)f' = lim S~<l>(i\)dE(i\)f' = s~cp(i\)dE(i\)f' = lim s~:~i\)dE(i\)(E(b)-E(a))f' = S:O<l>(i\)dE(i\)(E(b)-E(a))f'.

This proves the inclusion and the equality is proved by a similar

argument.

LEMMA 2. For a f'iXed f' E fi, the f'ollowing statements are equivalent.

7 4 VII. RESOLUTIONS OF TEE IDENTITY

(a) r.:. cp(l\)dE(l\)f' exists.

(b) r: lcp(l\)l 2d!E(l\)f'l 2 <cc.

(c) For every g,

F(g) = f.:: <!>(l\)d(E(l\)g,f')

exists and F(g) is a linear f'unctional.

We f'irst show that (c) implies (b). By Theorem I of' §3, Chap­ter II, there exists a constant. C such that IF(g) I { Clgl. Let g = S~<I>( I\ )dE( ll.)f'. Lemma 4 o~ the preceding secti-;;n implies that (E(b)-E(a))g = g and thus

F(g) = s.:.<1>(ll.)d(E(l\)f,g) = lim r~:<!>{ll.)d(E(ll.)f',g) = lim <r~:<l>{l\)d(E(ll.)f',g)) = lim (J~:<l>(l\)dE{ll.)f',(E(b)-E(a))g)

= lim ((E(b)-E(a))j~:<l>(l\)dE(l\)f',g) = (J~<!>(l\)dE(l\)f',g) = (g,g) = lg1 2-

Thus fgl 2 ~ C·fgl or lgl ~ C. Lemma 3 of' the preceding sec­tion now implies

S~l<P<"->l 2dlE<"->f'l 2 ~ c2•

Since this is true f'or every a and b, we have ( b) • Thus ( c) implies (b).

Now (b) implies (a). For if' a' <a< b < b 1 we have

IS~:<!>( ll.)dE( l\)f'-f ~cp(ll.)dE( ll.}f' 12

= s: I l<t>l 2dlE( l\)f' 12 +S~ I l<I> I 2d!E( l\)f' 12

by Lemma 3 of' the preceding section. This is easily seen to yield that (b) implies (a).

Finally the continuity of' the inner product insures that (a) implies (c). These three results yield the. equivalence of' the three statements.

'1'.HEORThl II. If' <1>( I\) is real and continuous f'or -cc < I\ < cc, H = f.:cp(l\)dE( I\) is a self'-adjoint tra.ns­f'ormation, whose domain consists of' all f' 1s such that

J~co l<1>1 2dlE(l\)f'l 2 < oo and 1Hf'f 2 = s:. l<Pl 2d!E(l\)f'! 2•

PROOF: Hf' exists if' and only if' f:. l<t>f 2d!E(l\)f'l 2 <ex>

§3. IMPROPER JFrEGRALS 75

since (a) and (b) in Lemma 2 are equivalent. Lemma 3 of the pre­

ceding section implies 1Hf'l 2 = s.:, l<1>1 2d!E(i\)f! 2 •

We next show that H is synnnetric. (Cf. Def. 1 of §3 of Chapter IV) If f € n, 3 of Definition 1 of the preceding section implies that there is an & and b such that lf-(E(b)­

E(a))fl < e. Lemma 4 of the preceding section implies that (E(b)-E(a))f is in the domain of H. Thus the domain of H is

dense. Lemma 3 of the preceding section and the continuity of the inner product imply that (Hf' ,g) = (f ,Hg) for every pair f

and g of the domain of H.

Thus H C H*. Now let g be in the domain of H*, and H*g = g*. Then for every f, ...

(f,(E(b)-E(a))g*) = ((E(b)-E(a))f,g*) = ((E(b)-E(a))f,H*g)

= (H(E(b)-E(a))f,g) = S~ (cj>)d(E(i\)f,g)

by Lemma 1 of this section and Lemma 3 of the preceding section. Thus

F(f) = s~CX>cj>(i\)d(E(i\)f,g) = lim s~cj>(i\)d(E(i\)f,g) = lim (f,(E(b)-E(a))g*) = (f,g*)

is a bounded linear :f'unctional and hence S::, l<1>1 2d!E(i\)gl 2 <co by Lemma 2 above. Hence if g is in the domain of H* it is in the domain of H. This and H C H* imply H = H*.

COROLLARY. If H = S~<l>(i\)dE(i\), where cj>(i\) is real and continuous for a < i\ ~ b, then H is a self-adjoint transformation defined for all f 's such that

S~l<l>l 2d(E(i\)fl 2 <co and 1Hf'l 2 = S~l<1>1 2d!E(i\)f! 2 •

The proof of the corollary is similar to the proof of the

Theorem.

LEMMA 3. Let H = §<l>(i\)dE(i\) be an improper integral. Let :D0 be a subset of the domain of H defined as follows.

If H = S~.,., cj>(i\)dE(i\), then D0 consists of all g 's

for which there exists an ag and a bg such that g =

(E(b)-E(a))g. If H = S~<l>(i\)dE(i\) for a cj>(i\) which is

continuous for a < i\ ~ b, then D0 consists of all g 1 s for which there is an ag) a such that ( 1-E(a' ))g = g.

VII. RESOLUrIONS OF THE IDENTITY

Let H0 be the contraction of H with domain D0 • Then

[H0 l = H.

Since by the preceding theorem, H = H*, H is closed. Hence

[H0 l C H. On the other hand if !f ,Hf I is in the graph of H,

then given e ) o, we can find an. a and a b such that

IHf-H(E(b)-E(a))fl = r;- l<1>1 2dlE(i\)fl 2+.f-oo l<i>l 2dlE(i\)fl 2 < e2/2 and

lf-.(E(b)-E(a))fl 2 ( e2/2

by Lennna. 1 of this section and the preceding Theorem. Now g =

(E(b)-E(a) )f is in the domain of H0• Thus the above inequal1-ties imply

l!f,HfJ-{g,H0gJl 2 ( e2 •

Since E is arbitrary, 1t follows that f is in the domain of

[H0 l. Hence H C [H0 l. The inclusions established in this para­graph show that H = [H0 l .

The proof is similar in the case of an improper integral of' the first kind.

I.ElvNA 4. Let H, = r.:. <I>, (i\)dE(i\), H2 = r.:: <l>2(i\)dE(i\) for <1>1 and <1>2 continuous. Then H1 •H2 is a transforma­tion with domain, those f 's for which f-"'00 I <1>2 I 2dl E( i\)f 12

< CC> and r:. I <I>, <l>2 I 2d IE( i\ )f 12 < CC>. For the contraction given in the preceding Lennna., we have H1 (H2 ) 0 = (H1 ) 0 • (H2 ) 0 = (~) 0 • Finally [H1 ·H2 l = H:3. Similar results hold in the case of an improper integral of the first kind.

If f is in the domain of H1 • H2, then both H2..f and H1 (H2f) exist. "H2f exists" is equivalent to

r_.:, l<1>2 I 2dlE( i\)f 12 < CC> by Lennna. 2 above. On the other hand,

when H2f and H1H2f exist,

H1 (H2f) = 11m j~<1>1 dE(i\)(f~"=<1>2dE(i\)f) = 11m S~ci>1 dE(i\)(E(b)-E(a) >S:O ci>2dE(i\)f

= l:rm <S~<1>,dE< "-> ><S~<1>2 (i\)dE(i\))f = lim S~ ct>, <1>2dE< ">f = S:"= ct>, ci>2dE< "->f

by Lemma 4 and 5 of the preceding section and Lemma. 1 of this lo

§4. COMMUI'ATIVITY AND NORMAL OPERATORS 77

section. Thus f' is in the domain of' ff~ and we must have

J~ l<1>111>2 12d!E(h)f'l 2 < co. On the other hand, ii' the f' 1 s sat­

isfy both conditions, .a similar argument shows that H3f' = H1 •

(H2f') and thus f' is in the domain of' H1 • (H2f'). We may sum

up by stating that the domain of' H1 ·H2 is the intersection of'

the domain of' H2 and H3 •

We note concerning the D0 of' the preceding Lemma, that if'

f' E ~, then by Lemma 1 above Hf'= J:"_ cj>(A)dE(A)(E(b)-E(a)f')

(E(b)-E(a))J.:,<l>(h)dE(A)f' = (E(b)-E(a))Hf'. Thus f' E D0 implies

Hf' E D0 • It f'ollows then that H1 (H2 )0 is precisely (H1 ) 0 (H2 )~ Furthermore Lemma 5 of' the preceding section and Lemma 1 above

yield that if' f' E D0 , then (H1 ) 0 • (H2 ) 0f' = (H3 ) 0f'. Thus

( H, ) o • ( H2 ) o = ( H3 ) o. The f'irst paragraph of' this proof' shows that H1 ·H2 C H3 •.

Thus [H1 ·H2 ] C ~. On the other hand [H1 •H2 ] ::> [ (H1 ) 0 • (H2 ) 0 ]

= [ (H3 ) 0 ] = n3 by the preceding lemma. Hence [H1 ·H2 ] = n3 •

One notes that if' <1>2 is bounded the condition

J~ao I <1>2 I 2d IE( h)f' 12 < co is always satisf'ied. Under such circum­

stances, one has H2 ·H1 C H1 ·H2 = H3 •

LEMMA 5. If' a continuous ~(x) is ) o f'or a < x ~ b

and E(a) = o, E(b) = 1, then H = J~<1>(7'.)dE(h) has an

inverse K = J~<l>(h)- 1 dE(A).

PROOF: In Lemma 4, let H1 = H, H2 = K, H3= J~dE(?I.) = E(b)-E(a) = 1. The remarks immediately preceding the present

Lemma show that BK C KH = 1 • Now Lemma 6 of' the preceding sec­

tion shows that H- 1 exists and KH = 1 shows that H- 1 C K.

BK C 1 shows that K has an inverse since Kf' = e implies

1 • f' = HKf' = He = 9. Since the range of' H- 1 is fi however,

H- 1 has no proper extension and thus H- 1 = K.

§4

DEFINITION 1. If' T is bounded and H is self'-adjoint,

and TH C m' then T is sa:id to commute with H.

LEMMA 1 • If' T coDl!l!Utes with H then

(a) T* commutes with H.

(b) if' H is bounded, T commutes with H, H2 etc.,

78 VII. RESOLurIONS OF THE IDENTITY

and a:ny linear combination of' these. (c) if' H has a:n inverse, T commutes with H- 1 •

PROOF OF (a): Theorem V of' Chapter IV, §2 and its corollary

tell us that HT* = (TH)* J (HT)* J T*H.

PROOF OF (b): SinGe TH= HT when H is bounded, this is

obvious. PROOF OF (c): TH C HT is equivalent to the statement that

f'or every pair {f' ,Hf' I in the graph of' H, {Tf' ,THf' I i.s in the graph of' H. This condition is symmetric f'or H a:nd H- 1 and

thus if' T commutes with H, T commutes with H- 1 •

LEMMA 2. If' T commutes with j~ll.dE(i\), where E(a)

o, E(o) = 1, T commutes with j~cp(i\)dE(i\) where

cp(i\) is continuous f'or a ~ x ~ b. Also T commutes

with E(i\) f'or a~ A.~ b.

PROOF: Let p(x) be a:ny polynomial aux11+an_ 1x11- 1 + ••• +a0, and let p(H) = BuHn+an_ 1Hn- 1+ ••. +a0 • Lermna. 5 of' §2 above and

1 = E(b)-E(a) = j~dE(i\) imply p(H) = j~p(i\)dE(i\). Lennna. 1(b) above shows that T commutes with p(H). If' cp(x) is contin­

uous on the interval a ~ x ~ b, we can f'ind a sequence of' poly­

nomials Pn(x) such that Pn(x) - cp(x) unif'ormly in the in­terval a~ x ~ b. Now j~cp(i\)dE(i\)-pn(H) = j~(cp(ll.)-pn(i\))dE(i\). Hence Lermna. 3 of' §2 above shows that Pn(H)f' - j~cp(i\)dE(A)f' f'or every f' E n. Thus Tj~<I>( i\)dE( i\)f' = lim Tpn (H)f' = J

lim Pn(H)Tf' = j~cp(i\)dE(i\)Tf' f'or every f' in n. Hence T com­mutes with j~cp(i\)dE(i\).

We next show the last statement of' the lermna.. Since T obvi­ously commutes with o = E(a) and 1 = E(b), we need only con­

sider the E(µ) with a ( µ ( b. For e ) o, let ~(x,µ,e) = 1

f'or x ~ µ, cl>(x,µ,e) = 1-(x-µ)/e f'or µ ( x ~ µ+e, cp(x,µ,e) o f'or x) µ+e. Now ~(x,µ,e) is continuous and hence H(µ,e) =

j~cp(i\,µ,e)dE(A) commutes with T. Since j~<l>(i\,µ,e)dE(i\) = E( µ)-E(a) = E(µ) and j~+ecp(A,µ,e)dE(i\) = o we have R(µ,e )­E(µ) = j~+ecj>(i\,µ,e)dE(i\). Then l(H(µ,e)f'-E(µ)f')l 2 = lj~~(ll.,µ,e )dE(A)f'l 2 = s~+e l<1>1 2dlE(i\)f'l 2 ~ S~*dlE(i\)f'l 2 == IE(µ+e)f'l 2-IE(µ)f' 2 • Since E(µ+e)--+ E(µ) as e--+ o, we

must have H(µ,e)f'---+ E(µ)f' f'or every f'. Since. T commutes with every H(µ,e), it is readily seen as in the preceding

§4. COMMUI'ATIVITY AND NORMAL OPERATORS

paragraph to commute with the l:ilJJit E(µ).

DEFINITION 2. A c.a.d.d. operator A will be said to

be normal if A*A = AA*.

LEMMA 3. Let A and B be normal and let A*A = B*B.

79

Let W be such that A = WB as in Theorem VII of §4,

Chapter IV. Then W commutes with A*A. Also the initial

set of W is also the final set of W.

PROOF: WA*A = WAA* = WEB* = AB* = AA"'W = A*NiN when one uses

the results of Theorem VII of §4, Chapter IV.

Now A*', which is defined in Theorem VIII of §4, Chapter IV

has precisely the same zeros as A*. For A*' is a contraction

of A* and hence '1A* I c '1*. On the other hand, if A*f = e, AA*f exists and hence f is in the domain AA* which is of

course the domain of A*'. Thus f € TIA* implies f € '1A* 1 •

This and our previous inclusion imply nA* = '1*· Now under our hypotheses AA* = A*A = B*B = BB*. But AA*f =

e is equivalent to A*'f = e. The latter obviously implies the

former. IT however we have AA*f = e, then f is in the domain

of A*' and o = (AA*f,f) = (A*f,A*f) = (A*'f,A*'f). Thus

'1A* = \* r = ~* = 7BB* = '1B*, = '1:8*. Since by Theorem VI of Chapter IV, §2, [:RA]= '1A*' we have [~] = [~B]. But these sets are respectively the final and initial sets of W.

LEMMA 4. If A and B are self-adjoint and A = WB

where W is partially isometric with initial set [:RB]

and final set [:RA] and W commutes with B, then W = W*.

PROOF: By the corollary to Theorem V of §2, Chapter IV, A*

= B"'W*. Since A and B are self-adjoint, this becomes A =

Brl*. Lemma 1 above shows that W* connnutes with B. Thus W*B

C Br/* = A = WB. Since W*B and WB have the same domain, we

have W*B = WB. This means that W*g = Wg for g in the range

of B. Since W* and W are continuous, we may extend the

equation W*g = _Wg to the closure of the range of B.

Now A = WB, A*A = B...(*WB = B*EB = B*B, where E is the

projection on [:RB] the init:!,al set of W (Cf. Lemma 1 of §.3~

80 VII. RESOLtJrIONS OF THE IDENTITY

Chapter VI). Since A and B are selr-adjoint they are normal. Thus the initial set or w is also the rinal set or w by Lemma 3 above. Lemmas 1 and 3 or §3, Chapter VI show that this set is also the initial set. or W*. Thus W and W* are zero on the orthogonal complement or the initial set [!RB] or W.

Thus W = W* on [~Bl and on [~Bl • Since W and W* are add1tive, we must have W = W*.

CHAPI'ER VIII

BOUNDED-SELF-ADJOINT AND UNITARY TRANSFORMATIONS

A self-adjoint operator H is said to have an integral rep­resentation if

H = s:: AdE( A)

for a resolution of the identity E(h). (Cf. Chapter VII, §3). A unitary U is said to have an integral representation, if U = S~neiµdE(µ) for a finite resolution of the identity. (Cf. Chapter VII, §2, Def. 1).

It is a f'undamental result in the theory of linear transfor­mations in Hilbert space, that every self-adjoint trans.f'ormation has an integral representation. Every unitary transformation has likewise. However both of these are normal and these results can even be generalized to the statement that every normal operator has an integral representation.

These general results will be established in Chapter IX. In the present Chapter, we will obtain the integral representation for bounded self-adjoint transformations and for unitary trans­formations.

§1

In this section, we will suppose that H is self-adjoint and bounded with lllHlll = C. If p(x) = ~r1+Bn__ 1 r1- 1 + ••• + a 0 is a polynomial, we let p(H) = Bn_Hn+Bn__ 1H - 1+ ••• +a0 • We consider only the case, when the a 's are real, and thus p(H) is self­adjoint. (Cf. Theorem V of §2, Chapter IV. Also Chapter IV, §3,

Lemma 4).

!EMMA 1. If for -C ~ x ~ C, p(x) ~ o then p(H) is definite.

Let '11 denote a:ny finite dimensional manifold and E the projection on m. Let H1 = EHE. By Theorem V of Chapter IV,

§2, H'* = (EHE)* = E*H*E* = EHE. Hence H' is self-adjoint.

81 ""

82 VIII. BOUNDED SELF-AD.JOINT TRANSFORMATIONS

Furthermore IH'fl = IEHEfl ~ IHEfl ~ C· IEfl ~ C· lfl. Thus H' has a bound { C.

Let p'(H 1) = 8nH'n+8n_1H1n-l+ •.• +a0E. Lemma 5 of §1,

Chapter VII shows that if f is E 71?, and p(x) ~ o for -C

~ x ~ C, then

o~ (p 1 (H')f,f).

If f is an arbitrary element of ~ let 7n be the manifold 2 (n) determined by f, Hf, H f, . . . , H f. 7n is finite dimen-

sional. Furthermore f = Ef, Hf = EHf = EHEf = H' f, H2f =

EII2f = EH·Hf = EII·E·Hf = H' ·H'f. A similar argument will show :akf = H'kf for 2 ~ k ~ n. Thus p(H)f = p' (H' )f. Thus for every f, (p(H)f,f) = (p'(H')f,f) ~ o. This proves the result •.

Lll:MMA 2. If for -C ~ x ~ C, lp(x) I ~ E, then Ill p(H) Ill ~ E.

PROOF: Under these circumstances, Lemma 1 states that p(H)+E

is definite and E-p(H) is definite. Thus o ~ ((p(H)+E)f,f) or -E(f,f) ~ (p(H)f,f). Hence C_ for· p(H) is ~ -E. (Cf.

Chapter IV, §3, Definition 4). Similarly c"" ~ E. Lemma 10 of Chapter IV, §3 now implies that lllp(H)m ~ E.

Lll:MMA 3. If P(x) is a continuous function on the interval -C ~ x ~ C, then there exists a unique bounded operator, P(H) such th3.t for every E) o and every polynomial p(x) such that IP(x)-p(x)I < E for -C ~ x ~ C then m P(H)-p(H) DI < E.

PROOF: The existence of one such P(H) can be established as

follows. Let E1, ~, ••• , En) o be a sequence of positive numbers such that En -> o as n -----.. co. Since P(x) is con­tinuous, we can find for each E~ a polynomial Pn (x) such

that I P(x)-pn(x) I < En for -C ~ x ~ C. This implies that for

-C ~ x ~ C, lpn(x)-pm(x) I ~ IPn(x)-P(x) l+IP(x)-pm(x) I < En+~. Lemme. 2 above shows that llpn(H)-pm(H) II ~ En+~. Hence for

every f E ~' lPn(H)f-pm(H)fl ~ (en+~)·lfl and the sequence Pn(H)f is convergent to some f*. Let P(H)f = f*. The sequence Pn(H) is easily shown to be uniformly bounded by Lemma 1 above. Hence P(H) is bounded.

§ 1 . FUNCTIONS OF A BOUNDED H

We prove that P(H) has the property given in the Lemma. For suppose IP(x)-p(x) I ( E for -C~ x ~C. Since P and p are continuous, we can find a 6) o such that I (P(x)-p(x)) I (

E-6. Let n 0 be such that for n ) n 0 , En ( 6. Then if n ) n 0

we have IPn(x)-P(x)I (En ( 6 for -C·~ x ~ C. (See preceding paragraph). Since we also have IP(x)-p(x)I ~ e-6, it follows

that IP:r,{x)-p(x) I ( e. Thus for every f, I (pn (H)-p(H))f" I (

e If" I • This holds f"or every n ) n 0 • Letting n -+co we get I P(H)f"-p(H)f" I ( EI f" I. This implies Ill P(H)-p(H) Ill ( e.

P(H) is unique. For suppose we have two distinct operators

P1 and P2 with the given properties. Since given e ) o, we can f"ind a p(x) with I P(x)-p(x)I ( e we must have II Pi (H)­

p(H)ll ( e f"or i = 1, 2. Hence MP;(H)-P2 (H)DI ( 2e. Since

e is arbitrary, this implies m p, - p 2• = 0.

LEMMA 4. If" P(x) and P(H) are as in Lemma 3 above

then (a): If" P(x) = p 0 (x) is a polynomial, then P(H)

=p0 (H). (b): If" P(x) ~ o for -C ~ x ~ C, then P(H)

is def"inite. ( c): If" I P(x) I ~ k for -C ~ x ~ C, then I P(H) DI ~ k.

(a) is a consequence of" the uniqueness of" P(H).

We next show (b). Let Eo: and Po:(x) be as in the t'irst paragraph of" the proof' of' Lemma 3 above. Let qo:(x) = po:(x)+eo:· Then qo:(x) = Po:(x)+eo: ~ P(x) ~ o. Hence qo:(H) is def"inite by Lemma 1 above. For every f", lim qo:(H)f' = lim Pa(H)f'+ lim eo:f" = P(H)f". Hence P(H) is the limit of" def"inite operators and can be shown to be symmetric. self"-adjoint and def"inite.

(c) is proved.in a manner analogous to the proof' of" Lemma 2

above.

LEMMA 5. Let P(x), Q(x) and R(x) be ·continnous functions and let P(H), Q(H) and R(H) denote the cor­

responding operators in Lemma 3. Then (a): If" R(x) P(x)+Q(x) f"or -C ~ x ~ C, then R(H) = P(H)+Q(H): (b): If" R(x) = P(x )"Q('x) then R(H) = P(H) •Q(H).

PROOF OF (a): Given e) o, we can f"ind polynomials p(x)

and q(x) such that IP(x)-p(x)I ( e/2, IQ(x)-q(x)j ( e/2 f"or

-c ~ x ~ C. Then IR(x)-(p(x)+q(x))I = (P(x)-p(x)+Q(x)-,(x)f (e

84 VIII. BOUNDED SELF-ADJOINT TRANSFORMATIONS

on the same interval. LeilllIIB. 3 now implies that HI R(H)-(p(H)+

q(H))RI < e, DI P(H)-p(H) ID < e/2 and II Q.(H)-q(H) m < E/2.

These inequalities imply lllR(H)-(P(H)+Q.(H) )I< 2e. Since e is

arbitrary we must have I R(H)-(P(H)+Q.(H)) I = o, R(H) = P(H)+

Q.(H).

PROOF OF( (b): Let k be such that I P(x) I+ 1 < k, IQ.(x) I < k

f'or -C ~ x ~ 1 • Let e be such that o < e ~ 1 • We can f'ind

polynomials p(x) and q(x) such that klP(x)-p(x) I ( e and

klQ.(x)-q(x) I < E f'or -C ~ x ~ C. Since k ) 1, we have

IP(x)-p(x) I < efk < e which implies jp(x) I < IP(x) l+e ~ IP(x) I +1 < k, f'or the given x -interval. LeilllIIB. 4 above then implies

that II p(H) HI < k. LeilllIIB. 4 and the other inequalities imply

m Q.(H) II < k, k 81 P(H)-p(H) II < E and k I Q.(H)-q(H) I < e. Now

f'or -C ~ x ~ C, IR(x)-p(x)q(x) I = IP(x)Q.(x)-p(x)q(x)I = IP(x) •

Q.(x)-p(x)Q.(x)+p(x)Q.(x)-p(x)q(x) I ~ IQ.(x) I· IP(x)-p(x) l+lp(x) I·

IQ.(x)-q(x)I < k!P(x)-p(x)l+k!Q.(x)-q(x)I < 2e. Thus LeilllIIB. 3

implies that RI R(H)-p(H)q(H) ~ < 2e. We also have II P(H)Q.(H)­

p(H)q(H) 11 = HI P(H)Q.(H)-p(H)Q.(H)+p(H)Q.(H)-p(H)q(H) ID ~ Bl ( P(H)­

p(H) )Q.(H) HI+ Hlp(H)(Q.(H)-q(H)) II ~ m P(H)-p(H) Bl• lllQ.(H) 11 +

HI p(H) Bl• Ill Q.(H)-q(H) Ill < k II P(H)-p(H) HI +k DI Q.(H)-q(H) Ill < 2e.

This and the preceding result imply I R(H)-P(H)Q.(H) 11 < 4e.

Since e is arbitrary, we must have R(H) = P(H)Q.(H).

If' A is c.a.d.d., we let ~A denote the zeros of' A. If'

~A is the range of' A and A is self'-adjoint then [!RA] = ~A. (Cf'. Lemma 8 of' §3, Chapter IV). For def'inite operators, we also

have the result.

I..EM"1A 6. If' A and B are self'-adjoint b01mded and

def'in.ite, then ~A+B C ~A and consequently [~A] C [~A+B].

PROOF: Let f' be such that (A+B)f' = e. Then CF( (A+B).t',f')

= (Af',f')+(Bf',f'). Since (Af',f') ~ o and (Bf',f') ~ o, this

implies (Af',f') = o.

For every f' and g, we also have the result . I (Af' ,g) 12 ~ (Af',f')(Ag,g). Inasmuch as A is def'inite, we have f'or every

real l\ and every complex z with I z I = 1 ,

(A(f'+l\Zg),f'+l\Zg) ~ o.

Expanding and using the S1Jll!lletry of' A, we get

§2. (Hf',g)

(Af',f)+27\R(z(Af',g))+h2(Ag,g) ~ o.

We may take z so that R(z(Af' ,g)) = I (Af' ,g) 1. Hence

(Af',f)+2hl(Af',g)l+h2(Ag,g) ~ o.

If' (Ag, g) = o, this implies I (Af', g) I = o and our desired

inequality holds in this case. If (Ag,g) t o, let h = -l(Af',g)V(Ag,g) and the above inequality becomes

(Af' ,n- I~~;~~ 12 ~ o.

Hence (Af',f)(Ag,g) ~ l(Af',g)l2.

85

From the preceding two paragraphs, we see that if (A+B )f=9,

(Af' ,f) = o, and hence (Af' ,g) = o for every g E 8. Hence

Af' = 9. Hence f E ~+B implies f E 7lA or 7lA+B C 7lA. The orthogonality relations mentioned before the Lemma, yield

[~A+B]) [~A].

§2

We now consider two f'unctions Pµ(x) =max (x-µ,o), Qµ(x) =

max (-(x-µ),o). Pµ(x) and Qµ(x) are continuous, non-negative,

Qµ(x)Pµ(x) = Pµ(x)Qµ(x) = o and Pµ(x)-Qµ(x) = x-µ. Applying

Lemmas 3, 4 and 5 of the preceding section, we obtain:

LEHnA 1. Let Pµ(x), Qµ(x) be as above and let

Pµ(H), Qµ(H) be the corresponding operators as in Lemma

3. Then Pµ(H) and Qµ(H) are definite, Qµ(H)Pµ(H) =

Pµ(H)Qµ(H) = o and Pµ(H)-Qµ(H) = H-µ.

LEMMA 2. Let 71( µ) = 7lp (H) and E( µ) be the pro­jection on 71(µ). Then (a~: If' µ1 ) µ2, E(µ1 ) ~ E(µ 2 ).

(b): lime+OE(µ+e2 ) = E(µ). (c): E(O) = 1, E(-C-e) = o, for e) o.

PROOF: If' µ1 ) µ2, Pµ2(x)-Pµ1 (x) ~ o and Pµ1 (x) ~ o.

Lemmas 4 and 5 of the preceding section imply that Pµ2(H)-Pµ1 (H)

and Pµ1 (H) are definite. Lemma. 5 of the preceding section

implies that Pµ2 (H) = (Pµ2 (H)-Pµ1 (H))+Pµ1 (H). Lemma 6 of that

section now implies that

7lp J 7lp or E(µ1 ) ~ E(µ2 ). l-'1 f.12

86 VIII. BOUNDED SELF-ADJOINT TRANSFORMATIONS

We next prove (b). We f'irst show that f'or every f' E n, 11me~opµ+e2 (H)f' = Pµ(H)f'. One notes that o ~ Pµ(x)-Pµ+e 2(x) ~ e 2 • Lemma 4 of' the preceding section implies that Ill P µ (H )-

p 2 (H)lll. {. e 2 • From this, we may easily inf'er that li.m +O µ+e - e P 2 (H)f' = P (H)f' f'or every f'.

µ+e µ 2 We also need that lime+oE(µ+e ) exists a.nd is the projection

on TTe7l(µ+e 2 ). If' e1 ) e2 ) • • • is a sequence with l~ .. 00 en = o, then Lemma 1 o of' §1, Chapter VI implies that l~ ... 00

E(µ+~) exists a.nd is a projection E with range nn7l(µ+e~). If' e is such that e~+ 1 ~ e2 ~ ~, then f'rom the above, we know that E(µ+e~+ 1 ) ~ E(µ+e 2 ) ~ E(µ+e~). From Lemma 6 of' · Chapter VI, §1, we know that if' µ1 ) µ2 , then E(µ 1 )-E(µ 2 ) is

a projection a.nd our preceding result permits us to state that

E(µ+e~+1 )-E ~ E(µ+e 2 )-E ~ E(µ+e~)-E. Lemma 8 of' §1, Chapter VI

now implies that f'or every f', IE(µ+~+ 1 )f'-Ef'I ~ IE(µ+e 2 )f'-Ef'I

~ IE(µ+e~)f'-Ef'I. It f'ollows that lime~oE(µ+e2 ) exists and is

E. Sin~e 7l(µ+e~+ 1 ) ~ 7l(µ+e 2 ) C 7l(µ+e~), we also have

nE7l(µ+e ) = nn'1(µ+en) • Thus lim .. 0E( µ+e 2 ) = E( µ) if' ne'1(µ+e 2 ) = '1(µ). Since

71(µ) C 7l(µ+e~), we must have ne'1(µ+e 2 ) ::> '1(µ). But if' f' €

nE'1(µ+e 2 ) then f' E 7l(µ+e 2 ) a.nd Pµ+e2(H)f' = 9. Letting e-+ O,

we obtain by a result given above that Pµ(H)f' = e or f' E '1(µ).

Thus f' E nE7l(µ+e 2 ) implies f' E 7l( µ). This a.nd 7l( µ) c ne7l(µ+e 2 ) imply 71(µ) = nE7l(µ+e 2 ) and E(µ) = 11me ... OE(µ+e 2 ).

Pc(x) = o f'or -c ~ x ~ c. Hence Pc(H)f' = e f'or every

f' € n and E(C) = 1. If' E ) o, then P-C-e (x)""e ~ 0 f'or -C

~ x ~ C a.nd hence ((P_c-e(H)-e)f',f') ~ o, by Lemma 4 of' the

preceding section. Thus f'or every f', (P_c-e(H)f',f') ~ e(f',f')

and P-C-e(H)f' = 9 implies f' = 9. Thus '1(-C-eJ = !9! and E(-C-e) = o.

Def'inition 1 of' §2, Chapter VII now inf'orms us that the

E(µ) 's f'orm a resolution of' the identity.

COROLIARY. The E( µ) 1 s of' Lemma 2 f'orm a resolution of'

the identity.

LEMMA 3. The E( µ) 1 s of' Lemma 2 above commute with

H. Consequently if' µ2 ( µ1, H(E(µ 1 )-E(µ2 )) = (E(µ 1 )­

E(µ2))H(E(µ1)-E(µ2)).

§2. (Hf,g) 87

PROOF: Lemma. 5 of' the preceding section implies that HPµ(H)

and Pµ(H)H both correspond to X·Pµ(x) = Pµ(x)·x. Hence

HPµ(H} = Pµ(H)H and if' Pµ(H)f' = e, e = HPµ(H)f' = Pµ(H)Hf'. Thus f' € 71( µ) implies Hf' € '1( µ). It f'ollows that HE(µ) =

E(µ)HE(µ). Taking adjoints yields E(µ)H = E(µ)HE(µ) = HE(µ).

Thus H(E(µ1 )-E(µ 2 )) = HE(µ1 )-HE(µ 2 ) = E(µ1 )H-E(µ 2 )H = (E(µ 1 )­

E(µ2))H. Multiplying this equation on the right by E(µ 1 )-E(µ 2 )

will give the desired result.

It is convenient to denote the interval µ2 <x ~ µ1 by I and

then let E(I) = E(µ1 )-E(µ2 ).

LEMlVIA 4. If' f' € 71(µ), (Hf',f) ~ µ(f,f). If' f' € ~(µ)•,

(Hf',f') ~ µ(f',f).

PROOF: If f €. n(µ), Pµ(H)f = e. Since (Pµ(H)-Qµ(H))f =

Hf'-µf, we must have µf-Hf' = Qµ(H)f'. But Lemma 1 above also

tells us that Qµ(H)f' is definite. Hence (µf-Hf',f') = (Qµ(H)f,f) ~ o. Thus µ(f',f) ~ (Hf',f').

If' f € '1(µ)', f € [~Pµ(H)] by Lemma 8 of §3, Chapter IV.

Lemma 1 above states that Qµ(H)Pµ(H) = o. Thus '1Q (H) ::> [~Pµ(H)] and Qµ(H)f = e. Since ( Pµ(H)-Qµ(H))f = :ff¥-µf, we have Pµ(H)f = Hf'-µf. Lemma 1 above states that Pµ(H) IB-

definite and thus (Hf'-µf',f') = (Pµ(H)f,f') ~ o. Hence (Hf',f)

~ µ(f,f').

LEMIM. 5. For every f' € ~ and e ) o, we have

(Hf' ,f') = S ~C-e M( E( i\)f' ,f').

PROOF: Divide the interval -C-e ~ i\ ~ C into subintervals

by the points µ0 = -C-e, <~ ( ••• ( µn_ 1 ( µn = Cn. Let Icx

denote the interval µcx- 1 ~ x ~ µex and. E( Icx) = E( µcx)-E( µcx_ 1 ) • Lemma. 6 of §1, Chapter VI, tells us that the range of' E(Icx)

is 71( µcx)7l( µcx_ 1 )' • Lemma 4 above now implies that

µa-- 1 (E(Icx)f,f) = µcx- 1 (E(Icx)f,E(Icx)f) ~ (HE(Icx)f',E(Icx)f)

~ µcx(E( Ioc)f' ,E( Ioc)f) = µoc(E( Ioc)f' ,f).

We also have r~ = 1E(Ioc) = E(C)-E(-C-e) = 1-0, by Lemma 2

above. Lemma. 3 above then shows that

(Hf',.f) = (H( r~= 1E(Icx))f,f) = I~= 1 (HE(Icx)f,f)

88 VIII. BOUNDED SELF-ANOINTS AND TRANSFORMATIONS

~ 1 (E(I )HE(I )f',f') = f!: _ 1 (HE(I 'f',E(I )f'). a= <X <X <X - ot <X

The previous inequality on (HE(Ia)f',E(Ioc)f') now yields that

I:~= 1µa-1(E(Ia)f',f') ~ (H:f',f') ~ ~= 1µa(E(Ia)f',f').

Lemma. 8 of' §1, Chapter VI, shows that (E(µ)f',f') = IE(µ)f'l 2

is a monotonically increasing :function of µ. Thus the right and left hand expressions in the preceding inequalities can be ma.de to approach the desired integral simultaneously and our Lemma. is proven.

§3

If we let H1 = S~c-ei\dE(i\) by using Def'inition 3 of §2, Chapter VII, then LeDl!llS. 3 of §2, Chapter VII implies that

(H1f,f) = S~c-ei\d(E(i\)f,f). Lemma. 5 of the preceding section states that (H:f',f) = S~c-ei\d(E(i\)f ,f') and thus f'or every f € n, (H1f,f') = (H:f',f) or ((H1-H)f,f) = o. H1-H is sym­

metric with domain n and hence self-adjoint. From Definition

4 of §3, Chapter IV we see that C+ = o, C~ = o, f'or H1-H and Lemma lo which follows this definition ·yields 11 H1-H I = o or H = H1• We have proved:

THEOIIBM I. If' H is self-adjoint with bound C and

e) o, then there exists a finite resolution of' the identity E( i\) such that

H = S~c-ei\dE(i\).

We can refine our result somewhat by using the considerations . of §2, Chapter VII, which f'ollow Lemma. 4. If' C+ for H is

< C, we take a 6 so small that C++o < C. In the discussion referred to, we let "-, = C++o, "'2 = C. We then obtain that

S~c-ei\dE(i\) = s~~~:i\dE(i\) for every 6 > 0 an.de ~~C)-E(C++~)=O. S~ce E( C) = 1 , we have E( C + +o) = 1 • Now IS C + i\dE( i\)f' 12 =

sc:+ 6i\2dlE(i\)fl 2 ~ k 2 1(E(C++6)-E(C+))fl 2 where ~ = ma.x CIC+I• IC++61 ), when we use Lemma 3 of §2, Chapter VII. Since E( C + + o)f --+ E( C + }f when we let 6--+ o, 6 ) o, by Lemma. 2

of the preceding section, we must have that the lim o•o,o)o sg++cSi\dE(i\)f exists and = e. This and our preceding result

mPlY S~c-ei\dE(i\) = S~c-ei\dE(i\). We also have E(C+) = E(C++o) -1. Of course if C+ = C these results are still true.

§4. INTEGRAL REPRESENTATION OF A UNITARY OPERATOR 89

If we consider the lower limit of integration, we have to

consider that lim6~0 , 6)0Jg=_6hdE(h) = C_(E(C_)-E(C_-o)~ But the discussion which follows Leimna. 4 of §2, Chapter VII shows also that E(C_-6)-E(-C-e)=O for C_-6)-C-e and 6)0.

Since E(-C-e) = o, E(C -6) = o for 6) o and hence

E(C_-o) = o. Thus 11m6=o,6)oS~=-6hdE(h) = C_E(C_). A proof

similar to that of the preceding paragraph will now show that

S~c-ehdE(h) = sg~hdE(h)+C_E(C_).

COROLLARY. If C+ and C_ refer to nition 4 of §3, Chapter IV, then

H = S8~hdE(h)+C_E(C_) E(C_ -o) = o.

§4

H as in Defi-

In this section, we will deal: with Unitary Transformations.

(Cf. Definition 1 of §2, Chapter VI). From Lemma 1 of §2, Chap­

ter VI, we know that UU* = U*U = 1 • Let A = ~(U+U*), B = ~i(U*-U). Then 1B = ~(U-U*) and U = A+1B. Theorem V of' §2 of Chapter IV implies that A and B are self-adjoint an<!__A-

1B = U *. We also have that AB = ii(U+U*) (U*-U.) = ii(U*2-u2) = -k1(U*-U)(U+U*) = BA. Thus 1 = (A+iB)(A-1B) = A2 + B2 •

We shall obtain an integral representation f'or U. by using the integral representation f'or A given by Theorem I of' the

preceding section. It will however be convenient to assume that Uf = f or Uf' = -f' implies f' = e. We shall see that this assumption can be ma.de without an essential loss of generality.

let 7Jl1 denote the Set Of' f' IS f'Or WhiCh Uf=f' 1 and 7Jl_ 1

denote the set of f' Is f'or which Uf' = -f'. m, and m_, are

mutually orthogonal since if' f' € m, and g € m 2, ( f, g) (Uf,Ug) = (f',-g) = -(f',g) or 2(f,g) = o. If E1 and E_ 1 are the projections on 7111 and 711_ 1 then E1E_ 1 = o and

E1 +E_ 1 is the projection with range U( 711 1 u 711_ 1 ) by Lemma 5 of' §1, Chapter VII. Let F = 1-(E1+E_ 1 ).

By the def'inition of m1, we have UE1 = E1• Since U*= u-! f' € 7n 1 implies U*f' = f' and hence U*E1= E1 • Taking adjoints

we obtain E1U = E1= UE1 • Similarly E_ 1u = -E_ 1= UE_ 1 • These results also imply that FU= UF. Multiplying on the lef't by F yields FU = FUF.

90 VIII. BOUNDED SELF-AD.JOINT TRANSFORMATIONS

Thus U = U(E1+E_ 1+F) = UE1+UE_ 1+UF = E1-E_ 1+FUF.

If' we contract FUF to the range of' F, we see that f'or every f' and g in this range, (FUFf' ,FUFg) = (UFf' ,UFg) = (Uf' ,Ug) = ( f', g) • Thus FUF is isometric on the range of' F.

Furthermore the range of' FUF on the range of' F is also the range of' F. Otherwise there would be a g + e in the range of' F such that f'or every f' E ~. o = (g,FUF(Ff')) = (g,FUFf') =

( g,FUf') = ( Fg,Uf') = ( g,Uf') • Thus g is orthogonal to the range of' U and since U is unitary, g = e. This is a contradiction. Thus we have proved that the range of' FUF on the range of' F is the range of' F.

Def'inition 1 of' §2, Chapter VI, is now satisf'ied by FUF re­garded as a transf'ormation on the range of' F i.e., in this sense FUF is unitary. For g in the range of' F, FUFg = g is equivalent to UFg = g or Ug = g. Thus g E m1• Since g is also in the range of' F, g = e. Hence FUFg = g, g in the range of' F, implies g = 9. Similarly FUFg = -g, g in the range of' F, implies g = 9.

Thus we have proved:

LEMM 1 • Let U, m1 , m_ 1 , E1 , E_ 1 , and F be def'ined as above. Then UE1 = E1 = E1U, E_ 1u = -E_ 1 = UE_ 1, FUF=UF'=FU. The ranges of' E1, - E_ 1 and F are ortho­gonal. We also have U = E1-E_1 +FUF. Let U, denote FUF considered as a transf'ormation on the range of' F. Then u, is un1tary and u,g = g or u,g = -g implies g = e.

In applying Def'inition 1 of' §2, Chapter VI, we have assumed that the range of' F is inf'inite dimensional i.e., a Hilbert space. (Cf'. the corollary to Theorem II of' §2, Chapter III). The cases in which the range of' F is f'inite can be easily taken care of' by elementary methods. We consider then only the case in which the range of' F is inf'inite dimensional.

For a time we will consider only those U 's f'or which Uf' = f' or Uf' = -f' implies f' = e. Le,t A and B be as in the f'irst

para.graph of' this section. Since A = ~( U+U*), I All ~ ~(Ill I + I U* I ). Thus Theorem I of' the preceding section shows that there

§4. INTEGRAL REPRESENTATION OF A UNITARY OPERATOR 91

is a resolution of' the identity E( i\) such that f'or e ) o, we

have

A= S~1-ei\dE(i\). As a proof' of' the corollary of' that theorem shows, we may write

this

A= S~ 1 ME(i\)-E(-1 ).

But if' f' is in the range of' E(-1 ), Af' = -f'. Since A2+

B2 = 1, we have (Af',Af') = (f',f') = ((A2+B2 )f',f') = (A2f',f')+

(B2f',f') = (Af',Af')+(Bf',Bf'). Hence !Bf'! 2 = (Bf',Bf') = o and Bf'=

9. Thus Uf' = (A+iB)f' = Af'+1Bf' = Af' = -f'. But this implies

f' = 9. Thus if' f' is in the range of' E( -1 ) , f' = 9 and

E( -1 ) = o. The expression f'or A also shows that if' f' is in the range

of' E(1 )-E(-1-0), Af' = f'. An argument similar to that of' the

preceding paragraph now implies f' = 9. Thus E( 1 )-E( -1 -o) = o.

But E( 1) = 1 and hence E( 1-0) = l. We have shown:

LEMMA 2. If' U is unitary and such that Uf' = f' or

Uf' = -f' implies f' = e·, then if' A = ~(U+U*) there is a resolution of' the identity E(i\) such that

A= S~ 1 i\dE(i\), and E(-1) = o, E(l-0) = 1.

From Lemma 5 of' §2, Chapter VII, we have that B2 = 1-A2 = S~ 1 (1-i\2 )dE(A). Let C = S~ 1 (1-i\2 ) 1 / 2dE(A); Lemma 5 of' §2,

Chapter VII implies c2 = S~ 1 ( 1- A2 )dE( i\) = B2 • C and B are self'-adjoint and thus we may apply Theorem I of' §4, Chapter VI to

obtain that there is a partially isometric W such that B = WC.

For C = S~ 1 (1-A2 ) 1 /2dE(A), the discussion which f'ollows

Lemma 6 of' §2, Chapter VII shows that the zeros of' C f'orm the

range of' 1-E(l-O)+E(-1) = O. Thus ~c= (9l and C-l exists.

A discussion similar to the proof' of' Lemma 5 of' §3, Chapter VII

will show that c- 1 = S~ 1 (1-i\2 f1 1 2dE(i\). The corollary to

Theorem II of' §3, Chapter VII now shows that the domain of' c- 1

is dense.

Lemma 5 of' §2, Chapter VII shows that AC= S~ 1 A(1-i\2 ) 1 /2dE(A) = CA. Thus A commutes with C. (Cf'. Def'inition 1 of' §4, Chap­

ter VII). LeillDS. 1 ( c) of' §4, Chapter VII implies that A commutes

92 VIII. BOUNDED SELF-ADJOINT TRANSFORMATIONS

with c- 1 • In the first paragraph of this section it was shown that AB = BA.. Thus ABC- 1 = BA.C- 1 C BC- 1 A. Since WC = B,

BC- 1 f = Wf for f in the domain of C- 1 • Thus for f in the domain of C 1 , !Wf = WM. Since W and A are continuous and the domain of c- 1 is dense, we must have NII =WA.

Since W and A connnute, Lemma 2 of §4, Chapter VII implies that W connnutes with E( i\) for -1 ~ i\ ~ 1. Since W com­mutes with E(i\) for -1 ~ i\ ~ 1, it is readily seen to com­mute with any rn1cj>AE( i\) as defined in Definition 2 of §2,

Chapter VII. W will then connnute with any limit of these,

S~1cj>(i\)d:E(i\). In particular w will connnute with c = S~1(1-i\2)1/2dE(i\). W will also connnute with B =WC.

Thus we have proved:

LEMMA 3. Let A, B, C and E(i\) be as above. There exi·sts a partially isometric W with initial set

[~0 ] and final set [~BJ such that B =we. w connnutes with A, B, c and E(i\). C~cl = ~.

With regard to the last statement, we recall that we have shown that ~C which is the domain of c- 1 is dense.

When we apply Lemma 4 of §4, Chapter VII we obtain:

COROLIARY 1 • W = W*.

COROLIARY 2. If T is bounded and connnutes with E(i\)

for a resolution of the identity, T connnutes with s~cj>(i\)dE(i\) for every cj>(i\) for which this integral exists.

LEMv1A 4. Let W be partially isometric, W = W* and 1 1 let E = W--W. Then F1 = 2 (E-!W) and F2 = 2 (E-W) are

projections such that F 1+F2 = E, F1-F2 = W.

F1 and F2 are self-adjoint, since E and W are. Since

W = W*, E = W-llW = W2 and EW = w3 = WE = W since E is the initial set of W. (Cf. Lemma 1 of §3, Chapter VI). Thus F2 =

t(E-iW)(E-!W) = *(E2-IWE+EW-!W2 ) = t(2E+IM) = ~(EtW) = F. Similarly F~ = F 2 • Lemma 2 of § 1 , Chapter VI now implies that F 1 and F 2 are projections.

§4. INTEGRAL REPRESENTATION OF A UNITARY OPERATOR 9:5

For W in Lermna. 3 above, E is the projection on [!Re] = n and thus E = 1. Since W and 1 commute with E(i\) f'or -1

~ i\ ~ 1, F1 and F2 also commute with E(i\) f'or -1 ~ i\ ~ 1. Now

U= A+1B = A+iWC = (F1+F2 )A+i(F1-F2 )C = F1 (A+iC)+F2(A-iC)

= F1s~1<"-+i(1-11.2) 1 /2 )dE( i\)+F 2s~1(i\-i(1- ,.,2) 1 /2 )dE( i\)

= S~1(i\+i(1-11.2), /2 )dF1E< i\)+S~, (i\-i(, _ ,.,2), /2 )dF 2E( i\).

For <I> between o and n let us def'ine F(<j>) = F1-F1E(coscj>

-o). Since F 1 and the E( i\) commute F 1 E( cos <I> -o) is a

projection. (Cf'. Lemma 4 of' § 1, Chapter VI). Since F 1 E( cos <I> -o) ~ F 1, Lermna. 6 of' the same section shows that F(cj>) is a projec­

tion. One easily verif'ies that if' <1>1 < <1>2 , F(<1>1 ) ~ F(<!> 2 ) because of' the analogous property of' the E(i\). One also has

2 11.me•OF(<j>+e ) = F(<j>), F(O) = F 1-F1E(1-0) = F1-F1= o, F(n) = F 1-F1E(-1-o) = F 1 •

It remains to show that S~ei<l>dF(<I>) = S~1<"-+i(1-11.2 ) 1 1 2 )dF,E(i\~ There is at most a denumerable set of' points at which E( i\) + E(i\-o). If' we avoid these points in f'orming partitions and use

i\= cos <j>, the partial sums f'or each integral are the same and

thus the limits are equa1 or s~ei<1>dF<<1>l = s~,<i\+i<1-11.2 ) 1 / 2 lcilitE<11.l For <1> between n and 2n, we let F(<j>) = F1+F2E(cos <1>-+-­

The discussion of' the preceding paragraphs can be extended and

we see that F(<I>) is a projection, F(<1>1 ) ~ F(<j>2 ) if' <1> 1 < <1>2 ,

11me-+OF(<j>+e2 ) = F(<j>), F(n) = F1, ( F(n) is def'ined in two ways)

F(2n) = F1+F2 = 1 and

s~, (i\-i( 1-i\2) 1/2)dF2E(i\) = s~, (i\-1 ( 1-i\2) 1 /2)d(F,+F2E(i\))

= s!nei<l>dF( <I>).

If' we combine our results, we obtain,

u = S~ei<l>dF< <1> l.

We may sum up as f'ollows:

LllMN\ 5. Let U be a unitary transf'orma.tion such that Uf' = f' or Uf' = -f' implies f' = e. Then there exists a

resolution of' the identity F(cj>), with F(o) = o, F(2n) = 1,

such that

94 VIII. BOUNDED SELF-ADJOINT TRANSFORMATIONS

To remove the restrictions on U, we recall the situation described in Le:mma. 1 above. We apply Lemma. 5 to U1 = FWi' con­sidered on the range of F and obtain a family of projections F(cj>) of the range of F. Now either Definition 1 or Lemma. 2 of §1, Chapter VI, can be used to show that F(<f>)F is a projection of n. The orthogonality of F, E1 and E_ 1 insures that E_ 1+F(<f>)F and E_ 1+E1+F(<f>)F are projections. We define G(cj>) = F(cj>)F for o ~ <1> ( n, G(<j>) = E_ 1+F(<j>)F for n ~<I> ( 2n and G(2n) = E1 +E_~+F = 1. One can easily verify that the G(<f>) form a resolution of the identity with G(O) = o, G(2n) = 1. Fur­thermore we have that

S2nei<l>dG(.+.) = <S2nei<l>dF("') )F+E +E = FUF+E -E u. 0 'I" 0 'I" 1 -1 1 -1

Thus we have established:

THEOREVI II. If U is unitary, there exists a resolution of the identity G(<j>) with G(O) = o, G(2n) = 1, such that

U = S~nei<l>dG(<j>).

CHAPI'ER IX

CANONICAL RESOLurION AND INTEGRAL REPRESENTATIONS

In this Chapter, we obtain.the canonical resolution of a c.a. d.d. operator T and the integral representations of self-ad­joint and normal operators.

The discussion of Chapters VIII, IX and X is essentially based on the two papers of J. von Neumann to which refere~ce is made at the end of Chapter I. In the present Chapter, however the use of (1+H2)- 1 to obtain the integral representation of an unbounded self-adjoint operator was suggested by the~esz­Lorch paper also listed in Chapter I. The canonical resolution of a normal operator is used to obtain the integral representa­tion by K. Kodaira.*

§1

In this section, we obtain the canonical resolution of a c.a. d.d. operator T. (Cf. Theorem I of this section), For a c.a. d.d. operator T, Theorem VII of §4, Chapter IV tells us that A·= (1+T*T)- 1 is a bounded definite self-adjoint operator with a bound ~ 1. The Corollary to Theorem I of §3, Chapter VIII shows that there exists a resolution of the identity E(h) with E(o-o), E(1) = 1, such that

A= S6ME(h)+o•E(o) = JbhdE(h).

Lemma 6 of §2, Chapter VII shows that the zeros of A are 1-E(1 )+E(o) = E(o). Since A- 1 exists, ~A= !0! and hence E( o) = o.

Lemma 5 of §3, Chapter VII implies that A- 1 = Jb(1/.i\)dE(.i\). We make the change of variable µ= 1/11., F(µ) = 1-E(1/µ-0) for 1 ~ µ < CD. Since E(O+O) = E(o) =O and limh•O,h)O~(h-0) = limll.+o,71.)oE(h) = limµ+OF(µ) = 1, F(1) = 1-E(1-0). As in the discussion of the proof of Lemma 5 of §4, Chapter VIII, one can show that F(µ) is a resolution of the identity and that for ~ b < cc,

J~/b(1/h)dE(h) = j~µdF(µ)+F(1 ).

*Proc. Imp. Acad., Todyo 15, pp. 207-2to, (1939). 95

IX. CANONICAL RESOLUrION

When we let b -- o:;,1 we see that corresponding improper 1nte­grals are equal and thus

A- 1 = S6(1/i\)dE(i\) = f~µdF(µ)+F(1 ).

Thus

1+T*T = f1""µdF(µ)+F(1)

or T*T = _\," (µ-1 )dF( µ ).

We make the f'urther change of' variable µ12 = µ-1, F1(µ 1) =

F( µ12+1) f'or o ~ µ' < o:>1 F1 ( µ') = o, f'or µ' < o. We then obta1n

T*T = f~ µ' 2dF 1 ( µ) •

Let B= f;µ 1dF1(µ 1). f;µ 14d!E(i\)f'! 2 <o:;, implies l~ µ' 2dl E( h)f' 12 < o:;,. Hence Lemma. 4 of' §3, Chapter VII shows that B2 = f;' µ12dF1 ( µ' ) = T*T. B is self'-adjo1nt by Theorem

II of' §3, Chapter VII. Furthermore B*B = B2 = T*T. Thus Theorem I of' §4, Chapter VI states that there is a partially

isometric W with initial set [~Bl and f'inal set [!RT] and such that T = WB, T* = Bo!IW* = Bil*, B = W*T = To!!W.

,19 LElVMA 1. Let T be c.a.d.d. Then there exists a resolution of' the identity F1(µ) with F1(o) = o and

such that if' B = f;'µdF 1 (µ), then B2 = T*T, T = WB, T* = Bil*, B = W*T = 'lW. Furthermore if' E( i\) = 1-F1((1/ i\-1 )1/ 2-o), then (1+T*T)- 1 = fbi\dE(i\).

The equation T = WB is called the canonical resolution. For completeness, we must still consider the corresponding results f'or T*. Lemma 1 above when applied to T* shows that there is a resolution of' the identity F2(µ) and C = ~µdF2 (µ) such

that C2 = TT*.

I..EMi.llA 2. If' D is a bounded transf'orma.tion, which coI1DIIUtes with B2 = T*T, then D coI1DIIUtes w1 th F 1 ( i\) and

and B. A similar result holds f'or c.

PROOF: If' D COIIDI!Utes with B2 , it also COIIDI!Utes with B2+1 and (B2+1 )- 1 • (Cf'. Lemma 1 of' §4, Chapter VII). Lemma 2 of' §4,

Chapter VII now shows that D coI1DIIUtes with E(i\) f'or o ~ i\ ~ 1.

§ 1 • THE: CANONICAL RESOLillION 97

By Lenu:na 1 above, we see that this implies that D coIIDnUtes

with F 1 (µ). By Corollary 2 to Lemma 3 of Chapter VIII, §4, we

obtain that D COIIDnUtes with S~µdF1 (µ) for every a. Thus if

f is in the domain of D, we have s~µdF1 (µ)Df = D(S~µdF1 (µ))f. D is continuous and hence the 111llit as a ----+ oo of the right

hand side exists. Hence Df is in the domain of B whenever

f is, and DB C BD.

LEMMA 3. Let T, B, C, W, F 1 (i\) and F 2 (i\) be

as above. Then WF1 ( i\)W* = F2 ( i\)-F2 ( o), W*F2 (i\)W =

F 1 ( i\)-F 1 ( o), WBN* = C, B = W*OIV.

Let E1 = W*W, E2 =WW*. These are projections on;-respec­

tively, the initial and final sets of W, i.e., [~B] and

['tr]· Now if we use [!nA]• = [~A], (Theorem VI of §2, Chapter

IV) and B = B*, C = C*, '1T = 71B' '1c = 'nT*' we ,-obtain [~B] = 71B 1 ['tr] = '1T = '1C. Thus E1 is the projection on '1:8, E2 on 710• From the expression for B and C given in Lemmas 1

and 2 above, we have that '1B has the projection F 1 (O) and

'1c has the projection F2(o) (Cf. proof of Lenu:na 6 of §2, Chapter VII). Hence E1 = 1-F1(o), E2 = 1-F2(o).

Thus E<X coIIDnUtes with F<X, <X = 1, 2. From Lemma 3 of §3, Chapter VI we see that W = WW*W = E2W = WE1 •

Now WF 1 ( i\)W* is bounded and self-adjoint. Furthermore

WF1 (i\)W* = WE1F1 (i\)W* = WE1Fi( i\)W* = WF1 (i\)E1F 1 (i\)W* =

WF 1 ( i\)W*WF 1 ( i\)W* = (WF 1 ( i\)W*) 2 • Thus F 2 ( i\) = WF 1 ( i\)W* is a

projection. (Cf. Lenu:na 2 of § 1, Chapter VI). Since E2F 2 ( i\) =

E2WF1 (~)W* = WF1 (i\)W* = F2(i\), we have F2(i\) ~ E2 • (Cf. Lemma 8 of § 1 , Chapter VI ) •

If we define F { ( i\) = W*F 2 ( i\ )W, a slln1lar argument will show

that F{(i\) is a projection with F{~i\) ~ E1 •

We next observe that C2 = T*T = WB•BN* = WB2w*. Since E1 is the projection on [~B] and on '1f,, we have W*C2w =

W*WB2w*W = E1 B2E1 = B2 •

Since C2 = WB2w*, we have F2(i\)C2 = WF1 (i\)W*WB2w* =

WF1(i\)E1B2w* = WF,_(i\)B:?vi* C W(S~µ2dF1 (µ))W* = WB2F 1(i\)W* =

WB2E1F1 (i\)W* = WB"W*WF1 (i\)W* = C2F2(i\). (Lemmas 1 and 4 of §3,

Chapter VII aee used here). TP.us F2(i\)C2 C c2F2(i\) and the latter has bound i\2 • S11llilarly F { ( i\)B2 C B2F { ( i\) and the

98 IX. CANONICAL RESOLUTION

latter has bound i\2 •

LeIIl!IIS. 2 above now shows that F2(i\) commutes with F 2 (µ). We

note that F2(i\) ~ E2 = 1-F2(o). Consider F2(i\)(1-F2(i\)).

Lemma 4 of §1, Chapter VI shows that this is a projection.

Suppose that an f =lo e is in the range of this projection. For

a. resolution of the identity, we have limµ+ooF 2(µ) = 1 and

lime~ 0F2 (i\~e2 ) = F 2(i\). Since f = F2(i\)(1-F2(i\))f =lo e, it

follows that we can find a /\. and a i\0 with i\ < i\0 < /\. such

that g = F2(i\)(F2 (/\)-F2 (i\1 ))f + e. Owing to the commutativity

of F2(i\) and F2(µ), we a.lso have g = (F2 (/\)-F2 (i\0 ))g = F2(i\)g. Hence IC2gl 2 = IC2 (F2(/\)-F2(i\0 ))gl2 = S~i\4d!F2 (i\)g! 2~ ~l(F2 (i\)-F2 (i\0 ))gl 2 = i\6lgl 2 • (Cf. Lemma 3 of §2, Chapter VII

and Lemma 1 of §3, Chapter VII). Also IC:2gl 2 = IC2F21 (i\)gl 2 {. 4 -

i\ lgl 2 • Since lgl + o, i\0 ) i\~ o, these statements contradict

ea.ch other and thus f = e. Thus we have shown that f in the

range of F2(i\)(1-F2(i\)) implies f = e. It follows that

F2(i\)(1-F2(i\)) = o or F2(i\) ~ F 2(i\). (Cf. Lemma 8 of §2, Chap­

ter VI). Si.nee we aiso have F2(i\) ~ E2 = 1-F2(o), we have

F2(i\)(F2 (i\)-F2(o)) = F2(.i\)(F2 (.i\)-F2 (.i\)F2 (o)) = F2(.i\)F2 (.i\)(1-F2(o).

= F2(.i\)(1-F2(o)) = F2(.i\), and F~(i\) ~ F2 (.i\)-F2(o). (Cf. Defi­

nition 1 of §2, Chapter VII and Lemma 8 of §2, Chapter VI).

Now WF1(o)W* = WE1F 1(o)W* = W(1-F1(o))F1(o)W* = o. Similar­

ly W*F2 (o)W = o. Thus F2(i\) ~ F 2(i\)-F2(o) becomes F 2 (.i\)­

F2(o) ~ WF1(i\)W* = W(F1(i\)-F1(o))W*. Multiplying by W* on the

left and W on the right, we obtain F{(.i\) = W*(F2 (.i\)-F2 (o))W

~ W*W(F1 (i\)-F1(o))W*W = E1 (F1(i\)-F1(o))E1 = (1-F1(o))(F1(i\)­

F1(0))(1-F1(o)) = F 1(i\)-F1(o), or F{(.i\) ~ F 1(i\)-F1(o). (Cf.

Definition 1 of §2, Chapter VII). But a. proof' analogous to that

of the preceding paragraph will show that F{(i\) ~ F 1(i\)-F1(o).

Thus we have established that F{(.i\) = F 1(i\)-F1(o).

This last result ma.y be written W*(F2 (.i\)-F2(o))W = F 1(i\)-

F1 ( o). Multiplying by W on the right, and on the left and

proceeding as above we obtain F 2 ( i\)-F 2 ( o) = W(F 1 ( i\)-F 1 ( o) )W* = F2(.i\) = WF1 (i\)W*.

If we forni partial sums, use this last equation and pass to

the limit, we obtain

S~dF2(µ) = ~<S~µdF, (µ) )W*

since either side is defined everywhere. Taking limits, we get

§ 1 • THE CANONICAL RESOLUTION 99

S;'µdF2(µ) = wcs~µdF, (µ))W*

or C = WBN*. Multiplying on the left by W* and on the right

by W yields W*Oli = W*WBN*W = E1BE1 =B. Our lemma is now dem­onstrated.

LEMMA 4. Let T be c.a.d.d. There is at most one B and W such that W is partially isometric with initial

set [~BJ and final set [~TJ and such that B is definite self-adjoint and possesses a resolution of the identity

F1 (i\) such that B = S;'µdF 1 (µ) and furthermore such that

T = WB.

Suppose that the pair, W, B = S;' µdF 1 ( µ) and the pair, W 1 ,

B1 = fo"'µdG 1 (µ) both satisfy the given conditions. ~ can suppose that W and B are as in LemmB, 1 above. By the corol­

lary to Theorem V of §2, Chapter IV, T* = Bj'W* = B1W* and since

W*W is the projection on [~BJ, we must have T*T = B~. (Cf. Lemma 1 of §3, Chapter VI).

Thus B~ = T*T = B2 and S;'µ2dG1 (µ) = S;'µ2dF1 (µ) by Lemma 4 of §3, ChB.pter VII. Now G1(i\) connnutes with B~ by Lemma 1

of §3, Chapter VII and thus with B2 • Lemma 2 above, shows that G~(i\) connnutes with F1(µ) and if we consider the. bound of B1G1(i\) we see that precisely the same argument as that used in the proof of Lemma 3 above will show that G1 ( i\)-G1 ( o) ~ F1 (i\)-F1 (o).

We also have iri. the above that F 1(µ) connnutes with G1(i\) and thus we may proceed to obtain the symmetric result F 1(i\)­F1(o) ~ G1(i\)-G1(o). Thus we have shown F 1(i\)-F1(o) = G1(i\)­

G1(o). Since '1B2 = '1g2, we also have F 1(o) = G1(o). We may conclude that F 1 ( i\) = G1 ( i\) and B = B1•

Since [~BJ = [~B, J, the initial sets of W and w1 are the same and both W and W 1 are zero on [~B ]' • (Cf. Defini­tion 1 of §3, Chapter VI). The equation WB = T = W1B shows

that W = W 1 on ~B. Continuity implies W = Wt on [~BJ. We also have W = W 1 = o on [~BJ• and since these transformations are linear we must have W = W 1 •

COROLLARY. A similar result holds for T = ON.

100 IX. CANONICAL RESOLUTION

We now complete our discussion: C = WBW* implies CW = WB = T. The corollary to Theorem V

of' §2, Chapter IV shows that T* = W*C = BW*.

THEOREM I. Let T be c.a.d.d. Then there exists

opera.tors W, B, C and resolutions of the identity F1(.>..) and F2(.>..) such that (a): W is partially

isometric with initial set [~BJ and final set [~T].

(b): B and C are self-adjoint and def'inite. (c):

T = WB =CW. (d): T* = BW* = W*C. (e): C = WBll*,

B = W*CW. (f) = B = r:11.dF1 <"->, c = s; 11.dF2c11.>. (a), (b), (c) and (f) determine W, B and C uniquely.

We shall show in the next section that ( b) implies ( f) . We can then state that (a), (b) and (c) determine w, B and C uniquely.

§2

We now obtain the integral representation for a self-adjoint operator H. H is c.a.d.d. and thus we may apply Theorem I of' the preceding section to obtain that there is a definite self­adjoint B with an integral representation, So"°µdF 1 (µ) a definite, self-adjoint C and a partially isometric W such that H = WB = CW. Since H*H = H2 = HR* we see from Lemmas 1 and 2 of' the preceding section that B = C and thus H = WB = B'V.

Thus W colllllllltes with C and Lemma 4 of' §4, Chapter VII shows that w = W*. Let E = w:11-w. Since w = W*, w =WE= W;= EN. Since W cOlllllllltes with B, W collllll\lte_s with B2 and thus Lemma 2 of the preceding section shows that W colllllllltes with F 1 ( 11.) • In the proof of the same Lemma 2, it was also shown that E = 1-F1 (O) and consequently E also colllllllltes with F1 (.\).

1 1 Thus if F1 = 2'(E+W), F2 = 2'(E-W) then F1 and F 2 com-mute with F 1 (I\). F 1 and F2 are projections by Lemma 4 of' §4 Chapter VIII. F1(h)F1 and F 1(ll.)F2 are projections by Lemma 4

of' §1, Chapter VI. We also have F1F2 = *(E+W)(E-W) = *(E2+WE­

EN-W2) = *(E+W-W-E) = o. We have then:

§2. SELF-ADJOINT OPERATORS 101

H = BW = .f;'µdF 1 (µ)(F 1-F2 ) = fo"µdF 1 ( µ)F 1- fo°µdF 1(µ)F 2 .

Let 7'.= -µ, G(i\) = (1-F1(-i\-O))F2 f'or -oo < i\< o. Then

limi\+-.,..G(i\) = limµ (1-F1(µ-0))F 2 = F 2-11mµ .. .,..F1 (µ)F 2 = F 2-F2 = 0 and limE•OG(-7) = lim (1-F1(e 2-o))F2 = limE•0(1-F1(e2))F2

= (1-F1(o+o))F2 = (1-F1(o))F2 = EF2 = (F1+F2 )F2 = F 2 • As in the

proof of Lemma. 5 of §4, Chapter VIII, we have that for b ) e2

)o,

Letting E - o, we have rb-0 -e 2 o-o·

-J 0µdF 1(µ)F 2 = lime .. ol-b AdG)'(A) = J_b i\dG1 (A).

Let G1(o) = F2+(1-E) = G1(o-o)+1-E. Since (1-E)F2 = (1-(F1+

F 2 ))F2 = o, G1 (o) is a projection by Lemma. 5 of §1, Chapter VI.

Also ~

sg_ 0Aa.G1(i\) = O•(G1(o)-G(O-O)) = O.

rO r-E 2 rO s-e2 rO r0-0 For a:rry integral J-b = J-b +J_e2 = limE•O( -b +J-e2) = J-b + sg_ 0 , when these last limits exist. Hence

l~bi\dG1 (A) = l~boi\dG1 (A)+Jg_ 0Aa.G1 (i\) = -s~~dF 1 (µ)F2+o. Letting b - oo, we obtain

-s~µdF1 (µ)F2 = J~.,..MG1 (i\).

For o < i\ < oo, let G1 (i\) = F2+1-E+F1F(i\). Our previous orthogonality relations and Lemma. 5 of §1, Chapter VI, shows

that G1 (A) is a p;c\ojection. One has also lim,._ .. .,.. G1 (i\) = F 2+

1-E+ 11mi\ .. .,..F1F 1 (A) = F 2+1-E+F1 = 1, since E = F 2+F1 •

By a familiar reasoning, we obtain that

s; i\dF 1 ( i\)F = s; MG, ( i\).

This and our preceding results imply that

H = s~µdF,(µ)F,-J;µdF,(µ)F2 = r:..MG,(i\).

Further G1 ( i\) is a resolution Qf the identity. For as we have

seen above each G1 (i\) is a projection and limi\ .. -co G1 (i\) = o, limi\ .. .,.. G1 (i\) = 1. The other properties of a resolution of the

identity are the results of known inclusion relations on the

given projections and the corresponding properties for F 1("µ).

we have proved:

102 IX. CANONICAL RESOLUTION

TREOREVI II. If' H is self'-adjoint, then there exists a resolution of' the identity G1(h) such that

H = S:°00 MG1 (A).

COROLLARY 1. IT C_ > -=, we have that

H = sg~AdG1 (h)+C_G1 (C_), while il c_ = - =, we have

H = S~!. MG, (A).

This is shown by a discussion similar to the proof' of' the

Corollary to Theorem I of' §3, Chapter VIII.

For a def'inite operator A, we have c_ ~ o. In any case,

we may take o, = as our limits of' integration and A = S~G1(h). In Theorem I of' §1 above, we now have that (b) implies (r).

This demonstrates:

COROLLARY 2. In Theorem I of' § 1 , above (a), ( b) and

( c) determine W, B and C uniquely.

COROLLARY 3. G1 (A) f'or -=

If' T commutes with H,

< A< +=. T commutes with

PROOF: If' T commutes with H, it commutes with H2 = B2 • It f'ollows f'rom Lemma 2 of' the preceding section that T commutes with F 1 (A) and with B.

We next show that T commutes with the projection 1-E on

'le = '1_a:· For il :r is in '1H, Hr = e and HTf' = THr = Te = e or Tf' € '1_a:· Hence T(1-E) = (1-E)T(1-E). By Lemma 1 of' §4,

Chapter VII, H also commutes with T* and we obtain T*(1-E) = (1-E)T*(.1-E). Ta.king adjoints, we f'in~ that (1-E)T =

(1-E)T(1-E) = T(1-E). It follows that El'= TE= El'E.

In the proof' of' Theorem II above, we have shown that W = WE. Thus il f' is in the range of' 1-E, TWf' = TWEf = e and WTf' = WEI'f' = WTEr = 9. Hence f'or f' in the range 1-E, TWf' = 9 = Wl'f'.

For f' in the domain of' B which is also the domain of' H,

we have TWBf' = THr = HTf' = WBl'f' = WTBf'. Thus il g is in the

range of' B, TWg = WTg. Since W and T are continuous, we

may inf'er that TWg = WTg f'or g € [!RB], which is also the

§3. NORMAL OPERATORS 103

range of E. This and the result of the preceding paragraph per­

mit us to conclude that TW = WT.

If F 1 and F 2 are as in the proof of Theorem II above, we

have FcxT = TFcx for ex = 1, 2, since 'IW = WT and El' = TE. In the above, we have shown that T collllIIUtes with F 1 (I\). From

the definition of G1 (1\), we may now infer that T collllIIUtes

with G1 (I\).

COROLLARY 4. The resolution of the identity G1 (1\) of

Theorem II above is unique.

Let us suppose that H also equals S::0AdG2 (1\) for a reso­

lution of the identity G2 (1\). We prove that G1 (I\)= G2 (1\).

For a given i\, G2 ( I\) collllIIUtes with H and thus from Coro­

llary 3 above, it must collllIIUte with G 1 ( µ) for -oo < .µ----( oo.

Thus G2 (1\)(1-G1 (1\)) is a projection by Lemna 4 of §1, Chapter

VI. Suppose now that f =Fe is in the range of G2 (1\)(1-G1(1\)).

Since G1 (1\+0) = G1(1\), 11mµ .. 00 G1(µ) = 1, 11mµ-+-coG2 (µ) = o, we can find three numbers ", < i\ < '\ < 1\3 such that g = (G2 (1\)-G2 (", ))(G1 (13)-G1 (1\~)f =f e.

Now g is easily seen to be in the domain of H. (Cf.

Theorem II of §3, Chapter VII). Let a be defined by the equa­

tion a(g,g) = (Hg,g). (We recall that (g,g) =F o). Since G1(1\)

and G2 (1\) collllIIUtewehave g= (G2 (1\)-G2 (",))g= (G1<13>­

G1('\))g. Lemna 1 of §3, Chapter VII now shows that (Hg,g) = (H(G2 (1\)-G2 (", ))g,g) = ~µd(G2 (µ)g,g) ~ i\((G2 (1\)g,g)-G2 (1\1 )g,g)

= i\((G2 (1\)-G2 (", ))g,g) = i\(g,g). Thus a~ i\. On the other hand

g = (G1(1\3 )-G1(1\2 ))g yields by a similar argument that a~ 1\2 •

Since i\ < '\• this is a contra.diction. It follows that the range of G2 (1\)(1-G1(1\)) must contain only e. Hence G2 (1\) ~

G1 (I\). In the preceding discussion, the collllIIUtativity of G1(1\) and

G2(µ) and the expressions for H, were sufficient for the result.

result. It follows that a similar discussion will also show that

G1 (>..) ~ G2(1\). Hence G1 (1\) = G2(1\).

We consider in this section, a normal opera.tor A. (Cf. Defi­

nition 2 of §4, Chapter VII). When we apply Theorem I of §1

104 IX. CANONICAL RESOLUTION

above to A, we obtain as usual W, B and C with the usual properties (a) - ( f). Since however A*A = AA* we have B = C

and thus A = WB = CW = BW. We also introduce E = W*W, the projection on the initial set

of w which is [~BJ= nB. (Cf. Lem:na 8 of §3, Chapter IV). Lem:na 3 of §4, Chapter VII shows that WW*, the projection on the final set of W is also E. Thus ( 1-E)W = o = W( 1-E).

Taking adjoints we obtain W*(1-E) = o = (1-E)W*. Since E is

the projection on [~BJ = ne, we have ( 1-E)B = B( 1-E) = o. Consider U = 1-E+W. Now UU* = ( 1-E+W)( 1-E+W*) = ( 1-E)+WW*+

W(1-E)+(1-E)W* = 1-E+E+o+o = 1. Similarly UU* = 1. Now UU* =1

shows that the range of U is n and tl*U = 1 implies that for every f and g, (Uf,Ug) = (U*Uf,g) = (f ,g). Definition 1 of

§2, Chapter VI shows that U is unitary. We have then that UB = (W+l-E)B = WB =A= BW = B(W+1-E) =BU.

LEMMA 1. If A is normal there exists a unitary oper­ator U and a self-adjoint definite B = J';µdF 1 (µ) such

that A = UB = BU.

LEMMA 2. Let U be unitary and let m1, '1?_ 1, E1, E_ 1, F and U1 be as in Lem:na 1 of Chapter VIII, §4. Then if D is a bounded self-adjoint operator which com­mutes with U, then D conmrutes with E1, E_ 1, F and

U,. Furthermore D = E1 DE1 +E_ 1DE_1 +FDF, where FDF com­mutes with ~, when both are regarded as contracted to the range of F 1 •

If f is in 7n1 then Uf = f and Df = DUf = UDf. Thus f E '1?1 implies Df E m1 and DE1 = E1 DE1 • Taking adjoints

yields E1D = E1DE1 = DE1 . Similarly E_~D = E_ 1DE_ 1 = DE_ 1• These results imply FD= DF and FD= F D = FDF. Lem:na 1 of §4, Chapter VIII implies

DE1-DE_ 1+Dt.li = D(E1-E_ 1+U1 ) =DU= UD

=(E,-E_,+u, )D = E,D-E_,D + u,D.

Since DE,= E,D, DE_, = E_,D, This implies DU, = U,D. Inas­much as 1 = E1+E_ 1+F, D = D(E1+E_ 1+F) = DE1+DE_ 1+DF = E1DE1+ E_ 1DE_ 1+FDF. The statement concerning the contraction of FDF is obvious.

§3. NORMAL OPERATORS

LEMMA 3 • Let U be unitary and let G( <I>) denote the resolution of' the identity f'or U given by Theorem II of

§4, Chapter VIII. Then if D is a bounded self-adjoint transformation which commutes with U, then D commutes

with G(<I>) for o ~<I> ~ 21t.

105

PROOF: We know f'rom Lemma 2 above that D1 = FDF commutes with ~. Let us confine our attention to the range of' F and

thenwemaydef'ine A1 =~(U,+U,*), B1 =~i(Uf-U1 ). Let Wand c1 be as in the proof of' Lemma 3 of' §4, Chapter VIII. D com­mutes with A, and B, by Lemma 1 of' §4, Chapter VII and E(1..)

by Lemma 2 or' §4, Chapter VII and with C by Corollary 2 to Lemma 3 of §4, Chapter VIII.

-1 Lemma 1 of' §4, Chapter VII shows that D1 commutes with c1 • -1 -1 -1 -1 Thus D1B1c1 = B1D1c1 c B1c1 D1• Since B1c1 = w on the range

of C, we have D1Wf' = WD1f' on the range of' C and since--this last set is dense and D1 and w1 are continuous, we obtain

D1W = WD1 • (Cf'. Lemma 3 of §4, Chapter VIII) Let F 1 = ~( 1 +W J, F 2 ':' ~( 1-W), as in the discussion follow­

ing Lemma 4 of §4, Chapter VIII. Since D1 commutes with W, D1 commutes with F1 and F2 • From the definition of F 1 (<I>)

preceding the statement of' Lemma 5 of §4, Chapter VIII, we see that D1 must commute with F1 (<j>) since D1 commutes with F 1, F 2 and the E(<j>).

We now return to the more general situation. We see firstly that F(<j>)FD = F(<j>)FDF = F(<j>)D,F = D,F(~)F = DFF(<j>)F = D•F(<j>)F.

Thus D commutes with F(<j>)F. We have snvwn above in Lemma 2

that D commutes with E1 and E_ 1 • It follows f'rom the def'i­n;ttion of' the G(<j>) preceding Theorem II of §4, Chapter VIII, that D cnmmutes with G(<j>). This proves the Lennna.

Returning to the result of Lemma 1, we recall that A= UB = BU. Thus if B = fo"µdF 1(µ), U conmrutes with F 1(l\). Now if'

S21t i<I> U = 0 e dG1 ( <j>), we see from Lemma 3 above that F 1 ( l\) and G1(<1>) commute. Thus

106 IX. CANONICAL RESOLUI'ION

A = s; j~itpei<l>dG1 ( <j>)dF 1 ( p) = j~it j~ pei<l>dF 1 ( p )dG1 ( <j>).

This suggests that we may introduce the notion or a planar reso­lution or the identity and a corresponding planar integral. Our discussion will be simpler U we modify G1(<1>) by introducing G2(<!>) = G1(<1>)+1-G1(2it-O) ror o ~<I>< 2n. It is readily seen that U = J~nei<l>dG2 (<i>)+G2 (o), G2(21t-O) = 1 and

A = s~ s~ pei<l>dF,, p )dG2(<j>)+ s; pdF, (p )G2( o,.

Ir p is a point or the plane with polar coordinates (p,<j>), p ~ o, o ~<I> < 2n, (the origin has the coordinates p = o, <I>= o,) let E1 (P) = F1 (p)G2(<j>). Since F1 (p) and G2(<j>) com­mute, one can show that E1 ( P) has the rollowing properties:

(a): Ir P1 and P2 are two points with coordinates. (p1,<1>1 ), (p2 ,<j>2 ) respectively and ir Q is the point whose coordinates are (min(p1,p2), min(<j> 1,<j> 2 )) then E1(P1)-E1(P2 ) E, (Q).

(b): Ir we let E1(p,<j>) = E1(P(p,<j>)) then E1(p+O,<j>+O) =

E1(p+O,<j>) = E1(p,<j>+O) = E1(p,<j>). (c): Ir we now let F1(p) = E1(p,2n-o), ror p) o, F1(o)

E1 (O,O) and F1 (p) = o U p < o then F1 (p) is a resolution or the identity. Similarly 1r we let G2(<j>) = ·11mp .. = E1 (p,<j>) ror o ~ <I> < 2n, G2(<1>) = o ror <I> < o, G2(<!>) = 1 ror <1> ~ 2n, then G2(<j>) is a resolution or the identity with G2(21t-O) = 1.*

DEFINITION 1. A ramily or projections E1 (P) having the properties ( a) , ( b ) and ( c ) above, will be called a planar resolution or the identity.

Let us consider a sector S or a circular ring, S not containing the polar axis. Thus S is the set or points (p,<j>) with p1 < p ~ p2, <1>1 <<I> ~ <j>2 • We can associate with it a pro­jection E1(S) = (F1(p2 )-F1(p1))(G2 (<!>2)-G2 (<!>1)) = E1(P1)-E1(P2 )­E1 (P3 )+E1 (P4), where P1, P2, P3 and P4 are the points

* These statements are redtmda.nt. For instance in ( c) it is only necessary to show that 11mp .. _F1 (p) = 1 and that G2 (2n-o) = 1, since the otaer properties are consequences or (a); (b) and the derinitions or F1 ( p) and G2 (c1>). For example a consideration or projections associated wltn the areas as in the rollowing dis­cussion will show that G2 (cj>+O) = G2 (<j>).

§3. NORMAL OPERATORS 107

(p2 ,<1>2 ), (p2 ,<1> 1 ), (p 1 ,<1>2 ) and ( p1 ,<1>1 ) respectively. (E1 (S) is a projection by Lemma 4 of' §1; Chapter VI).

Any such S can be expressed as the logical sum of' mutually

exclusive smaller Sa, i.e. S = s 1 u ••• u Sn, and we will call this a partition of' s. We can then f'orm f'or any f'unction "1,

the partial sums I: '41(Qa)E( Sa) where Qa € Sa. If' "1 is con­tinuous, it can be shown that f'or every sequence of' partitions

such that the maximum diameter of' the Sa 1s approaches zero, these partial sums approach a limit which we will deno~e:

5Ss"1(P)dE1 (P).

Furthennore a f'a.miliar discussion will show that

SSs"1(P)dE, (P) = s:~ 5~"1(P(p,<j>) )dG2(<1>)dF, (p).

If we close S and call the result S', we def'ine

E1(S 1 ) = (F1 (p2 )-F1(p1-o))(G2 (<j>2 )-G2 (<1>1-o)).

We can partition S 1 into closed and partially closed sec.tors

and again obtain f'or every continuous 141( P),

Hs ,14i(P)dE1 (P).

This equals

SP2 r<l>2 P1-0J<1>,-o (P(p,<j>)),dG2(<1>)dF1 (p).

One may also introduce the notion of' an illlproper integral, so that if' s 0 denotes the entire plane, we may def'ine the planar integral over s 0 and obtain:

Hs "1(P)dE1 CP) = ro=. 0s~~014i(P(p,<1>) )dG2 C<1>)dF1 (p) 0

5~ 5~n'41( P( p,<j>) )dG2 (<j>)dF1 ( p)+ 5~ '41( P( p, o) )dF1 ( p)dG2( o )+

"1(P(o,o))G2(0)F,(O).

In particular if' we let 14i(P(p,<j>)) = z(P) = pei<I>, we obtain:

A = SSs zdE1 (P). 0

THEOREl\11 III. If' A is a normal operator, there exists a planar resolution of' the identity E1(P) such that

A= Hs zdE1 (P). 0

Lemma 3 above can also be used to show

108 IX. CANONICAL RESOLUTION

I.:m.NA 4. If' U is unitary, the equations U= s~nei<l>dG(<j>), G(O) = o, determine the resolution of' the identity G(<I>) uniquely.

Let G be as in Theorem II of' §4, Chapter VIII and suppose that we also have U = S~eilf>dG1 (<j>), G1(o) = o.

Suppose now that f'or o <<I> < 2n, G1 (<!>) commutes with U and hence by Lemma 3 above, G1 (<!>) commutes with G(\jl). We will prove that G 1 (<I>)( 1 -G( <tiil ) = o. Lemma 4 of' § 1 , Chapter VI shows that G1(<1>)(1-G(<1>)) is a projection. Now suppose that there is a f' + e such that f' = G1(<j>)(1-G(<j>))f'. Since G1(o+o) = G1(o) = o and G(<j>+O) = G(<j>), we can f'ind an ex and a <1>1

such that g = (G1(<1>)-G1(ex))(1-G(lj>))f' + e and o <ex< <1> < <1>1• Since G1 (<!>) and G(\jl) commute, we have that g = (G1 (<l>)­

G1 (ex) )g = (1-G(<j>1))g. Let a be def'ined as the equation a(g,g) = (Ug,g) which is possible since g + e. The argument of' a is the same as that of' (Ug,g) = (U(G1(<j>)-G1(ex))g,g) = S~e14>d(G 1 (<j>)g,g). A consideration of' the partial sums will show that their arguments always li~ between <I> and ex and it f'ollows that the argument of' their limit (Ug,g) is in this interval closed. Thus the argument of' a is { <I>·

However we also have (1-G(<j>1 ))g = g and U-= S~e14>dG(<j>) and a similar argument will show that the argument of' a is ~ <1>1 • Since <I> < <1>1 , this yields a contradiction. Thus f' in the range of' G1(<1>)(1-G(<I>)) implies f' = e and hence G1(q,) ~

G(<j>). In this discussion, we have used simply that G1(<1>) and G(ljl)

commute, G1(o) = o and the two expressions f'or U. A similar argument will therefore show that G(<I>) ~ G1(<j>) and hence G1(<1>) = G(<!>). Since G( o) = o, we now have G1 (<1>) = G(<j>) f'or o ~ <I> < 2n. But G1 ( 2n) must be 1. For if' f' is in the range of' 1-G1 (2n), Uf' = S~nei<l>dG1 (<1>)f' = e and o • IUf'I = lf'I. Thus 1-G1 ( 2n) = o and G1 ( 2n) = G( 2n). This completes the proof' of' the Lemma.

We may also establish.

I.:m.NA 5. If' A is c.a.d.d. and F is a projection such that FA C AF, then F comm:utes with B, C, W, W*W = E1, WW* = E2, where B, C, and W are as in

§3. NORMAL OPERATORS

Theorem I of §1 above. In particular if A is normal and equals UB = BU where U is unitary, then F com­mutes with U, B, F1 (p), G2(<1>) and E1 (P) discussed in the proof of Theorem III above.

, 09

By the Corollary to Theorem V of §2, Chapter IV, FA C AF implies A*F = (FA)*:::> (AF)* :::> FA*. Thus FB2 = FA*A C A*FA C

A*AF = B2F, or F commutes with B2• Lemma 2 of §1, above shows that F commutes with B.

A discussion similar to that given in the proof of Corollary 3 to Theorem II of §2 above, now shows that F commutes with E1 and W. Since A and A* are now interchangeable, we obtain that F also commutes with C, E2 and W*.

For a normal A, .F commutes with U = W+1-E. Lemma 3 above and Corollary 3 to Theorem II of §2, above shows that F com­mutes with G1(<j>) and F1(p). Since G2(<j>) = G1(<1>)+1-G1(2n-o), F also commutes with G2(<1>). Since E1(P) = G2(<1>)F1(p), F commutes with E1 (P).

We conclude our discussion of normal operators by showing \

COROLLARl1. The planar resolution of the identity E1(P) of Theorem III above is unique.

Let F 1 ( p) and G2 (<I>) be as in property ( c) of Definition 1 above. A proof similar to that of Corollary 4 of the preceding section will show that F1(p) depends only on A. Similarly, a discussion similar to the proof of Lemma 4 above will show that G2(<j>) is unique. Property (a) of Definition 1 above can now be used to show that E1 (P) is determined.

· coRoLIARY 2. A* = Ss zcm1 ( P >. 0

This is a consequenc of A* = BU* = BU- 1 • For if U = s~ne1<1>dE(<!>) and v = s~ne-i<l>dE(<!>), we have by Lemma 5 of §2, Chapter VII, UV = VU= 1 . and hence u-1 = V. If we apply the discussion preceding Theorem III above to A* = VB, instead of A= UB, we will get the result stated in the Corollary.

CHAPI'ER X

SYMMErr'RIC OPERATORS

In this Chapter, we discuss syn:metric transformations; i.e. those for which H CH*. (Cf. Definition 1 of §3, Chapter IV). We will be concerned 1n particular with the notions of a symmetric extension and of maximality and its relationship with the property of being self-adjoint. (Cf. Definition 3 of §3, Chapter IV).

§1

In this section, H will be considered to be closed and syn:metric.

LEMMA 1 • If f and g are in the domain of H, then ((H+i)f,(H+i)g) = ((H-i)g,(H-i)g) = (Hf,Hg)+(f,g).

Expanding the expressions, will yield this result since (f,Hg) = (Hf,g). (Cf. Definition 1 of §3, Chapter IV).

LEMMA 2. (a): The set 'D of pairs !(H+i)f,(H-i)f! for f in the domain of H is the graph of a transformation V. (b): V is closed and isometric. (c): (H+i)- 1

exists and V = (H-i)(H+i)-1• (d): If v1, V 2 corres­pond to H1, H2 respectively as in the above, then H1 is a (proper) syn:metric extension of H2 if and only if v1 is a (proper) isometric extension of V2 • (e): If 7)1 i is the set of f E ~ such that H*f = if, the domain of V is 7'1j_ • (:f) ! If m_1 is the set of f E fi such that H*f =-if, the range of V is m~i . (g): The range of V-1 is dense. (h): (V-1 )- 1 exists and H = -i(V+1)(V-1 )- 1• (i): If E is a projection such that EV C VE, then EH C HE.

11 0

§ 1 • THE CAYLEY TRANSFORM

Proof' of' (a). If' f' is such that (H+i)f' = e then o = i(H+i)f'i 2 = ((If+i)f',(H+i)f') = (Hf,Hf)+(f',f') = 1Hfl 2+ lf'i 2 by Lemma 1 above. Thus (H+i)f' = 0 implies lf'l 2 = o and f' = e. Hence (H-i)f' = 8. Thus {e,hl E 71 implies h = e. 71 is easily seen to be additive and hence Lemma 3 of' §1, Chapter IV shows that 71 is the graph of' a transfor­mation.

11 1

Proof' of' (b). We first show that inasmuch as H is closed, 71 is closed. Let f cj>,ljl! be a pair in the closure of' 71. Let [ f (H+i)f'n,(H-i)f'nl] be a sequence in 71 such that (H+i)1'n -- <1>, (H-i)f'n---+ ljl. Let f' = ii(ljl-cj>) = ii(lim (H-i)f'n-lim (H+i)f'n) = ii 11.m (-2ifn) = 11.m f'ljl-. Similiarly f'* = i(cj>+ljl) = 11.m Hfn. Thus if we let f' = 2i(lji-<j>) then Hf exists and= i(<l>+ljl), since H is closed. We also have if'= i(cj>-ljl), and thus (H+i)f' = <1>, (H-i)f' = ljl. Hence f<j>,ljll is in 71. Thus 71 contains its limit points and thus V is closed. It follows by definition that V is closed.

Since 71 is a linear manifold, V i.s additive. Further-more, if <1>1 and <1>2 are in the domain of' V then cp1 = / (H+i)f'1, <!>2 = (H+i)f'2, Vcp1 = (H-i)f'1 and Vcp2 = (H-i)f'2 f'or some f' 1 and f' 2 in the domain of' H. Hence Lemma 1 above implies ( cj>1 ,<!>2) = ( (H+i )f' 1, (H+i )f' 2 ) = ( (H-i )f' 1, (H-i )f' 2 ) = (V<l>1' V<1>2 ). Thus Definition 2 of' §2, Chapter VI, shows that V is isometric.

Proof' of' (c). In the proof' of' (a) above, we have shown that (H+i)f' = e implies f' = e. Lemma 4 and Definition 2 of' §1, Chapter IV, now show that (H+i)- 1 exists. Now if' <I> is in the domain of' V, cf> = (H+i )f' f'or f' in the domain of' H. Hence (H+i)- 1<1> exists and equals f'. Also Vcj> = (H-i)f' = (H-i)(H+i)-1<1>. Thus V C (H-i)(H+i)- 1• On the other hand, if'

<I> is in th~ domain of' (H-i)(H+i)- 1, we let f' = (H+i)- 1<1>

and ljl = (H-i)(H+i)- 1<1>. Since <1> = (H+i)f', ljl = (H-i)f', we have V<1> = ljl. This shows that V ::> (H-i)(H+i)- 1 and with our previous inclusion proves the equality.

H1 a proper extension of' H2 is equivalent to H1+1 a proper extension of' H2+i, which in turn is equivalent to (H1+i)-1 being a proper extension of' (H2+i)- 1• Now the do­main of' V = (H-i)(H+i)- 1 is precisely the domain of' (H+i)- 1, since (H-i)f' is defined on the range of' (H+i)- 1• Hence

112 X. SYMMF.l'RIC OPERATORS

(H+i)- 1 being a proper extension of (H2+i)- 1 is equivalent

to v1 being a proper extension of v2. These equivalences

are surficient to prove (d).

Proof of ( e). Lemma. 4 of §2, Chapter VI shows that since

V is closed, both its domain Dv and range !Rv are closed ~ -1 linear manHolds. Now "'v is also the domain of (H+i)

which is the range of H+i. But :DV = !RH+i must be the orth­

ogonal complement of the zeros of (H+i)* by Theorem VI of §2,

Chapter IV. Now, by Theorem V of §2, Chapter IV, (H+i)*:>H*-i

and H* = (H+i-i)* :> (H+i)*+i or H*-i :> (H+i)*. It follows

that (H+i )* = H*-i. Thus Dv is the orthogonal complement of

the set for which (H*-i)f = e or for which H*f = H.

(r) is proven in a similar way.

Proof of (g). If f is in the domain of H, there is a <I>

in the domain of V , such that (H+i)f = cj>, (H-i)f = Vcj>. 1 Subtracting, we get 2H = ( 1-V)<I> or f = 2i(V-1 )cj>. Thus the

range of ~i(V-1 ) includes the domain of H, which is dense.

The statement (g) follows easily from this.

Proof of (h). We first prove that if V is isometric and

!R,-v is dense, then ( 1-V)- 1 exists. Now (V-1 )- 1 exists 1r

and only H (V-1 )<I>= e implies <I>= e. (Cf. Lemma 4 of §1,

Chapter IV). Let us suppose that (V-1 )<I> = e or Vcj> = <I>· For

ljl in the domain of V, we have

o = (Vcj>,Vljl)-(cj>,ljl) = (cj>,Vljl)-(cj>,ljl) = (cj>,Vljl-1!1).

Thus <I> is orthogonal to !RV-l and since this last set is

dense, we must have cj> = e. Hence (V-1 )<I>= 0 implies cj> = e. Furthennore 1.f f is in the domain of H, we have for a

cl> EDv• -2jf = (V-1)<1> and 2Hf = (V+1 )cj>. It follows that <1> = -2i(V-1 )f' and Hf= -i(V+1 )(V-1 )- 1f. Thus H C -i(V+1 )(V-1 )- 1 •

On the other hand, in the above, we have shown that 1r g = (V-1 )cj>,

g is in the domain of H. Thus !RV-l C DH. This is equivalent -1

to D(V-1 )-1 = !Rv-1 c DH. If T = -i(V+l )(V-1) , DT = D(V-1 )-1

since V+l is defined everywhere on the range o'f (V-1 )- 1

Thus DT = D(V-l )-1 C DH, and with our previous inclusion H C T, this shows T = H.

Proof of (k). If EV C VE, we see that the domain of these

transformations include ny. We also have E(V-1) C (V-1 )E.

If f is in the range of v-1, 1.e. f = (V-1 )cj>, for a <I> 1n

the domain of V, then Ef = E(V-1 )f = (V-1 )Ef, and Ef is also

§ 1 • THE CAYI.EY TRANSFORM 113

in the range of' (V-1). Furthermore Ef' = (V-1 )E<I> = (V-1 )E (V-1)- 1f', or (V-1 )- 1Ef' = E(V-1 )- 1f'. Since this holds f'or ev-

-1 -1 ery f' in ~V-l = ll(V-l )-l, we must have E(V-1) C (V-1) E.

We also have E(V+1) C (V+1 )E. Hence E(V+1 )(V-1 )- 1 C (V+1)E

(V-1)- 1 C (V+1)(V-1 )- 1E. The expression f'or H obtained in the

above now shows that EH C HE.

DEFINITION 1. If' H is closed s;ymmetric and V is as

in (a) of' Lemma 2 above, then V = VH is called the Cayley

transf'orm of' H.

In the proof' of' (h) above, we have shown:

LEMMA 3. If' V is isometric and such that ~V-l is dense, then (V-1)- 1 exists.

LEMMA 4. Let V be closed and isometric and such that

~V-l is dense. Lemma 3 above shows that (V-1 )- 1 exists.

Let H = -i(V+1 )(V-1 )- 1 • Then (a): H is closed s-ymmetric.

(b): The Cayley transf'orm of' H is V. ( c): Let 71i =

Dir, 71_i =~v, then the domain of' H and 71i or 7!_i have only e in common. ( d) : The domain of' H* consists

of' elements in the f'orm f'+g1+g2 where f' € 11J:• g 1 € 71-i' g2 € 71i and H*(f'+g1+g2 ) = Hf'-ig1+ig2 •

PROOF OF (a): If' <I> is in the domain of' V , let f' =

li(V-1)<j>. Then <I>= -2i(V-1 )f', and Hf'= (V+1 )(-i(V-1)-l )f' =

~ (V+l )<j>. Thus if' f' 1 and f' 2 are in the domain of' H and <1>1 and <1>2 denote the corresponding <I>' s, we have (Hf' 1,f'2 ) =

((V+l)cj>1 ,i(V-1 )<1>2 ) = -i((V+l)cj>1,(V-1 )<1>2 ) = -i[(Vcj>1'V<j>2 )+(<j>1'

V<1>2 l-0'<1>1 ,cj>2 )- (<1> 1 ,<j>2 )] = i [ (V<1>1 ,<j>2 )-(<1> 2 ,V<j>1 ) ] because f'or an

isometric V, (V<j>1 ,V11>2 ) = (lj> 1 ,1!>2 ). Similarly (f',Hf'2 ) =

i[(V<j>1 ,1j>2 )-(1j>1'Vlj>2 )]. Thus f'or every f' 1 and f'2 in DH,

(i' 1 ,Hf'2 ) = (Hf' 1,f'2 ). Furthermore 11J: = D(V-l )-1 is by hypothe­sis dense. Thus Def'inition 1 of' §3, Chapter IV shows that H is s-ymmetric.

The proof' that H is closed is analgous to the proof' of'

the closure of' V, in (b) of' Lemma 2 above.

11 4 X. SYMMEI'RIC OPERATORS

PROOF OF (b): If' cf> is in the domain of' V, f' = li(V-l)cf> 1 21

is in the domain of' H with Hf' = 2 (V +l )cf>. Thus cf> = 2 (V +l )cf> 1 1 1 - 2(V-1 )cf>= (H+i)f and Vcp = '2(V+1 )cf>+2(V-1 )cf> =(H-i)f'. Thus V

is included in the Cayley tranaf'orm of' H. If', however, VH

were a proper extension of' V, (VH-1)- 1 would be a proper ex­

tension of' (V-1 )- 1 since ~(V- 1 )-1 = Dv· Hence Dcv- 1 )-1 is

included in but not equal to D(VH- 1 )- l • However we see f'rom our hypotheses and (b) of' Lemma 2 above that these sets are both

D_a:. Thia is a contradiction and we have V = VH.

PROOF OF ( c). Let us suppose that g is in '\ • D_a: and g f e. Then g = i(V-1 )cf> f'or cf> E Dv· Since '1i = D\r• we must have o = (g,cp) = i(Vcf>-cf>,cf>) •. Thia implies (Vcp,cp) = (cp,cf>) = 14>1 2

= IVcf>l • lcf>I. Thia is only possible if' Vcf> = laj> f'or a constant

k. If' g f 0, cf> f e and (Vcp,cf>) = (cp,cp) implies k = 1. Thus

V<j> = cp or (V-1 )cp =0. Thia implies g = i(V-1 )cf> =0 contrary

to our supposition. Thus g = e and '1i ·D_a: = le!. The proof'

of' '1_ 1 ·DH= le! is similar.

PROOF OF ( d) • Let ~ denote the graph of' H and "* de­

note the set of' pairs !f',H*f'!, i.e. the graph of' H*. Since

H is symmetric, we have ~ C ~. Consider ~··~* and let us

suppose that !h,H*h! is in ~··~*. We have f'or every f' in

the domain of' H,

(f',H*h) = (Hf',h),

o = ( !f',Hf'J, !h,H*h!) = (f',h)+(Hf',H*f').

Thia implies that f'or every cf> in the domain of' V,

(~i(V-1 )cf>,H*h)-(~(V+l )cp,h) = o,

(~i(V-1 )cf>,h)+(~(V+l)cf>,H*h) = o.

A simple calculation will show that these equations are equiva­lent respectively to

(Vcp,H*h-ih)- (cf> ,H*h+ih) = o,

(Vcf>,H*h-ih)+(cp,H*h+ih) o,

and these equations are equivalent to

(V<j>,H*h-ih) = o,

(cf>, H*h+ih) = o.

Thus if' -2ig1 = H*h-ih, 2ig2 = H*h+ih, g 1 is 1n '1_i = ~~ ,

g 2 is in '1i = Dv , h = g1 +g2, H*h = -ig1 +ig2 • Thus if'

{h,H*h! € ~*~, h = g 1+g2 , g 1 E ~-i' g2 E '1i' H*h = -ig1+1g2 • On the other hand if' H = g 1+g2, h* = -1g+ig2 , revers~

§ 1 • THE CAYLEY TRANSFORM 115

the above discussion, will show that g1 E n_i and g2 E ~i 1:mplies

(f',h*) = (Hf',h),

0 = ( lf',Hf'l, {h,h*l ),

f'or every f' in the domain of' H. Theorem II of' §2, Chapter IV, shows that H*h exists and equals h*. Thus we may con­clude that lh,h*l E ~*~1 •

If' k is in the -domain of' H*, lk,H*k l is in ~* and lk,H*kl = {f',Hf'l+{h,R'*hl .where {f',Hf'l E ~. {h,H*hl E ~'.~*. by Corollary 1 to Theorem VI of' §5, Chapter II. From the above, we obtain k = f'+h = f'+g1+g2 where f' E 1>ii:• g1 E ~-i' g2 E n1 ,

and H*k = Hf'+H*h = Hf'-ig1+ig2 • Thus every element k in DH* is in the desired f'orm and the converse is also readily shown when our previous results are used. Furthermore, the given f'or­mula f'or H*k holds.

This completes the proof' of' the Lemma. We may now state:

THEOREM I. If' H is closed symmetric, there exists a closed isometric VH called the Cayley Transf'orm, having the properties (a) to (i) of' Lemma 2 above. If' V is closed isometric and such that ~V-l is dense, then there exists a symmetric H having properties (a) to (d) above.

COROLLARY 1. A closed symmetric H is self'-adjoint if' and only if' VH is unitary, i.e. lltr = !Rv = l0l.

If' vH is unitary and Dv = lel, ~\r = lel, (d) of' Lemma 3 shows that the domain of' H* is s1:mply that of' H. Since HEH*, we must have H = H*.

If' VH is not unitary, either Dv or !Ry.; lel. Let us suppose that g1 1' e is in D{r. By ( c) of' Lemma 2 above, g1 is not in DH. However (d) of' Lemma 2 above shows that g1 is in DH*' Thus H* f H.

COROLLARY 2. If' H is closed symmetric, H has a maximal syiometric extension. (Cf'. Def'inition 3 of' §3, Chapter IV). H has a closed self'-adjoint extension if' and only if' Dy has the same dimensionality as !RV.

116 X. SYMMEIT'RIC OPERATORS

PROOF: If' VH is such that either !R{r or Dy = le!, then VH has no isometric extension and it f'ollows f'rom (d) of' Lemma 2 above that H is maximal SJ'Illllletric. Thus we may consider the case where both ~V and Dy are not f0l. For convenience let us assume that Dv has dimensionality less than or equal to !Rv. By using Lemma 7 of' §2, Chapter VI, we can f'ind an iso­metric V' with domain Dv and range included in !Rv· Lemma 8 of' §2, Chapter VI shows that v1 = V ~ V' is an isometric trans­f'ormation such that V1 ::> V, Dy1 = n. Since V1 ::> V, !Rv1_ 1 ::> :Rv- 1• Since the latter is dense, !Rv1_1 is also and Lemma 4 above shows that there is a SJ'Illllletric H1, whose Cayley tra.ns­

f'orm is V1• Since Dv1 = n, H1 must be maximal SJ'Illllletric as we remarked above. Since V 1 ::> V, Lemma 2 ( d) above shows that H1 is a proper SJ'Illllletric extension of' H. A similar argument

holds if' the dimensionality of' D{r is greater than that of' !R~l'

with however the result that :Rv1 = n. V has a unitary extension v1, 1i' and only if' the dimension­

ality of' :Ry is the same as that of' :DV. ( Cf'. Lemma 1 o of' § 2, Chapter VI) • Since !Rv _ 1 is dense, we see f'rom Corollary 1 above, that H1 can be taken as self'-adjoint if' and only if' the

dimensionality of' :Rv equals that of' llv· We have also shown:

COROLLARY 3. H is maximal SJ'Illllletric if' and only if' at least one of' the D{r or !Ry consists of' e alone.

§2

In this section, we present an analysis of' ma.ximal SJ'Illllletric opera.tors, obtaining both structural and existential results.

DEFINITION 1. Let cp0 , cp1, cp2 , •.••• be a complete orthonormal set in n. (Cf'. the end of' §6, Chapter II.) Let V 0 be the transf'ormation defined by the equation,

V0 (I;:,0 acxcpcx) = r::0 acx<l>cx+l" Let E denote the projection on m = m(!cp1,cp2, ... }), i.e. the range of' V0 • (Cf'. §6, Chapter II, Theorem XI).

LEMMA 1 (a):

m0 : (b): V0

v 0 is isometric with domain n and range is partially isometric with initial set n

§2. STRUCTURE OF MAXIMAL SYMMEI'RIC OPERATORS 1 1 7

-1 -1 and f'inal set m. V0 * = V0 E, V0 *V = V0 V = 1, V0V0 *

= E: (c): ~-l and !Rv_ 1_1 are dense: (d): (V0 -1 )-1 and (~9- 1 -1 )- 1 exist and (V0 - 1+l)(V0 - 1-1 )- 1 = -(V0 +1)·

(V 0 -1 ) •

(a): is a consequence of' Lemma 7 of' Chapter VI §2.

(b): f'ollows f'rom Def'inition 1 and Lemma 1 of' §3,

Chapter VI.

( c) : We f'irst show that !Rv 0_ 1 is dense. By Theorem

VI of' §2, Chapter IV, we have (!Rv 0_ 1 )• = '1v 0 *- l • Thus if' f' E (!Rv0 _ 1 )•, (V0 *-1 )f' = e or V0 *f' = f'. Since f' E ~. f' = a 0$ 0 + a 1$ 1 + .•••. (Cf'. Theorem XII of' §6, Chapter II). We have V0 *f' =

-1 -1 Vo Eo(ao$o+a,~,+ ••• ) =Vo (a1$1+a2$2+ ••. ) = a1$o+a2$1+ ••.. Thus V0 *f' = f' implies a 0 = a 1, a 1 = a 2 , •.•• etc. Since

I:CD 0 I a 12 <co, lim a = o, and hence a = o f'or er: = o, ex= ex a-+ CD ex ex 1 , . • • • Hence f' = 9. Thus f' E (!RV _ 1 )' implies f' = e

0 and hence !Rv 0_ 1 is dense.

!Rv -1_ 1 is also dense. Consider !Rv *-E and suppose g E 0 -1 0 -1

!RVo*-E" Then g = (V0 *-E)f' = (V0 E-E)f' = (V0 -l)Ef'. Thus

g E !Rvo*-E implies g E !Rv0 -1_ 1 or !RV*-E c !Rv -1_ 1 • Hence (!Rv -1 ) ... c (!Rv "'- )• =7lv -E by Theorem VI o~ §2, Chapter o -1 o*-""'O o IV. Thus f' E (!Rv0 -1_ 1 )1 implies f' E '1vo-E or (V0 -E)f' = e or V0 f' = Ef'. Letting f' = a 0 $ 0 +a1$ 1+ •••• as bef'ore we obtain

V0 f' = a0~1+a1$2+.... while Ef' = a 1$ 1+a2$ 2+.... Since V0 f' = 2 Ef' we must have a 0 = a 1, a 1 = a 2 , • . • • etc. Thus I:exlaexl

< co again implies a = o, ex = o, 1,... and f' = e. Thus

f E (!Rv0 -1 _1 )• implies f' = 9 and !Rv0 -1 _1 is dense. -1 (d): Lemma 3 of' §1 above shows that (V0 -1) and

(V 0 - l -1 )- 1 exist. Now V 0 _, -1 is def'ined only on m and thus V - 1-1 = V - 1-v V -i = (1-V )V _, since V V -l = 1 on

0 0 00 9 9 90 "1. Thus (V0 - 1-1 )-l = ((1-V0 )V0 - )- = V0 (1-V0 )- , and (V0 - 1

+1 )(V0 - 1-1 )- 1 = (V0 - 1+l)V0 (1-V0 )- 1 = (l+V )(1-V )-l = -(V +l)• -1 0 0 0

(V0 -1) and this completes the proof' of' the Lemma.

LEMMA 2. Let H0 correspond to V 0 as in Theorem I

of' the preceding section. Then H0 is maximal symmetric and -H0 corresponds to v 0 - 1 •

Since !Rv0 _ 1 is dense, there is a symmetric H whose Cayley

11 8

transf'orm is V 0 • maximal SJ'1Illlletric. corresponds to v;1

X. SYMMEI'RIC OPERATORS

Corollary 3 of §1, above shows that H0 is Lemma. 1 ( d) and Lemma. 4 of § 1 show that - H0

LEMMA 3. Let V be maximal isometric, i.e. either Dv= n or !Rv= n. Then we can f'ind a f'inite or infinite set of

manifolds mo, ml, m2, • • • with corresponding projections

E0 , E1, E?, • • • having the properties: 1. Eel = VEa = EaVEa

1 = Iet=OEa, V = Ia=OEaVEet. 2. If m0 =f !9!, then E0VE0 when considered on 77\, alone is unitary. 3. If m0 + n and (a) : Dy = n, then for mcx' a ~ 1 , we can find an orthonormal

set <l>a, 0 , <l>a, 1 , • • • such that m( l<l>a, o•<l>a, 1, ••• J) = ma

and EaVEa(r~0al3<1>a,~) = r~0a~<l>a.~+l' that is Vac on ma is precisely analogous to V0 above. If m0 + n and (b): !Rv = n, then for mcx' a ~ 1 , we can find an orthonormal set

<l>a,o•<l>a,1' ••. such that m(!<l>a,o•<l>a,l' ••• !)=ma' EaVEet is def'ined for m( !<I> 1 ,<1> 2 , .•• ! ) and EVE (r"'=1a,,,<1> ,,,) a, a, a a ..- . ,. a, 1•

~1 a13<1>a,~-1 •

We notice first that if v is such that !Rv = n then Dv-i = n. Furthermore one can readily show that 1 , 2, and 3 (a) for v- 1

imply 1, 2, and 3(b) for V. Thus we will consider only the case in which Dv = n since the other case is a consequence of this result.

Suppose then that Dv = n. If also !Rv = n then V is uni­tary and we let '110 = n. ma is undefined for et ~ 1 • We must still consider the case in which !Rv + n. Let 7l = !Rv + !9!. We can find an orthonormal set "'1 • lji2 , • • • such that 7l= m( !lji1, lji2, ••• J). Cf. Theorem XI of §6, Chapter II. We note also that V is partially isometric, with initial set n and final set !Rv =7'1~. Thus if F istheprojectionon 7'1, V*=V- 1(1-F) (Cf. Definition 1 and Lemma. 1 of § 3, Chapter VI) •

Let <1>a,l3 = v131jia for 13 = o, 1, and every liia• Since <l>a,~ is in !Rv for 13 = 1, 2, • • • , we must have (<l>a,l3' liia)=O for 13 = 1, 2, • • • • If 13 2 1 and 6 2 1, we also have ( <l>et,l3'

) -(~ 6 _ ~1- 6-1 _- ) <l>-y, 6 - v ilia,v "1-y) - (V "'°''v lii-y) - (<1>a,~- 1 ,<1>-y,c5-l • This

implies that if 13 > 6, (<l>a,13•<1>-y,6) = (<l>cx,13-6'<1>-y,o) = (<l>cx,~-6' "1-y) = o, while if 13= 6, (<1>a,(3'<1>-y,c5) = (<l>a,o'<l>-y,O) = (liia, "1-y) 6cx,-y. Thus the <l>cx,~ form an orthonormal set.

§2. STRUCTURE OF MAXIMAL SYMMEll'RIC OPERATORS 119

Let mcx = m( l 4>a,o' 4>cx, 1' .•. ! ) for every ex such that 'i'cx is defined. From the above, it is readily seen that the ma's

a.re mutually orthogonal. Since V( Ij3=0~4>«,~) = I:(;'=0a.134>cx,f3+l' VECX = ECXVECX. Furthem.ore V*( r~=oa.134>cx,13> = v- 1 (1-F)( I:B=Oa.(34>cx,(3) = v-l I:f3'=la(34>cx,(3 = I:~=la(34>cx,(3-l. Thus V*Ecx = EcxV*Ecx. Ta.king adjoints, we obtain EaV = EaVEcx = VEa.

By Lemmas 5 and 7 of §1, Chapter VI, I:cx=lEa is a projection with range 7n(m 1 um2u ... ) = m(l<t>a,(31). Since EaV=VEcx' forming sums and if necessary ta.king limits, we obtain ( :rcx=lEcx)V

= V( I:a=lEa). Let E0 = 1- I:a=lEcx. Then VE0 = E0V = E0VE0 ,

and 1 = I: a=OEa. F'urthermore V = l·V = ( I:a=OEcx)V = I:a=OEaV = I: cx=OEcxVEa: The properties listed in ( 1 ) have now been estab­lished.

The range or l-E0 is m( l4>cx,(3!) and hence includes m ( l4>cx,o I) = m( lqicx!) = '1 = !Rv . By Lemma. 3 or §1, Chapter VI, we see then that m0 , the range 01· E0, is included in !Rv· Hence E0V has the same range as E0 • Since E0V = VE0 , we

a.ea that !Rwo = m0 • Since J5QV = VE0 = E0VE0, we see that v regarded only on mo has range mo and hence is unitary with respect to m0 if the latter is not simply f 01. (Natur­ally mo can be finite dimensional, but the above discussion applies in that case also). Thus we have shown (2). (3) (a.) is quite obvious under these circumstances.

I.m.MA 4. Let V be isometric., with !RV- l dense and let H be the corresponding S'YDJlnetric opera.tor. Let E be a. projection with range m, such that EV = VE. Let V' be the contraction of V, with domain llv·m. Then V' is closed isometric and !RV'-l is dense in m. Re­garded a.s a. transformation within m, V' has a. corres­ponding symmetric transformation H' which is the contra.c­t ion of H with domain :DH. m.

Since »v·m is additive and closed, V must be additive and closed, since V is continuous. (Cf'. the comment preced­ing Def'. 6 of §1, Chapter IV). Since V1 is additive and a contraction of V , it muet be isometric. (Cf. Definition 2 of §2, Chapter VI).

Now !RE(V- 1 ) is dense in m, since !RV-l is dense in f;.

120 X. SY),f,4Eil'RIC OPERATORS

Also !RE(V-1) = ~-E = ~-E = !R(V-1 )E = !Rv•-1. Thus !Rv•-1 is dense in m.

Let H' denote the symmetric transf'ormation on m, which corresponds to V' regarded as a transf'ormation on m alone. Let f' be in the domain of' H' • Then correspondingly we will have a <!> in Dv·!m such that f' = ~i(V 1 -1 )<!> = ~i(V-1 )<j>. Hence f' is in the domain of' H and we a1 so have Hf' = -21 (V+l )<!> = 1 . 2(V 1 +1 )<j> = H'f'. Thus H' is included in the contraction of' H with domain DH·!m.

On the other hand if' f' is in ~·m, we have f'or a <I> E Dv, f' = li(V-1)<1> = Ef' = li(EV-E)<j> = li(VE-E)<j> = li(V-1 )E<j>. Since

2_1 2 2 2 (V-1) exists, (Cf'. Lemmas 3 and 4 of' the preceding section),

we must have <I> =E<i> and <I> E Dv·m. Thus f' = ~i(V-1 )<!> = ~i· (V'-1 )<!> is in the domain of' H'. Thus f' E DH.m implies f' E ~' and H contracted to DH·!m cannot be a proper ex­tension of' H'. This and the result of' the preceding paragraph imply the conclusion of' the Lemma.

We also have by ( i) of' Lemma 2 of' § 1 above., EH C HE, Thus

1f' f' E ~, Ef' is in DH and Ehf' = HEf'. If' f' E Dii·!m , this means EHf' = Hf' = H'f'. Combining these statements, we get that for eveey f' E ~, Ef' E ~·!In and EH(Ef') = H'Ef'. Furthermore EHf' = HEf' implie~ EHf' = E2Hf' = E(EH)f = E(HE)f = EHEf' = H'Ef'. Thus we have shown the corollary·:

COROLLARY 1 . If' E, H and H 1 are as in Lemma 4 above and f E DH, then Ef' E ~-m and EHf' = EHEf' = H'Ef' .

If' · H is maximal symmetric, VH = V is maximal isometric and either Dv = f> or !RV = f>. If' Dy = f>, we have m 0, m1, •.• and E0 , E1, .•.• as in Lemma 3 above, and if we apply the Corol­lary to Lemma 4 above, we get that· for every f E DH, Hf' =

( I:a=OEa) 0 H = I:a=oH(a)Ec/, where H(a) is H contracted to

DH. ma and is considered as a transformation oil !ma. ( 2) of Lemma 3 above, and Corollary 1 to Theorem I of the preceding section and Lemma 4 above show that H( 0 ) is a self-adjoint

transformation if'· !m0 =} 191. On the other band, H(a) for

ex = 1 , 2, • • • , when they are defined are each analogous to H0 of Lemma 2 above, with respect to the orthonormal set <l>a, o, •a. 1' ..•••

§2. STRUCTURE OF MAXIMAL SYMME.rRIC OPERATORS 121

If' !RV = n, the situation is similar except that H(<x) f'or a= 1, 2, ... , if' there are a:n:y such, are each analogous to -H0 of' Lemma 2 above with respect to an orthonormal set ~a,o•

~a, 1' Thus we may state

THEOREM II. Suppose H is maxilllal SJ'Dlllletric. Then there exists mutually orthogonal manif'olds, "lo' ml, m2, with projections, E0, E1, ... respectively, such that

r a=oEa = 1 and such that EJI C HEa f'or every Ea. Let H\aJ denote the contraction of' H with domain DH·m (a) and which is regarded as a transf'ormation on ma- Then f'or every f' E ~· Hf' = r a=OH( a)Eaf'. Furthermore if' m 0 f {9 l, H( o) is self'-adjoint. If' m 0 f n, we have at least one ma f'or an a ~ 1 and two cases are pos­sible:. (a) If' V is the Cayley transf'orm of' H, then Dv = n and H(a)f'or the a L 1, is such that there is an orthonormal sequence !o+o• ;a 1, • . • f'or which H(a l is analogous to the H0 of' Lemma' 2 or (b) !Rv = n and H( a)

f'or the aL 1 is such that there is an orthonormal se-quence, ~~o•~a,l•··· f'or which H(a) is analogous to -H0 of' Lemma 2.

The converse to this result can also be given. Let a n\Dllber of' Hilbert spaces, no, n,, n2, .•. be given,

and consider a self'-adjoint H(o) in n0 and realize H0 as a H( a) in each na f'or a= 1, 2, • . . We may f'orm n = n0 e n1 e .• as in Def'inition 1 or 2 of" §3, Chapter III. Let

(0) (1) H!f' 0 , f' 1, •.• j = IH f' 0,H f' 1, .•. l when both sequences exist and are in n. Now if' V(a) is the Cayley transf'orm of' H(a),

we know f'rom Lemma 8 of' §2, Chapter VI that V = V 0 • v 1 •· •.• is isometric with D;'f = n. One can also readily show that Vlf'0 , f'1 , ..• l = !V(o f'0 , vC 1 )f'1, ... j, that ·~-l is dense, and that H corresponds to V as in Theorem I of' § 1 above. Thus H is maximal SJ'Dlllletric.

If' we take H(a) as a realization' of' -H0 , we have !Rv = n and H is again maximal SJ'Dlllletric. Thus we have:

COROLLARY 1 . If' H is constructed as in the above, H is ma.xilllal SlJlll!etric.

CHAPI'ER XI

REFERENCES TO FURTHER DEVELOPMENTS

Our main purpose in this Chapter is to give rererences in a nmnber or topics ror rurther reading. We will also give a brier heuristic introduction to each topic.

Our rererences will be nmnbered as they are introduced. Two essential rererences are the rollowing:

( 1 ) M. H. Stone. "Linear Transformations in Hilbert Space". Amer. Math. Soc. Colloquimn Publications. Vol. XY, New York, N.Y. (1932).

(2) J. v. Neumann. Princeton Lecture Notes, ror the years, 1933 -34, 1934 - 35.

§1

In the rootnote to Derinition 1 or Chapter VII, §2, we in­dicated two dirrerent kinds or resolutions or the identity. Other types are possible; ror instance: Let E2(ll.) denote the example (b) in this rootnote and let $(h) denote a mono­tonically increasing continuous runction with $(h) = o ror h ~ o, lt>(ll.) = 1 ror II. ~ 1, and lt>(ll.) has variation zero on the complement or a closed set or measure zero, in the in­terval o ~II. ~ 1. Then F(ll.) = E2(1t>(ll.)) orrers another ex­ample or a resolution or the identity.

Furthermore combinations or these cases occur. For instance: Let E1 (ll.) denote the example (a) or the rootnote rererred to in the previous paragraph and E2(ll.) denote the example (b). Sup­pose that these are realized in the two spaces fii and n2 re­spectively. Then in n1 $ n2, we may consider the projections derined by the equation G(ll.)fr1 ,r2! = !E1 (11.)r1 ,E2(11.)r21. This again rorms a resolution or the identity.

Since E1 (ll.) varies only on a discrete set or points, E1 (ll.) is said to have pure point spectra. E2(ll.) and F(ll.) have what is termed continuous spectra, while G(ll.) has a mixed spectra.

122

§2. THE OPERATIONAL CALCULUS 123

However all the examples cited above have one property in common. Let Uf denote the set of elements g for which there is a function <j>(i\) such that g = f:,,<l>(i\)dE(i\)f. Now in each case, there is an f such that mf = [UrJ = n. Therefore, these resolutions are said to have simple spectra.

But this is not the case in certain other examples. Sup­pose we consider n1 e •.. $ fin as in definition 1 of §3, Chapter III, and let E2 (i\) be realized in each fioc, cc= 1, .• ,n. Let G2(i\)!f1 , ••• ,fnl = !E2(i\)f1 , ••• , E2(i\)f'nl· Then the least number m such that there are m elements, f 1 , •• ,fm

such that mf', e e mfm = n is n and thus G(i\) is said to have n'tuple spectra.

A complete analysis of these possibilities is given in Stone's treatise, reference ( 1 ) above in the following· places·: Chapter IV, §2; Chapter V, §5; Chapter VI, §1; Chapter VII. (It is believed that these can be read in this order, by a person familiar with the material of this book).

§2

The Operational Calculus

If p(x) is a polynomial and H is self'•adjoint we can de­fine p(H) = 8nHn+8IJ._,Hn-1+ ••. +a0 = S~00 p(i\)dE(i\) where E(i\) is the resolution of the identity corresponding to H. (Cf. Theorem II of §2, Chapter IX). If <I> (x) is continuous, we can define <l>(H) = S:O<l>(i\)dE(i\). (Cf. Theorem II of Chapter VII, § 3) • Lemma 4 of § 3, Chapter VII and Lemma 5 of' §2, Chapter VII show the connection between the properties of these operators and the corresponding properties of the function <j>(x) itself'.

We have considered these only for cont±nuous <j>(x). However the equation

(Hf,g) = s.:, <l>(i\)d(E(i\)f,g)

offers certain possibilities for generalizations. (Cf'. Lemma 2 of §3, Chapter VII). For f fixed, this determines the conju­gate of' an additive functional of g. When the functional is botmded, there corresponds to this functional an element Hf. (Cf. Theorem IV of §4, Chapter II). Thus far we have considered only the possibility of a Ri.ema.nn-Stieltjes integral. However,

124 XI. REFERENCES TO FURTHER DEVELOPMENTS

if' we consider Radon-Stieltjes integrals, we can def'ine the above integral expression f'or a wider class of' f'unctions ~(x).

This is done in Ref'erence (1) in Chapter VI. For bounded operators these questions are considered f'rom a

dif'f'erent point of' view in

(}). J. v. Neumann. operatoren" • 226 (1931).

"Uber Funktionen von Funkt1onal­Annals of' Ma.th. Vol. 32, pp 191~

A direct generalization is given in the f'ollowing two papers of' F. Maeda. Maeda does not interpose the numerical integral.

(4). F. Maeda. "Theory of' Vector Valued Set Functions". Jour. of' Soc. of' the Hiroshima Univ. Vol. 4. pp. 57-91, and pp 141-160.

§3 Commutativity and Normal Operators

Def'inition 1 of' §4, Chapter VII applied only to the case in which one of' the operators is self'-adjoint. F0r linear opera­tors, an obvious extension is possible but f'or unbounded opera­tors, certain dif'f'icult1es in the domains appear. In the more general· case, the notion of' commutativity has been discussed f'rom a number of' points of' view. For instance, if' A is linear, we may def'ine commutativity by the inclusions, AB C BA, A*B C BA*. This is discussed in Chapter 1 4 of' ( 2) above and also in

( 5). J. v. Neumann. "Zur Algebra der Funktionalop­eratoren". Ma.th. Annalen. B. 102, pp. 370-427. (1929).

In this connection, we would also like to ref'er the reader to §1 of' Chapter VIII of' the ref'erence (1).

A similar situation holds with respect to normal operators in the general unbounded case. Various def'initions are given in (5), (p. 406), (1) Def'inition 8.3;

(6). J. v. Neumann. "On Normal Operators". Proc. of' the Nat. Acad. of' Sc. Vol. 21, pp. 366 - 369. (1935),

( 7). K. Kodaira. "On Some Fundamental Theorems in the Theory of' Operators in Hilbert Space". Proc. of'

§5. INFINITE MA.TRICES

the Imp. Acad., Tokyo. Vol. 15, pp. 207 - 210. (1939). (It is this def1n1tion which we have used.)

125

These def1n1tions are all equivalent, since they can be shown to be equivalent to A having the integral representation A = Ss0ZdE1(P) of Theorem III of §3, Chapter IX.

The theory of nonna.l operators can be developed much further. We have an operational calculus for the .ftmctions of a single nonna.l operator. (Cf. (1), Chapter VIII, §3).

§4

S-ymmetric and Self-adjoint Operators

There are a number of topics in the study of symmetric oper­ators, which we haven't discussed. We refer the reader to the matters discussed in §2 and §3 of (1), Chapter IX. These deal with the abstract significance of "realness" as applied to oper­ators and also the possibility of approxilllating symmetric oper­ators by bounded s-ymmetric operators.

Another develop!llent having practical significance is the re­sult given in:

(8). K. Friedrichs. "Spektraltheorie Ha.lbbeschrt!nkten Operatoren". Ma.th. Ann. B. 109, pp. 465 - 487. (1934).

This paper describes a general method for obtaining a self­adjoint extension of a symmetric operator, which is bounded be­low. Le. C_ ) - ex>.

It has also been shown that one can construct two s-ymmetric operators H1 and H2 so that their domains have only 0 in common. This significant result is given as Satz 15 in:

( 9). J. v. Neumann. "Zur Theorie der unbeschrt!nkten Ma.trizen". Jour. f. reine u. a.ngewa.ndte Ma.th. B. 161, pp. 208 - 236. (1929).

§5

Inf1n.1te Ma.trices

If T is a c.a.d.d. transfonna.tion, we can find a complete orthonormal set S, ~1 , ~2 , .•• in D.r (Cf. Theorem X of §6,

126 XI. REFERENCES TO FURTHER DEVELOPMENTS

Chapter II). We have f'or each such orthonormal set S an in­finite matrix (aa,~) with aa,~ = (T<l>a~<I>~). Furthermore, the inrinite matrix and the orthonormal set, will determine f'or each

a, the value of' T<!>a = r.~aa,~<1>~ U r.alaa,~1 2 < oo ._ However the f'ollowing possibility may occur. Let S denote

the set of' pairs !<l>a,T<l>a! in ne n. For a given S, T de­term1nes the matrix (aa,~) · and thus S, nevertheless '1!(S) may be a proper subset of' the graph of' T. An example can eas­ily be given. Let '.D consist of' all functions f' in S::2 in the f'orm a0+ So xg( Ud~ where g( ~) is also in S::2 • Let Tf' = g. One can readily see that T is c.a.d.d. The complete orthonormal set {exp (i21tnX)j,n = o, ± 1, ± 2, ••• is in D but the pair {ex-el-x,ex+e1-x!, in the graph of' T is orth­ogonal to the corresponding s.

This cannot happen for bounded operators T and f'or these a satisf'actory matrix theory exists. The reader is ref'erred to ref'erence (9) f'or a more general discussion and to (1) Chapter III, §1, which also contains an interesting historical co111111ent.

§6

Operators of' Finite Norm

A specialized but nevertheless interesting class of operators is the set of T's whose matrices possess the property that r: a,~ laoc,~1 2 < oo. These are said to be of' finite norm and are discussed in (1) Chapter II, §3, Definition 2:15 et seq., Chap­ter III, §2, and Chapter V, Theorem 5.14.

§7

Stone's Theorem

If' we have a f'amily of' unitary operators, U ( i\) def'ined f'or -= < i\ < oo having the properties that U(i\1 ) ·U(.11. 2 ) = U(.11. 1 +~)

and that (U(i\)f',g) is a continuous function of' i\ f'or every f' and g, then there exists a self'-adjoint H such that U(i\) = exp (i!UI). This result, which has many important appli­cations, is due to Stone:

( 1 o). M. H. Stone. "On One Parameter Unitary Groups in Hilbert Space". Annals of' Math. Vol. 33, pp. 643-648. (1932).

§8. RINGS OF OPERATORS 12'7

J. v. Neumann pointed out that we may replace the cond.1 tion of continuity for (U(h)f,g) by measurability.

( 11). J. v. Neumann. "Uber einen Satz von Herrn. M. H. Stone". Annals of Math. Vol. 33, pp. 567 - 573. (1932).

§8

"Rings of Operators"

We have been concerned up to now with matters which depend on the structure of a single operator, indeed on the structure of a single normal operator. The structure of certain sets of transformations is also interesting and has been studied recent­ly.

These investigations have been concerned with the infinite dimensiona:J_ equivalent of semi-simple matrix algebras. These sets are also the generalization of group algebras and are of interest because of Haar's result that any locally compact sep­arable topological group can be represented as a set of Uill.tary transformations in Hilbert space.

(12). A. Haar. "Der Massbegriff in der Theorie der kontinuierlichen Gruppen". Annals of Math. Vol. 34, pp. 147 - 169. (1933).

In reference ( 5) above, J. v. Neumann introduced the notion of ring of operators. A set of linear operators, ·M, is said to be a ring of operators if A E M and B E M imply aA, A*, A+B and. AB E M and if, furthermore, M is closed in a certain topology. This topology insures that the limit of a weakly con­vergent sequence of operators of M, is in M. Whether clos­ure in this topology is equivalent to this property, if M has the algebraic properties of a ring, is not known. If M does not have these algebraic properties, this topology is not equiv­alent to sequential closure.

This topology is desired, however, since rings closed under this topology have the following very interesting property. Let M' denote the set of linear operators which comnute with all A E M. Then if M is a ring containing 1, then (M') 1 = M. This result is proven in ( 5) in a slightly more general form.

128 XI. REFERENCES TO FUR'1'BER DEVELOPMENTS

Also important is the result proven in ( 5) and ( 3), that if' M is abelian, i.e. M' CM', then there exists a self'-adjoint H, such that evecy transf'onnation in M is a f'unction of' H.

In an unpublished work, Prof'essor von Neumann has also estab­lished what might be called the resolution of' a ring with re­spect to its center, M·M•. M·M' is abelian. Suppose its cen­ter contained only a f'inite number or mutually orthogonal pro­jections E1 , .•• , E\i and suppose that each Ea is minimal in M·M'. If' M contains 1, 1 = I: aEa and if' A E M, A = (I: ~a)A

= I:aEPa since Ea E M·M'. If' we consider the EP.a on ma, the range of' ElX, then these transf'ormations f'oI'lll a ring of' operators on ma whose center is simply fa· 1 I·

A ring f'or which M·M' = fa·1l is called a.f'actor. Thus M can be called the e sum of' f'actors. In the general case, the possibility of' a continuous spectrum f'or the H which deter­mines M·M' of'f'ers dif'f'iculty, .but Prof'. v. Neumann's result is that f'or a suitably generalized def'inition of' e sum, evecy ring M is the e sum of' f'actors.

Thus the analysis of' rings of' operators in general can be re­f'erred to in the study of' f'actors. From the corresponding re­sult in the f'inite dimensional cases, one would suspect that a f'actor must be isomorphic to the set of' all linear operators on Hilbert space or on a f'inite dimensional unitacy space.

But this is not the case in general as is shown in:

( 13). F. J. Murray and J. v. Neumann. "On rings of' operators". Annals of' Math. Vol. 37, pp. 116 -229. ( 1936).

Here one considers a relative dimension f'unction J:M(E) de­f'ined f'or the projections in M and having the properties that J:M(E1) ~ J:M(E2) if' and only if' there exists a partially iso­metric W in M, whose initial set is the range of' E1 and whose f'inal set is included in the range of' E2• When J:M(E1) = J:M(E2), W can be chosen so that the f'inal set is the range of' E2.

If' M is a f'actor isomorphic to all the operators on a Hil­bert space, J:M(E) takes on the values o, 1, 2, .•• , oo f'or various E's in M and only these values. A similar result holds if' M is isomorphic to the operators on an n-dimensional space. These cases are called respectively I 00 and. In.

§8. RINGS OF OPERATORS 129

There are, however, essentially, the range of'. :I\!(E); case II1 values a such that o ~ a ~ 1 , and IIIco where a = o or =.

three other possibilities f'or 1n which IMCE) asstnnes all

IIco 1n which o ~ a ~ =

Examples of' f'actors 1n case II1 and IIco were given 1n

( 13). An example of' a III00 was f'irst given 1n

(14). J. v. Neumann. "On rings of' operators III". Annals of' Ma.th. Vol. 41; pp. 94 - 161. (1940).

Let M be a f'actor 1n a case II1 and H = LgME(i\) be a self'-adjoint operator 1n M. The expression

T,. CH> = S~0~CE(i\) >

was def'ined 1n ( 1 3) and shown to have a ntnnber of' the properties of' a trace in the f'1n1te dimensional case. The property, T,. (H1 +H2) = TA(H1 )+T ,.(H2 ) was established in

( 15). F. J. Murray and J. v. Netnnann. "On rings of' op­erators II". Trans. of' the Amer. Ma.th. Soc. Vol. 41, pp. 208 - 248. (1936).

For a f'ixed a= 1, 2, ••. , co, all M in case Ia are isomorphic. However, in a f'orthcoming joint paper of' Prof'essor von Neumann and the writer it will be shown that not all II1 's are isomorphic.

The f'ollowing paper is also of' interest in connection with rings of' operators:

(16). J. v. Neumann. "On 1nf'1n1te direct products". Compositio Ma.th. Vol. 6, pp. 1 - 77. (1938).

An application of' the Rings of' Operators theory to a new development is given in:

( 1 7). F. J. Murray. "Bilinear transf'ormations in Hilbert space." Trans. of' the Amer. Ma.th. Soc. Vol. 45, pp. 474 - 507. (1939).

CHAPTER XII

REFERENCES TO APPLICATIONS

In this Chapter, we give a brief list of references to the applications of the theory which we have discussed.

§1

We wish again to refer the reader to reference (1) of the preceding Chapter, in particular to Chapters III and X. There the following topics are discussed: Integral Operators (in­cluding the Fourier transforros), Differential Operators, Oper­ators corresponding to flows (briefly), Jacobi matrices and Moment problems •

In connection with integral operators, mention should also be made of:

( 1 8). J. v. Neumann. "Charakter1s1erung des Spektrums eine Integraloperators". Actualites Sci. et In­dustrielles, 229, Herman et Cle, Paris. (1935).

This paper is concerned with self-adjoint operators, which may be represented as integral operators. It is characteristic or these operators that there exists no E) o such that E(e)­E(-E) has a finite dimensional range.

§2

Differential Operators

The nature of the domain of differential qperators and the possibility of s-ymnetric extensions have been investigated from a number of viewpoints. Perhaps the most natural continuation of the work of von Neumann and Stone is to be found in the papers,

(19). I. Halperin. "Closures and adjoints of linear differential operators". Annals of Ma.th. Vol. 38, pp. 880 - 919", (1937).

130

§4. CIASSICAL MECHANICS

(20). J. w. Calkins. "Abstract SJ'lilllletric boundary conditions." Trans. or the Amer. Math. Soc. Vol. 45, pp. 369-442. (1939).

131

The notions or ~ererence (8) or the preceding Chapter have been extended and applied to dirrerential operators in:

(21). K. Friedrichs. "Spektraltheories halbbeschr§.nk:­ten Operatoren. Zweiter Teil". Math. Annalen. B. 109. pp. 685 - 713, (1934).

(22). K. Friedrichs. "Uber die ausgezeichnete Randbe­dingungen in der Spektraltheorie der halbbe­schN!nk:ten gewabnlichen Dirrerentialoperatoren zweiter Ordnung". Math. AnnalenB. 112, pp. 1

( 23).

- 23, (1935}.

K. Friedrichs. Hilbert space". LXI, pp. 523 -

"On dirrerential operators in Amer. Jour. or Ma.th. Vol.

544, (1939).

Another paper dealing with dirrerential operators is:

(24). F. J. Murray. "Linear Transrormations between Hilbert spaces". Trans. or the Amer. Ma.th. Soc. Vol. 37, pp. 301 - 338. ( 1935).

§3

The very important applications or the notions or linear transrorma.tions in Hilbert space to quantum mechariics are ex­plained in:

( 25). J. v. Neumann. "Mathematische Grundlagen der Quantenmechanik". J. Springer, Berlin. ( 1932).

§4 The connection between the theory or Operators and the usual

Hamiltonian mechanics was pointed out in:

( 26). B. O. Koopman. "Hamiltonian systems and trans­rorma.tions in Hilbert space". Proc. or the Nat. Acad. or Sci. Vol. 17, pp. 315 - 318. (1931 ).

This connection is based on Stone's Theorem (Cr. rererence ( 1 o) and ( 1 1 ) , and has led to many interesting and important results. These are given in:

132

( 27).

XII. REFERENCES TO APPLICATIONS

B. O. Koopman and J. vonNeuma.nn. "Dynamical systems or Continuous Spectra." Proc. or the National Acad. or Sci., Vol. 18, pp. 255-263. (1932).

(28). J. vonNeuma.nn. "Proor or the quasi-ergodic hypothesis." Proc. or the National Acad. or Sci., Vol. 18, pp. 70-82 (1932).

(29). J. vonNeumann. "Zur Operatorenmethode 1n der klassichen Mechanik." Annals or Math., Vol. 33, pp. 587-642 (1932).

REFERENCES

References to the literature. are given on pages 3, 7, 15 and in Chapters XI and XII.

The following indices refer only to Chapters I to X.

14; 34.

* , 11; 34. - 1 , 31.

43. c , 32. a , 26.

, 4, 33.

U( ) , 1 6

c , 9.

c+, 41. c_ , 41.

c.a.d.d. , 36.

D , 34.

D0 , 75. d( , ) , 7.

IScc,(3 , 17.

J) , 4.

H0 , 76 or 117.

H( cc) , 1 21 •

INDEX OF SYMBOLS Non-litteral Symbols

, ) , 4.

I , 4.

] ., (the closure or a set or transformation), 32.

+ , 4, 33. m Ill , (the bound or a

transformation), 81.

Litteral Symbols

133

I , 87. I , 87.

£2 , 27. 12 , 23. lp , 23.

me ) , 16.

m_i' 110. mi , 11 o.

7l, 37. 7!(µ) , 85.

p 1 (H 1 ), 82.

INDEX OF SYMBOLS

P(H) , 82.

p(H) , 81.

Pµ(H) , 85.

Pµ(X) , 85.

Qµ(H) , 85.

Q.µ(x) , 85.

~ > 84.

S<t>(i\)dE(i\) , 69.

I , 31. e , 4.

INDEX OF TERMS

Adjectives are given in connection with the associated noun.

Adjoint, 34.

Bound, "C", 9, 33, 41.

c.a.d.d., 36. Category, rirst, 49. Closure (or a tra.nsrormation), 32. Commutes (tra.nsrormations), 77. Compactness,

local, 47. weak, 47.

Complement, orthogonal, 14. Completeness, 5.

weak, 45. Contraction, 32. Convergence (or a sequence), 5.

weak, 45.

Dimensionality, 59, D:>main, 8, 31.

Extension, 32 additive, 32. closed, 32. S'Ylillllet ri c , 4 o •

maximal s., 4o.

Function, additive, 8. continuous, 8.

c. at r 0 , 8. Functional, 8.

linear, 11.

space or l.r., 11.

Graph, 31.

Identity, (1·), 67. resolution or i., 67.

Integral, 69. improper, 73.

Inverse, 31 •

Ma.nif'old, linear, 9. m. or zeros (~), 37.

Norm, 6.

Operator (see transrorma.­tions).

INDEX OF TERMS 135

Partition, 68. marked, 68.

Perpendicular, 34. Postulate, 4.

A, 4; B, 4; B', 6; C, 4; D, 5; E, 5.

Process, Gram-Schmidt, 17, 18. Product, 33. Projeption, 51.

orthogonal, 53.

Range, 31. Representation, integral, 81. Resolution, canonical, 96.

Set

r. of the identity, 67. planar, 106.

additive, 9. final, 60. initial, 60. orthonormal, 16; complete 0.,21.

Space Banach s., 7. Hilbert s. , 4.

of linear f'unctionals, 11. Sum

of elements, 4. of tra.iisforma.tions, 33.

Symbol, Kronecker, 17.

Transformation additive, 32. adjoint, 34. bounded, 33. c.a.d.d., 36. closed, 32. closure, 32. commutes, 77. domain, 31. extension, 32. graph,, 31. inverse, 31 • isometric, 57.

partially i., 60, linear, 36. normal, 79. perpendicular, 34. product, 33. range, 31. self-adjoint, 38.

sum, 33. symmetric, 38. unitary, 56.

Transform, Cayley, 113.