An Introduction to Linear Transformations in Hilbert Space. (AM-4), Volume 4 9781400882267

Table of contents :
TABLE OF CONTENTS
Preface
Chapter I. INTRODUCTION
Chapter II. HILBERT SPACE
§1. The Postulates
§2. Linear Normed Spaces
§3. Additivity and Continuity
§4. Linear Functionals
§5. Linear Manifolds
§6. Orthonormal Sets
Chapter III. REALIZATIONS OF HILBERT SPACE
§1. Preliminary Considerations
§2. l2
§3. h1 ⊕ ... ⊕ hn and h1 ⊕ h2 ⊕ ...
§4. L2
Chapter IV. ADDITIVE AND CLOSED TRANSFORMATIONS
§1. The Graph of a Transformation
§2. Adjoints and Closure
§3. Symmetric and Self-adjoint Operators
§4. C.a.d.d. Transformations
Chapter V. WEAK CONVERGENCE
§1. Weak Completeness
§2. Weak Compactness
§3. Closed Transformations with Domain h
Chapter VI. PROJECTIONS AND ISOMETRY
§1. Projections
§2. Unitary and Isometric Transformations
§3. Partially Isometric Transformations
§4. C.a.d.d. Operators
Chapter VII. RESOLUTIONS OF THE IDENTITY
§1. Self-adjoint Transformations with H Finite Dimensional
§2. Resolutions of the Identity and Integration
§3. Improper Integrals
§4. Commutativity and Normal Operators
Chapter VIII. BOUNDED SELF-ADJOINT AND UNITARY TRANSFORMATIONS
§1. Functions of a Bounded H
§2. (Hf,g)
§3. Integral Representation of a bounded H
§4. Integral Representation of a Unitary Operator
Chapter IX. CANONICAL RESOLUTION AND INTEGRAL REPRESENTATIONS
§1. The Canonical Resolution
§2. Self-adjoint Operators
§3. Normal Operators
Chapter X. SYMMETRIC OPERATORS
§1. The Cayley Transform
§2. Structure and Existence of Maximal 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
Chapter XII. REFERENCES TO APPLICATIONS
§1. Integral and Other Types of Operators
§2. Differential Operators
§3. Quantum Mechanics
§4. Classical Mechanics

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ANNALS OF MATHEMATICS STUDIES NUMBER 4

AN IN TRO D U CTIO N TO LINEAR TRANSFORMATIONS IN HILBERT SPACE BY

F. J. MURRAY

PRINCETON P R I N C E T O N U N I V E R S I T Y PRESS LO N D O N : H U M P H R E Y M IL FO R D O X FO R D U N IV E R S IT Y PRESS

1941

Copyright 1 9 4 1 PRINCETON

UN IVERSITY

P rinted

PRESS

i n u . s .a .

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

PREFACE The theory of operators in Hilbert space 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 Imag­ ination, it is also satisfying because due to its present abstract methods, questions of necessity and sufficiency are satisfactorily 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 struc­ ture 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. An attempt however has been jnade 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 integral 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 offer so much. The second purpose was to emphasize those notions which seem to be proper to linear spaces and in partic­ ular to Hilbert space and omitting other notions as far as pos­ sible. 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 por­ tions of this work, except §k of Chapter III, can be read with­ out a knowledge of Lebesgue integration.

PREFACE On the other hand, for these very reasons, the present work cannot claim to have supplanted the well-known treatise of M. H. Stone or the lecture notes of J. von Neumann. It is simply hoped that the student will find 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 he neces­ sary to point out that the influence of Professor von Neumann 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 am also deeply grateful to my brother, Mr. John E. Murray, whose valuable assis­ tance in typing these lecture notes, was essential to their prep­ aration. Columbia University, New York, N. Y. May, 19^0 F. J. Murray

TABLE OF CONTENTS Page Preface Table of Contents . . . ...................

1

Chapter II. HILBERT SPACE ..................... §1 . The Postulates..................... . §2 . Linear Normed Spaces..................... §3 . Additivity and Continuity................ §*u Linear Functionals....................... ....................... §5 . Linear Manifolds §6 . Orthonormal Sets .......................

k b

Chapter I. INTRODUCTION

Chapter III. REALIZATIONS OF HILBERT S P A C E ........ §1 . Preliminary Considerations........... . . . §2 . 1 2 ....................... ............. §3 . frj $ ... 9 and ^ 9 ... ...... C2 .................................... Chapter IV. ADDITIVE AND CLOSED TRANSFORMATIONS . . . . §1 . The Graph of a Transformation............. §2 . Adjoints and Closure..................... §3 . Symmetric and Self-adjoint Operators ...... §1+. C.a.d.d. Transformations ................... Chapter V. WEAK CONVERGENCE..................... §1. Weak Completeness . ...................... §2. Weak Compactness ........................ §3. Closed Transformations with Domain ft. . . . . . Chapter VI. PROJECTIONS AND ISOMETRY............. §1. Projections .............................. §2. Unitary and Isometric Transformations......

6 7 11 1k 16 22 22

23 26

27 31 31 3 *i

38 ^2

^5 kj

^8 51

51 56

TABLE OF CONTENTS

_______ Page rage

Chapter V I. (Continued) (Continued.) §3. PPaarrttiiaallllyy Iso Isom metric etric TTran ransform sformation ationss . .. .. .. .. .. .. . § 4 . CC.a .a.d .d.d .d,.,. O Operators perators .............................................................. ............................................................. Chapter VVIIII. . RESOLUTIONS RESOLUTIONS OF OF THE THE ID IDEENNTTIT ITYY............................... .............................. §1 . S e lf - a d jo in t T ran sform ation s w ith H F in ite D i m e n s i o n a l ............................................ .............................. §2. R eso lu tio n s o f th e I dI de en nt it t it y y and. and.I nIte n te g rgartio a tio nn . . . . §3. Improper I n te g r a ls ................. ....... ......................... ....................... .... ........................ § 4 . Com m utativity and Normal OO perators perators ............................ BOUNDED SELF-ADJOINT AND UNITARY TRANSFORMATIONS.......................................................... §1 . F u nction s o f a Bounded HH ................................................. ................................................ §2. (H f,g ) ........................................................................................ §3. I n te g r a l R e p resen ta tio n oof f aa hounded hounded H .................... §4 . I n te g r a l R e p resen ta tio n oof f aa UUnnita itary ryOOperator perator . . . .

6 o 60

62 64 64

67 67 73 73

77 77

Chapter V I I I .

81 81 81 85

88 89 89

Chapter IX.

CANONICAL RESOLUTION AND INTEGRAL REPRESENTATIONS...................... ....................................... 95 §1 . The The CCanonical anonical RReso esolu lutio tionn ............................ ............... * . . . *. . . . . 95 95 §2. S e lf - a d jo in t O p e r a t o r s .........................................................1001 00 §3.

Normal O perators ...................................................................... ...................................................................... 1021 02

Chapter X. X. SYMMETRIC SYMMETRIC OPERATORS..................................................... OPERATORS.................................................... 110110 §1. The The CCayley ayley TTrraannssfo forrm m.............................................................. .............................................................110 110110 §2. S tru c tu re and and. E x iste n c e o f Maximal Symmetric O p e r a t o r s ................................................................................1161 1 6 Chapter §1 . §2. §3. §4 . §5 . §6. §7, §7. §8.

X I. REFERENCES TO FURTHERDEVELOPMENTS...................122 DEVELOPMENTS.................1221 22 Spectrum ...................................................................................122122 O peration al C a lcu lu s ........................................................ 123 123 Com m utativity and Normal O p e r a t o r s .............................. 124124 Symmetric T r a n s f o r m a t i o n s ................................................125125 I n f i n i t e M a t r i c e s ................................................................. 1251 2 5 O perators o f F in it e N o r m s ................................................1261 2 6 S to n e ’ s Theorem ........................................ . . . . . . . 125 125 Rings o f O perators .............................................................. 127127

TABLE OF CONTENTS Page Chapter §1. §2. §3.

XII. REFERENCES TO APPLICATIONS ......................... 130130 Integral and Other Types of O p e r a to r s .....................130130 ........................................... 130130 D ifferential Operators Quantum M ech an ics....................................................... 131131 §k. Classical M ech an ics................................................... 131131

CHAPTER I

The expressions: T ,f = J k(x,y)fCy)dy or Taf - p ( x ) i ^ f ( x ) + q ( x ) f ( x ) or in the case of a function of two variables, T f . d 2f . d 2f 3 d x 2% i are linear operators. Thus the first two, when applicable, take a function defined on the unit interval into another function on the same Interval. Now if we confine our attention to functions f(x) continu­ ous on the closed unit interval and with a continuous derivative, we know that such a function can be expressed in the form, f(x) = I00

x_,exp(2ttI«x)

where x = / f(x)exp(~2rtiax)dx. oc

j 0

T±f where ya =

f

If T.f Is of the same sort,

ya exp(2ni«x)

T.f exp(-27iictx)dx = 0

1

/ QT1 (exp(2Tiipx))exp(-2Tdax)cbc = I ^ . . ao xpa ^ a .

Now for T^ 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 equa­ tions by substituting a finite system and then passing to the limit, for example one migiht take the first n equations and Ignore all but the first n unknowns. But this process is in-

I. INTRODUCTION

2

effective in general and introduces certain particular difficul­ ties of its own. Other methods must he sought. The choice of the functions exp( 2niooc) corresponds to a choice of a system of coordinate axes in the case of a finite number of unknowns. In the finite case for a symmetrical operator, the coordinate system can he chosen, so that, a«,P -

o£a,f3

6

=

0

l f

“

+

a, a

(Cf. Chap.VII, §1, Lemma k ) Correspondingly in the infinite case we would seek a complete set of functions a such that Ta(x) = Aa 0. (5 ) (f,f) - o If and only If f = 0 . The not-negative real number (f,f)1/2 will be de­ noted for convenience by |f|. POSTULATE C. For every n, n = 1 , 2 , 3 , ... , there exists a set of n linearly Independent elements 4

§1 . THE POSTULATES of 55 ; that is, elementsf1, ... , fn such that the equation a ^ 1+ ... +&n£n = €> istrue only when a1 = ... “ a^ — o. POSTULATE D.55 is separable; that is, there exists a denumerably infinite set of elements of 55 , f2, ... , such that for everyg in 55 and every positive e there exists an n = n(g,c) for which |fn*~gI < e. POSTULATE E.55 is complete; that is, if a sequence of elements of 55 satisfies the condition

\fn \

> 0> m ’ n — then there exists an element

f

|f-fn l — >• o,

of

S

*‘0° such that

n — >■ oo.

In this statement of the postulates, certain notations and conventions were introduced. These are (a) -f = (-1 )f, (b) f-g = f+ (-l)g, (c) a denotes the complex conjugate of a. 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) |a| is the absolute value of a . The properties B( 1) - B(5) Imply B(6)

(f,ag) = a(f,g)

B( 7 ) (f^T+gg) = (f,g, )+ (f,g2 ) B(8) |af| = |a|•|f|

We shall next prove that these imply B( 9 ) l(f,g)l i IfI•IgI > the equality sign holding If and only if f and g are linearly dependent. For if r\, A and ^ are real and

p p A +yi + 0

then

Af+exp(irj)|ag is 0 for some choice of r\, A and if and only if f and g are linearly independent. Thus by B(4 ) and B(5 )

6

II. HILBERT SPACE (Af+exp(irOMg,Af+exp(ir|)|ig) ^ 0

and equality can only occur when f and g are linearly depen­ dent. Expanding by means of B( 2), B ( 1 ), B( 7) and B( 6) and using, with B(3) the fact that .for any complex a, a+i = 2R(a), we obtain A2|f|2+2 AnR(exp(-iri)(f,g))+M2|g|2 ^ 0 with equality possible only if f and g are linearly depen­ dent . Now we can choose r\ so that exp(--irj)(f,g) = -|(f,g) I. Then the equation becomes in the linearly independent case A2|f|2-2AM|(f,g)|+n2|g|2 > 0. Now for linear Independence, |g| + 0 and thus if we let Igl, M= l(f,g)l/lgl, we get |f|2-lg|2-|(f,g)l2 > 0. On the other hand if f and g are linearly dependent it is easily seen that the equality holds. B(10) |f+g| £ |fl+|g|, with equality possible only If f and g are linearly dependent. Proof: |f+g|2 = (f+g,f+g) = |f|2+2R(f,g)+|g|2 £ |f|2+2|f|-|g|+|g|2 - (|f|+|g|)2

§2

A weaker restriction than B is the postulate: POSTULATE B1. There exists a real valued function |f| of elements of with the properties B(4 ), B( 5), B(8), B(10). The function

|f| is called the norm.

If a space satisfies

§3. ADDITIVITY AND CONTINUITY

7

postulates A, Bf, 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 B(11)

|f+g|2+|f-g|2 = 2(|f|2+|g|2)

This equation is an immediate consequence of the equation |f±g|2 = |f|2+2R(f,g)+|g|2 Thus in Hilbert space, we have B1 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 possi­ ble 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 |f|, then d(f,g) - |f-g| is a metric for th§ space, i.e., satisfies the conditions (i) (ii) (iii)

d(f,g) > o, d(f,g) = o if and only if f = g. d(f,g) = d(g,f). d(f,g) £ d(f,h)+d(h,g).

These conditions are consequences of B(4 ), B(5 ), B(8) and B(10). * These spaces have been investigated In a famous treatise "Theorie des Operations Linealres." by S. Banach (Warsaw (1 9 3 2 )). ** J. von Neumann and Jordan, Annals of Mathematics, vol.36 (1 9 3 5 ), PP. 719 -7 2 4 .

II. HILBERT SPACE

8

Thus we are invited to Introduce the notion of continuity in such a space. DEFINITION 1 . Let F(f) be a function defined on a subset of £. This subset is called the domain of F. Let fQ be an element of the domain of F. If for every e > 0, it is possible to find a 6 such that if f is in the domain of F and |f“f0l < then |F(f)-F(fQ)| Since the values of F (fn ) are In a complete space, they must converge to an f*. Any two sequences ff^J and ff^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 taken alternately from one and. then the other sequence of F(f) fs would 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 con­ ditions (a) and (b) determine [F] precisely. To complete our proof It is only necessary to show that [F] 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 |F(f )| £ CI f | for every f in the domain of F, then I[F3(f )| • oo and furthermore, this approach to zero is uniform on those f ’s for which |f| = 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 |F(f)| £ C-|f| for every f € £ . Now given € , take N so large that for n and m )> N, |Fn“Fm l < € . This means that we have |Fn(f)~Fm(f)i ' i €|fi* Let us fix m, and let n — > oo . We then obtain |F(f)-Fm(f)| £ «|f|. Thus for m > N, |F-Fm | < e. This implies that F is such that Fm — »-F as m — ► oo and hence that £* is complete. THEOREM III. The set £* of linear functionals on

II. HILBERT SPACE

12

a Banach space £ is again a Banach space.* Now one of the essential facts concerning Hilbert space is that fi* is equivalent to . The specific relation Is given by the following theorems. THEOREM IV. If F is a linear functional defined on the Hilbert space , then there exists a g € such that for every f € fi, F(f) = (f,g). Proof: If F =* o, we can let g = 0 . Suppose then that IF| y o. if we are given a sequence of positive numbers [en ! with €n — > o, we can find a sequence {g^i of elements such that IFI-Ig^l > iP(g^) I > (1-*^) |P[ -IfiE^I and F(g^) + 0. If we multiply g^ by l/lFCg^)! we obtain a sequence g^ with F(gn) = 1 and iF M g J

Now consider

I

1

^ O -^ M F M g J .

|8n+gm I • We have

IFI •I gjj+gmI i lp(gn+Sm)l = 2

1

lp l •(1 -«n ) •Ig J + IFI (1 ~€m)

• Ig^l

or ^^n+sm^ ^ ^1~€n ^ ^ +^1~€m^ *

*

Thus ISn’Sm'2 = 2(IStll2+ISml2)-ISn+Sml2 i 2( lgnla-Hl.3m I2)-((1-en)* lgnl+(1-em)-|gml)2 * A proof of the fact that Postulate C for £ implies C for £* can readily be given if the Hahn-Banach Extension Theorem is shown (Cf. Banach loc. cit. pp. 27-29). This has the conse­ quence that If F Is additive and continuous on a linear subset G-| 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 o we have Ign-gJ2 — * o as n and m — > oo . Hence the g^’s form a convergent sequence. ttefine g so that g^— ► g. Then |g| = 1/|F|, F(g) = 1. Now if his such that F(h) = 0, we have that 1 = |F(g)| = |F(g+Ah)I £ IFI •Ig+Ah| = |g+Ah|/|g|

or for every A , |g| i

|g+Ah|.

Squaring we must have Igl2 i

|g+Ah12 = |g|2+2 R(A(h,g))+|A|2.|h|2.

Now we can choose A so that 2R(A(h,g)) - -2|A| •|(h,g) |. Thus Igl2 i lg|2-2r)*l(h,g)|+ri2|h|2 for every n > 0. But this is possible only if |(h,g)| = 0. Thu3 if F(h) = 0, (h,g) = 0. If h is arbitrary, h = F(h)g+hf where F(h!) = F(h-F(h)g)== F(h)-P(h)*F(g) = 0. Let gQ = (1 /Ig|2)g. Then (h,gQ) = (P(h)g+h',g0) = F(h)(g,g )+(h',gQ) = F(h)(l/|g|)(g,g)+(l/|g|2)*(h',g) = F(h), using the fact that (h*,g) = 0 since F(h!) =0. satisfies the condition of the theorem. The converse of Theorem IV is the following:

Thus gQ

THEOREM V. The equation (f,g) = F(f), f € defines for each g , a linear functionF with |FI*= |g|. Proof:

F is obviously additive.

Also

|F(f)l = l(f,g)l ^ |f|‘|g|. This implies that P is continuous and |F| £ |g|. Since however |F(g)| = |g|2 = IgMgl, |F| > Igl, and thus we obtain the theorem. Theorem V tells us that (f,g) is continuous in each variable, separately. But since |(f+a2+b2, ... j 12 if Ja^a^

... I is. Now it can be shown in precisely the same way as we we established B(9 ) In §1, Chapter II, that

IfVrfJ Ifn,p — -fL, J (a)1 ' ' n m' ^*= 1 m,p’ v for everyp. Thus if we fix p, since — * °> |f„ -r _ must be convergent, ii in, pJ— ► o and the sequence fn,p This means that for q { p, a. — ► b . We remark that since — ■ LLjq q p can be taken indefinitely large that bq is defined for every q. Let gp = fb1, ...,bp,0, ... |. Obviously fn^p— ► gp as p -- > oo . Furthermore for N = N(ti), we have that fN — ► f^ as p — k o o . For Ifjpf^pl2 - rS=p+1 l ^ a l 2* Thus given n > o, we can find a P(n) such that if p ^ P(n)>

lfN“fN,pl < n

III. REALIZATIONS OF HILBERT SPACE Now let n and m be ^ N(n)> P ^ P(n)* ((3 ), and (a) again, we have

25

Then by (a),

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

If we let n > N(ri),

m — > oo,

i

lfn-fNl + 'fr%,pl + l%,p-fm,pl

i

3 n.

then fffl p — > g^ and we obtain that

(Tf) for

p ^ P(n), |fn-gpl £ 3 n.

(-y.1)

If in particular we let n = N, wehave l%"Spl£ for every p ^ P(n). This implies |gj £ |fNl+3 n for P ^P(n). Thus < dfNl+5 n)2. We may let p — * oo and obtain £

Ibal2 * (|fN l+3 ri)2 < oo.

Thus we may let g = (b1, bg, ... !. We observe that g^ — > g. This and (7. 1) Imply |fn-gl i 3 n

(K.2)

for n ^N(rj). Thisimplies that fn — >■ g. The existence of a g with this property indicates the completeness of 1 and we have demonstrated Theorem I. When we recall Theorem XII of Chapter I, we obtain, THEOREM II. Every Hilbert space Is equivalent to lg, in the sense that to every f of ^>, there Is an element fa^i of lg, f ~ (a^l and this correspondence is oneto-one, and preserves the operations + , a* , 8 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 any two reali­ zations are in such a relations with lg, they must be related in this way to each other. Thus the axioms of Hilbert space are categorical. Now if a linear manifold, ffl is infinite dimensional, i.e.,

26

§3. DIRECT SUMS

satisfies postulate C, it satisfies all the postulates, with the original definitions of + , a* , 0 and (, ).Thus we ohtain: COROLLARY. Every infinite dimensional manifold in a Hilbert space is equivalent to the space itself. It is also equivalent to 1 . §3

DEFINITION 1. Let n = 2 , 3 , ...be a given integer and let us suppose that we have n Hilbert spaces, ... , fin. Now consider the n ’tuples of elements ff^ ... ,fn) with f^ € for i = 1, ... ,n. Define

,

If.,, ... ,fn !+ |g.,, ... jgJ = ff.j+g.,, ... ^n+Snl a|f1, ... ,fn! = faf1, ... ,afn )

6 = {e1, . . . ,en ! ( t ... * * ** We call this set of n !tuples, THEOREM III.

®

® 8n is a

” ^1 ... 9 fin«* Hilbert space.

The proof of this theorem is most elementary. DEFINITION 2. Let ^ ,fi2, ... be a sequence of Hilbert spaces. We consider the sequences jf^,fg, ... j such that fa € and 2Z^==1 |fal2< o o . We define 1^*1>-^*29

***

••• I = 1^*-]+£*1 2*^2* *** ajf^fg# ••• I “ f&faf2, ... I

^

0 = |0^,©2, *• • I ({f^,fgj, ... 19 lg ••• > Xn are a set mutually orthog­ onal non-zero functions 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 S1 denote the set of characteristic functions of the measurable sets F C E. Since every function of can be approximated by step functions, it follows that WUS., ) =C2. Let S2 denote the subset of these, in which the F is an intersection of an open set G with E. It is well known that Sg is dense in S1. Thus D7KS2) = Wl(S^ ) = Cg. Let S^ denote the subset of S2, in which G is the interior of an n -dimensional cube whose faces have the equation x^ = p^, where is a rational number. Now any open set is a denumerable sum of such n -dimen­ sional cubes regarded however as closed point sets, but whose interiors are mutually exclusive. The faces of the cubes in such a sum form a set of measure zero andthus it is possible to show that 3W(Sj) contains and thus ^(S^) - Wt(S2) = C2. Lemma 1 of §1 of this chapter now implies Postulate D since is denumerable. It remains to prove Postulate E for Let f1, fg, ... be a sequence of elements such that lfn~fm l — ► as n and

III. REALIZATIONS OP HILBERT SPACE

29

m — ► oo. Let €1, €2, ... be a sequence of positive numbers such that €a n ^ ^ fallows that |f -f I < €a. Thus a+1 a

^a=1^na+1 “^na ^ ^ *-a=o €a^ 00* We let Let

k=

I“ ,|f

-fn I. 0ffl “

Vp>- IS-,lfn„]-1'nt LEMMA 2. T2 is an extension of T1 if and only if for every f in thedomain ofT, Tgf is defined and T2f - T^f. LEMMA 3. A transformation T is additive. (Cf. Chapter II, §3, Definition 2.) if and only if its graph X is an additive set. LEMMA k. An additive set of a transformation from !©1,hj € X implies h = 0g.

X C ft,® ^2 is the graph to if and only if

By Lemma 1, the condition is necessary. It is also sufficient For suppose ff,g1 I and ff,g2l are in X. Then since X is additive, ff,g1 I-ff,g2j = (G^g^ggi € X. Our condition implies g1 = g2 and thus there is at most one pair ff,gi € X with f in the first place. DEFINITION 4. Let T be a transformation with graph X. If W(X) is the graph of a transformation, T , this latter transformation is called the additive exten­ sion of T. DEFINITION 5. A transformation T from ft., to ft2 will be said to be closed, if its graph is a closed set in ft,® ft2. If [X] is the graph of a transformation [T], [T] is called the closure of T. We note that 9tt(X) is the graph of [T] when this latter a transformation exists. In general, given T, T& will not exist (A necessary and sufficient condition that Ta exist can be obtained by applying Lemma b to U(X)). However 6ven if T& exists, [TaJ need not exist. We give an example of this. Let 1, -} i is in CM(I)]. To see this we notice that T& ( I^■ I ;=1 (1/n /n)a _1 (1 /n) , )4>a ) = 4>1 and thus [\ ix nn__1 (1/n) 1 i is in Now fI (l/n)a 1 /n)1/2— hTo0 as In W(I). M(I). (1/n)a l *- ((l/n)1/2— (l/n)4»a, 1 j —— ►► fe,^ !. ThusThusfQ,1 } ! n — — *►■ oo. Co. Hence |\ l£=1 (1 /n)a,4>1) {6,4^1. }©,1 is in [W(!)3 [M(l)3 and the latter is not the the graph graph of of a atransforma­ transforma­ tion. However Theorem II of Chapter II, § 33,, tells us that if T& is a continuous transformation [Tcl ] exists and has domain, the closure of the domain of T& . Thus if a continuous additive T has domain, a linear manifold, T is closed. DEFINITION 6. If T^ is a transformation from ^ T2 is a transformation from fig to fi^ to fig and Tg T gT 1 is the transformation whose domain consist then TgT1 of those f !s for which T ggC^^ff ) is defined and has the value T g ^ f ) , i.e., (TgT 1 )f - T, (Tgf). Since a continuous function of a continuous function is continuous, it follows that if T 1 and Tg Tg are are continuous, continuous, TgT1 TgT1 and additive is also continuous. If in addition and TTgg areare additive and C2 3, I.), then then with bounds C1 and Cg (Cf. (Cf. Chapter Chapter II, II, § § 3 , Theorem Theorem I.), TTgT.! 2T.j has bound CgC1 C2C1 . DEFINITION 77.. If T 1 and Tg T 2 are two transformations to fig, fi2,TT ^^ TT gg is from fi1 to is the the transformation, transformation, the the domain of which which is is the the set set of of those those elements elements for for which which T.jf and Tgf Tgf are are defined )f == defined and and for for which which (T1 (T1+Tg +Tg)f aT1 is the transformation, whose domain is T^+Tgf. ( T ^ ). that of T 1 and for which (aT1 (aT. )f = aa(Tnf The sum of two continuous transformations is again continuous and if in particular T 1 and Tg are "additive with bounds C1 and Cg, Cg, then the bound of the sum is £ C-j+Cg ^i+Cg

3^

IV. ADDITIVE AND CLOSED TRANSFORMATIONS §2

THEOREM I. If T is a transformation from 551 to with graph 1 *and domain D then IA is the graph of a transformation from to 5o1, if and only if 931(0 ) = Proof: XA is the graph of a transformation from Jo2 to , if and only if jh^Ggi € IA implies h = ©1. But }h,©2} Cl1 is equivalent to o - (fh1,©2I,ff,Tf|) - (h,f) for every f in the domain of I. fh^02i € IA is equivalent to hj€ S A. Thus XA is the graph of a transformation if and only if h € implies h = ©1. But h € Ba implies h = 0 1, if and only if SA= j. Thus l l is the graph of a transformation if and only if = f©1}. But Dx - 0 1 is equivalent to (BA)A = \ Q^ \ X ^ by Theorem VII of Chapter II, §5 . But since (D*)A = (7R(D)A)A= 771(D), we see that BA = j01j is equivalent to Tfl (D) - *5,. From a preceding statement we see that IA is the graph of a transformation if and only if 7ft(D) = Jo. If in particular T is additive, D is additive and M(D) « D. Thus 77KD} = [M(D)j ~ [D] and the statement D7t(D) is equivalent to D is dense. We have then: COROLLARY. If T in Theorem I is also additive, then XA is the graph of a transformation, if and only if 2) is dense. DEFINITION 1. If T is a transformation from ^ to fi2 and if X* is the graph of a transformation from ^2 to ^ , we, will denote the latter transformation by T1 and -TA by T*. * THEOREM II. Let T 6e such that TA exists. Then (a) A pair [g1,g2I £ f>2 is such that TAg2 = g1 if ♦ If T is a transformation between two Banach spaces ^ and £2, T1 can be regarded as a transformation from £| to £* (the conjugate spaces).

§2, ADJOINTS AND CLOSURE

35

and only if for every f in the domain of T, (f,gl)+(Tf,g2) = o; (b)

A pair

fg,,g2} € f5.,®f52

is such that

T*g2 = g1

if and only if for every f in the domain of T, (f,gl) = (Tf,g2). Since T* = -T1, (a) and (b) are equivalent.

Inasmuch as

(!f,Tfi,fgl,g2i) = (f,g1)+(Tf,g2), the condition in (a) is equivalent to

fg^gg! € X\

COROLLARY. Let T be such that TA exists. Then (a) A transformation T* is C T1 if and only if for every f in the domain of T and every g in the domain of T1 (f,T'g)+(Tf,g) - 0. (b) A transformation TT is C T*, if and only if for every f in the domain of T and every g in the domain of T!, (f,T'g) - (Tf,g). THEOREM III. Let T be a transformation from ft., to ft2 for which T A exists, i.e., 571(D) - ft. Then [T ] exists if and only if TA (or T*) has domain dense. By Definitions k and 5 , we see that [T ] exists If and only if 9ft(I) is the graph of a transformation. But Tfl( I) = (1 A)A. (Cf. Chapter II, §6). Furthermore since X1 Is a linear mani­ fold, TA Is additive. Thus the corollary to Theorem I of this section states that "Domain of TA dense” Is equivalent to "(XA)A is the graph of the transformation." Since (X1)4 = 771 (X), the first sentence in this paragraph shows that this latter statement is equivalent to " [T&] exists.” COROLLARY 1. [T ] exists if and only if (TA)A (— (T*)*) exists. When they exist [T&] = (TA)A (= (T*)*).

IV. ADDITIVE AND CLOSED TRANSFORMATIONS

36

COROLLARY 2. If T is a closed additive transforma­ tion with domain dense, (TA)A (= (T*)*) exists arid equals T. 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. As we remarked before Definition 6, in §1, a linear transformation is closed. THEOREM IV. If T is a continuous additive transfor­ mation, whose domain is dense and with bound C, (Cf. Chapter II, §3 , Theorem I), then TA (and T*) is a linear transformation with the same bound as T. PROOF: [T] exists by Theorem II of Chapter II, §3 . Since [T]a _ we suppose that T = [T] and has domain the full space. By Theorem III of this section, TA has domain dense. It is alsoc.a. since X1 is linear manifold. Thus Theorem II of ChapterII, §3, implies that TA is linear if it is contin­ uous. By Theorem I of Chapter II, §3 , we have for every f |Tf| £ C|f | (we have assumed that [T] = T). Hence for every f and g, I(Tf,g)| £ |Tf|•|gI £ C*|fI *Igl • If g is in the domain of TA, we have by (a) of Theorem II of this section that (TAg,f)+(g,Tf) = o. Hence I(T4g,f)| - |(g,Tf)| £ C»|f|•|g|. If we let f = TAg, we get |TAg|2 £ C*|TAg|*|gl which implies |TAg| £ C•|g|. Theorem I of Chapter II implies that TA is continuous, with a bound CA £ C. Since however (T*)* ~ t, we also have C £ CA and thus the bounds must be equal. THEOREM V. If T1 and Tg are additive transformations with dense domains, then if (T^T^ )* exists (or (T^+Tg)* ), we have that T*T* Is a contraction of (TgT1)*, (T*+T* is

§2. ADJOINTS AND CLOSURE

37

a contraction of (T.j+T2)*). (aT1)* = aT* if a + o. If T1 and Tg are linear, we have that T* T* = (TgT1)*. (Similarly T*+T* = (T1+T2)*). PROOF: If f is in the domain of TgT1 and g in the do­ main of T*T*, (t>) of Theorem II of this section implies (TgT^g) - (T^f,T*g) - (f,T*T*g). Now (b) of the Corollary of Theorem II of this section implies T*T* C (TgT1)*. In the case of T1+T2 the argument is similar. To show that (aT1)* = aT*, we note that if a + 0 (T1f,g< J) = (f,g2) is equivalent to (aT1f,g1) - (f,ag2). If T1 and T2 are linear T* and T* are also by Theorem IV above. Thus T*T* is everywhere defined and has no proper extensions and we must have T*T* = (TgT1)*. This argument also applies to the sum. COROLLARY. If T1 is c.a.d.d. and T2 is linear, then (T2T1)* - T*T*. PROOF: We know fhat (T2T1)* D T*T*. On the other hand let f be in the domain of (T2T1)* and let g be in the domain of T1 and hence in that of TgT1. Then (g, (T2T^)*f) - (TgTig,f) = (Tlg,T*f). Since this holds for every g in the domain of T1, we have that T*(T*f) exists and equals (T2T1)*f. This implies that T*T* D (T2T1)*. LEMMA 1. Let T be c.a.d.d. Let 71* be the set of f Ts for which T*f = 0. Let !R denote the range of T. Then R A = 7t*. Since T* is c.a., 71*

is closed.

Since

(}61,g},ff,Tf1 ) - (g,Tf), we see that {0 ,gl is in J x RA is the set of zeros of T A It is evident geometrically and T*~1 exist, then (T~1)*

if and only if g € 7*A. Thus = ~T*. that if T is c.a.d.d. and T~1 = T*_1. Lemma ^ of§1 and the

38

IV. ADDITIVE AND CLOSED TRANSFORMATIONS

preceding Lemma shows that T*~1 exists if and only if [DR] = and that T~1 exists if and only if [!R*] = THEOREM VI. Let 71 denote the zeros of T, ft* denote the zeros of T*, DR the range of T, DR* the range of T*. Then 71* ~ 7i l , 71 = (DR*)1. 'T_1 exists if and only if 71* = (DR*)1 = (0 1 . T*~1 exists if and only if 7\* = (DR)1 = (0 J. If T_1 and T*"1 both exist, (T“1)* = T*"1. §3

We now introduce certain notions which are fundamental in our discussion. DEFINITION 1. An additive transformation H within fi, will be called symmetric if (a) the domain of H is dense and (b) for 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. LEMMA 1. An additive transformaion H is symmetric, if (a) it has domain dense and,(b) H C H*. LEMMA 2. symmetric.

If H Is symmetric,

[H] exists and

is

PROOF: H* is a closed transformation. Since H C H*, we must have the graph of H in a closed set which is the graph of a transformation. 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 [H]1 - H1 and hence [H]* = H*. Lemma 2 permits us in general to consider only closed symme­ tric transformations. DEFINITION 2. If H* = H, H is called self-adjoint.

§3. SYMMETRIC AND SELF-ADJOINT OPERATORS

39

LEMMA 3. A self-adjoint transformation is symmetric. If H is closed symmetric and H* is symmetric, then His self-adjoint. If the domain of a symmetric trans­ formation H isthe full space, H is self-adjoint. A symmetric linear transformation is self-adjoint. The first sentence is a consequence of Lemma i. If H is closed, symmetric and H* is symmetric, 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 follows from Lemma 1 of this section since a transformation with domain the full space can have no proper extension. The fourth statement follows from the third. LEMMA 4 . If H1 and of H^Hg is dense, the If a is real, aH1 Is adjoint, a real, then

Hg are symmetric and the domain latter transformation is symmetric. symmetric and If H1 is selfaH1 Is self-adjoint.

This is a consequence of Theorem V of the preceding section. For if the domain of H1+Hg is dense, (H^+Hg)* exists. Then too, H1+H2 C H*+H* C (H.j+Hg)* by this theorem. The second sentence is an immediate consequence. LEMMA 5 . If H1 is self-adjoint and Hg linear sym­ metric (and hence self-adjoint by Lemma 3 above) then H^Hg is self-adjoint. PROOF. The domain of B^+Hg is the same as that of H1 and thus is dense. Hence Lemma 4 , tells us that K^+Hg Is symmetric and that -Hg is self-adjoint. Furthermore (H1+Hg )+(-Hg) has domain the domain of H1. Hence H1= (H1+Hg)+(-Hg) C (H1+Hg)*+(-Hg)* C ((R, +Hg)+(-H2))* - H*- H1 . This implies that the domain of (H^Hg)* which is the same as that of (H1+Hg )*+(-Hg )* is included in the domain of H1 which is also the domain of H^Hg. Since H.,+H2 is symmetric, H1+H2 C (H1+Hg)*. Since the domain of (H^+H2)* is included in that of H1+H2, we must have (H1+Hg)* « H1+H2.

IV. ADDITIVE AND CLOSED TRANSFORMATIONS

AO.

H~ 1

LEMMA 6 . If H is symmetric, and. if H~ 1 exists, is symmetric if H* 1 exists, i.e., if [DR] = 5o.

This is a consequence of Theorem VI of the preceding section. We note that if H*” 1 exists, since H C H*, we must have H" 1 C H*"1. LEMMA 7. If H is self-adjoint and H~ 1 is self-adjoint.

H~ 1exists, then

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 Lemma 8 . LEMMA

8.

If H

is closed symmetric, then

C 7}* =

!RA.

Suppose H 1 and H2are closed symmetric and H 1 C H2. Then we have H1 C H2 C H* C H*. Now if H 1 is self-adjoint since H 1 = H*, we see that H2 must equal H1 and H 1 has no proper symmetric extension. On the other hand, it Is also conceivable that H 1 is symmetric with graph ^ and 6 *6 * Is one dimensional. If H2 is then a symmetric closed exten­ sion of H1, we have H 1 C H2 C H*. This last inclusion and Chapter II §5, Corollaryl to Theorem VI imply that either H2 = H 1 or H2 = H*. But H* Is not symmetric because (H1 )** = H1 is C H*, but H 1 ^ H*. Under these circumstances then H 1 would have no proper symmetric extension and yet not be self-adjoint. We shall show later the existence of an H 1 having these properties and give a complete discussion of this phenomena. But for the present, we simply introduce the defini­ tions. DEFINITION 3. If H 1 and H2 are closed symmetric transformations such that H 1 C Hg, then H2 Is called a symmetric extension of H1. If in addition H 1 ^ H2, H2 is called a proper symmetric extension of H 1 . If H1 is closed symmetric and has no proper symmetric ex­ tensions, H1 is called maximal symmetric.

§3. SYMMETRIC AND SELF-ADJOINT OPERATORS LEMMA 9 . A self-adjoint transformation is maximal sijmmetric. If H is symmetric and fis in the domain of H, then (Hf,f) = (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 (+ ©) in the domain of H, C(f,f) 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(f ) is convergent.

V. WEAK CONVERGENCE

46

tradiction. Hence the Cn ’s are uniformly bounded. THEOREM I. Let Tn be a sequence of continuous additive functions, whose domain is the full space ft and whose values lie in a linear space. Suppose that for every f inTnf is convergent. Then the bounds of Tn !s are bounded. PROOF: Let us suppose that the Theorem does not hold for a specific sequence fTn !* Then Lemma 1 above implies that the |Tnf| are unbounded in every sphere. Now suppose that for i - 1 , ... , k we have specified a function Tn^ a sphere with a center f^ and radius r^ and such that if f € 6 ., |T„,f| ^ i. Suppose also that r. £ * j l ~ 2ri-1 £ 1/S 8114 Ri+1 C We know that the T f ’s are not bounded In the sphere with center f,k, and radius -2-r,K . We can therefore find a Tnk+1 and an f^ +1 within this sphere, with lTnk+1 *k+l^ i 2 (k+l). Since T is continuous, we can find a closed sphere with nk+i .1 y , k+f k+1 center f^ +1 and radius rk+1 £ —r^ 1 1 /2 , for which 'Tnk+lf' ^ k+1

f°r f€(W

~

If S 13 in < W

|fk-gi i iv^+i+^+rs' i ifk-fk+il+ifk+rsi i rk or g In 8 ^. Hence C and we see that we may define a sequence of T , f^, r1, which have the properties given In the preceding paragraph for every i. Since each contains all that follow and rp— ► 0 . as n — ► co, the fn ’s form a convergent sequence, whose limit f is In every sphere f^. Consequently lTnif I — ► ooas i— >co and the T f 's cannot converge. This contradiction shows that the Tn ?s must be uniformly bounded. COROLLARY 1 , If in Theorem I, for each f, is bounded, then the result still holds. COROLLARY 2 . A weakly convergent sequence of elements ffn J must have the norms M fn lI bounded. Let fn be a weakly convergent sequence. We have a C such

§ 2 . WEAK COMPACTNESS___________ that |fn | £

F(g), where F(g) denotes the value of the limit* Since |(g,fn )| £ lfn l“ P(g) Igl £ C- |g| |gl for every g, we have |F(g)| £ C*|gl- Since F is obviously additive, Theorem I of Chapter II, § 5 , implies that FF is is aa linear linear functional. functional. Thus Thusthere thereisisa a f f€ €fifi such such that (g,f) (g,f) -- F(g) F(g) for for every every g, g, by by Theorem Theorem IV IV of of Chapter Chapter II, II, §4. Thus we have established: THEOREM II. weakly convergent sequence II. If If jf jf JJ is Is aa weakly convergent sequence of elements elementsof %ofthere % there exists exists an f ansuch f that such for thatevery for every g in fi, (g,fn ) — ► (g,f).

§2 Thus if a sequence [fn J is weakly convergent It it has a weak I.e., (fn ,g) — ► (f,g) for every g. Thus ft is limit f, i.e., complete for weak convergence too. Since (f,g) is continuous in f, we have

to

LEMMA 1 . If a sequence ff J is strongly convergent f, it is weakly convergent to the same limit.

The converse of this lemma does not hold. For let be an infinite orthonormal set. For every g we have Ia=‘ jlacJ2 ^ ^ where a a « (g,4>a). Thus for every g, (g>a ) -- * 0 anjt* !s ^orm a weakly convergent series. Since however aa ++ ((33,, they they are are not not strongly strongly convergent. This example also shows that there are bounded infinite sets of elements, which have no limit points. Thus Hilbert space is not locally compact. However for weak convergence, we have a kind of compactness. THEOREM III. If ff If I is a bounded sequence of elements, there exists a weakly convergent subsequence. PROOF: Let g-,, g2, ... ... bebea adenumerable set, dense dense in in f3f.3. are bounded and thus we can find a subse­ The numbers (g-j quence ff^! for which (g^f^) (g^f^) is isconvergent. convergent. Similarly, Similarly, we we can chose a subsequence jf”! of ff^i ff^i such such that that (g (g22>f^) >f^) is is

g

ke>

V. WEAK CONVERGENCE

convergent. By this process, we can continue to choose sub­ sequences so that (g^f^11^) is convergent for i £ n. The "diagonal sequence” ff^i then has the property that for each n (when the first n elements are ignored) it is a subsequence of ff^n ^l. Hence (gj^fi0^) is convergent for every i. The norms of the ff^ ! are bounded, and thus the linear functionals (g,f^°^) 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 lemmas. LEMMA 1 . Let T be a linear transformation from to The domain of T is the full space and we let ^ denote the set in of those elements in the form Tf, |f | £ n, [Tf | £ 1 . The set £n is closed. Since T is linear, T* is also linear. (Cf. Chapter IV, §2 , Theorem IV) Now let g be a limit point of We can find a sequence jg^J with g^ — ► g and such that g.^ = Tf^ for an fi with lf^ | £ n. Since the f !s are uniformly boun­ ded we can find a subsequence , which converges weakly to an f with |f| £ n (Cf. TheoremIII, §2 above). Let g£ = Tf|. Then for every h, (f,T*h) = lim (f»,T*h) a+oo - lim (Tf',h) CX+ao = lim (gi,h) CX-+00 - (g,h). Since (T*)* * T, (b) of Theorem II of Chapter IV, §2 implies Tf = g. Since |f| £ n, |g| £ 1 , g is in f$n. Thus is closed. LEMMA 2 . Let T be a linear transformation fromm ^ to fig for which T 1 exists. Let f$n be as in Lemma 1 . Then if for some n = nQ, contains a sphere 6 , then

_________ §3. CLOSED TRANSFORMATIONS WITH DOMAIN C jt_______ k£_ T-1

is bounded.

Let 6 haveradius r and center For g € 6 C fin, we have that g = Tf for an f with |f| £ n. In particular this is true for g1 = Tf1. Now if h is such that |h| < r, we have that g = g1 +h is in 6 and thus h = g-g1 = T(f-f1 ). Since If~f1 I £j f |+1f1 1 £ 2n, h is in 6 2n. Thus 6 2n con­ tains the sphere with center and radius r. This implies that T 1 is defined everywhere and has a bound £ k n / r . For if g € *$2 and g 4 02,let h = (r/2 |g|)g. Since h € 6 2n, (Cf. ^above) we have that |T~1h| £ 2n, which is equivalent to |T~1 g| £ (tei/r) *|g|. We introduce certain set-theoretic definitions which have played a very important r61e in the general theory of linear spaces. —

DEFINITION 1 . A set S is said to be nowhere dense if every sphere contains asphere of the complement of S. A set will be said to be of the first category if it Is a denumerable sum of nowhere dense sets. The following Lemma Is LEMMA 3 . A set tain any sphere.

important.

S of the first category doesnot

con­

Let S = S.,+S2+... where S^ is nowhere dense. Let 6 be any sphere. We shall show that S does not contain 6 . Within 6 , we can find a belonging to the complement of S1 and we can suppose that the radius r^ of Is £ 1 /2 . Within , we can find a sphere 6 2 of the complement of S2 with radius r2 £ 1 /2 2. Continuing, we can find a sequence of spheres each containing all subsequent spheres, in the complement of Si and whose radii approach zero. The centers ff^J of the spheres form a conver­ gent series, with a limit f, which Is in every sphere includ­ ing 6 . Since f is in every sphere, 6 ^, It is in the comple­ ment of every S^ and hence in the complement of S. Since f is in R, , S does not contain 6 .

50

V. WEAK CONVERGENCE LEMMA 4. Let T be a linear transformation from fito fig for which T-1 exists. Let 6 n be as in Lemmas 1 and 2 . If T" 1 is not bounded, each Rn is nowhere dense.

PROOF: Let R be any sphere of ft2. Rn does not contain by Lemma 2 . Thus 6 contains a point f of the complement of fin. Since the complement of £n is open by Lemma 1 and its intersection with 6 is not empty, 6 and the complement of must have a sphere in common. Thus is nowhere dense. THEOREM IV. Let T be a linear transformation from fij to fi2 whose inverse T 1 exists. Then if T has as its range T~ 1 is bounded. —

Under these circumstances, the sum of the 6 of Lemmas 1 , and. b contains the unit sphere of fi2. If T-1 were not bounded, then this sum would be of the first category by Lemma b and. hence could not contain a sphere by Lemma 3 . Thus T~ 1 is bounded.. 2,

THEOREM 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 1 (=fi1 ) to D (= fi2)> defined, by the equation Aff,Tf j = f. A has domain 1 and bound 1 . Thus A is linear. The inverse A-1 exists and we note A”1f = (f,Tfj. The range of A is fi2. Thus we may apply Theorem IV and. obtain that there Is a C such that for every f € D, C-|f| i |jf,TfII = (If|2 +|TfI2 ) 1/ 2 2 lTfl-

CHAPTER VI PROJECTIONS AND ISOMETRY In this chapter we will consider four special kinds of trans­ formations of particular interest in the theory that follows. §1

DEFINITION 1 . Let 307 be a linear manifold of ft and for every f, let f * f.,+f2, f, € HT,, f2 € 3R*. (Cf. Chapter II, §5 , Theorem VI) The transformation E which is defined by the equation Ef « f1 is called a projection. Lemma 1 . E is a linear self-adjoint transformation with C_ ^ 0 , C+ £ 1 . (Cf. Def. k of Chapter IV, §3) Furthermore E2 = E. PROOF: The uniqueness of the resolution of Theorem VI of p Chapter II, is easily seen to imply that E = E and that E is additive. We also see from the same Theorem that the domain of E is ft. Since |f| 2 = |f1 |2 +|f2|2, |f| ^ |Ef| and thus E is bounded. Hence E is linear. Now for f and g € ft, we resolve f ~ f1 +fp, g = g1 +g2 and then since f1 and g1 are orthogonal to f2 and g2 we obtain, (Ef,g) - (f^g^gg) = (f^g.,) - (fj+fg^) = (f,Eg). Thus E is symmetric. Since it is also linear, we know by Lemma 3 of §3 of Chapter IV that E is self-adjoint. Since we also have (Ef,f) = (f,,f) =

= IfJ2 ^ Ifl2

we see that C_ ^ o, C+ £ 1 . If both 7ft and. Tflx we easily verify that C+ - 1 and. C_ = 0 . Conversely, we have

are not

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

the set of zeros of E, includes all ele51

VI. PROJECTIONS AND ISOMETRY

52

ments g for 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 do­ main of E and Eg = E( 1 -E)f = (E-E2 )f = 0. By Chapter IV, §3, Lemma 8 , we know that ftCft* = [DR]-1 . Thus ft and DR are orthogonal and for f in the domainofE, f = Ef+(1 -E)f If*12 - |:Ef|2+|d-E)f|2.

Thus IEf I ((E,+ ... +En)g,g) = l£=1(Eag,g) = I ^ j E ^ g l 2 ^

|Elg|2+|B.jg|2 = IE±g12+1g12 > |g|2.

This contradiction indicates that E^E^ =0 . On the other hand if E^E^ = 0 for i + j, then (E1 + ... + l^ ) 2 = E^ + ... +Eq. Since E.,+ ... +EQ is self-adjoint, it is a projection by Lemma 2 above. If E^Ej = 0 , Ej - Ej-E^j = (1 -Ei)Ej. It follows from Lemmas 3 and k above that 77\. is in Tfl^ Tttj or 7Kj in Thus 371j is orthogonal to 771^. Obviously the range of E 1 + ... +En must be in M( 7Tl1 u ... u 5Wn). 0n the other hand if f€&(7tt1u ... u^Rn )> it is readily established that f = f.,+ ... +fn, where f1 € 7ft1, ... , fn € 7Hn. Now (E1 + ... + l^n ) ( f ^ + ... +fn ) = f^ + ... +fn since E^fj = EiEjfj “ i + 3• Thus f1 + ... +fn is in the range of E ^ ... and W(7ft1u ... u7ttn ) is included in this range.. This and the previous result prove the last statement of the lemma. DEFINITION 2 . Two projections E., and Eg are called orthogonal if E^E. = o, or what is equivalent, If 7711 Is orthogonal to 7Rp.

VI. PROJECTIONS AND ISOMETRY

5b

IiEMMA 6 . If E1 arid Eg are projections with ranges and. 7ftg, then E.j-Eg is a projection if and only if E2 « ^Eg. If E1 -Eg is aprojection then Tftg C 7ft,,, and the range of E.j-Eg is 7ft.,•7ft*. It follows from Lemma 3 , that E^Eg is a projection if and only if 1 ~(E1 -Eg) = 1 -E.j+Eg is a projection. By Lemma 5 , (1 -E^+Eg is a projection if and only if E (1 -E1 ) = 0 or Eg = EgE1. These two results imply the first statment of the Lemma, If Eg = EgE1, Lemma h implies Tftg - 7ft1 7)?2 or 7ft2 C 7ft1 . Since E^Eg = E-,“EgE1 = (1 -Eg)E1 Lemmas 3 and k imply that the range of E1 -Eg is 7R* 7ft1 . LEMMA 7 . Let E.,, Eg, ... be a sequence of mutually orthogonal projections, with range 7ft.,, Tftg, ... respectively. Let E = I -_ 1 Ba (i.e., Ef = f t l “=, y whenever this limit exists). Then E is a projection with range 7ft( 7ft.,u Tftgu ...) (where u denotes the logical sum). PROOF: We note that if n ^ m and f € ft then by Lemma 5 |f|2 i i u £ =m+1V f'2 = i ^ V - C i V 2 - l ^ , ¥ l 2 - ^ = m+i l V l 2If we let m = o, we obtain |f| 2 ^ Hence |f| 2 ^ £ * = 1 lEaf 12 ‘ This implies that I ^ m+i|Eaf |2— ► o as m and. n -- ►o o . Our first inequalitythen shows that |£^==1 --> 0 a s m a n d n -* o o . This shows that Ef exists for every f. Now E is symmetric, since for every f and g of ft, " (iiS> Z£-i V ' « > * Furthermore

- (f>(iis> C

=1 Ea)g>

-

§1. PROJECTIONS

55

Hence Lemma 2 above shows that E is a projection. We now prove the last statement of the Lemma. Let IN denote the set of elements f of in the form f = where fa € Since the 5fta fs are orthogonal, |f |2 = 1^*, |fal2. Now a proof similar to the completeness argument of Theorem I of Chapter III, will show that 71 is a closed set. Furthermore ...) C 7\ and every f € 71 is the limit of elements of U( 771^ uTftgU ...). Thus 71 is the closure ofM(5H1u5flgU ...) that is 2R( DWgU ...). One can easily verify that 71 is the range of E and this completes the proof of the Lemma. DEFINITION 3. We will write Eg £ E1, if

57*2 C m, .

LEMMA 8. The following statements are equivalent: (a) E2 £ E i5 (b) EgE1 « E2 (= E^g by Lemma 4); (c) For every f of fi, |E2f|2 £ \ E ^ f \ 2 . PROOF: (a) implies (b). For if f € 55, we knowthat f = f1+f2 where f1 € 5ft.,, f2 € 5ftj By Corollary 1 to Theorem VI of Chapter II, §5, we have = f1 1+fl , where f1 1 € 5ft2, f1 2 € 5ft**5ft.,. Thus f = f1 1+(f., 2+fg)/ Since 531* C'5ft* by Corollary 2 to Theorem VI of Chapter II, we have f2 € 5ft*. Thus f1 2+f2 € 5ft*, f1 1 € 5ftg and these imply Egf = f1 1. But by the method of definition of f1 1, = f-, Hence EgEjf = Egf and (a) implies (b). (b) implies (c) since|Egf|2 * IE^E1f|2 £ |E1f|2. (c) implies (a). For if f € jn £ , |.f|2 = |Egf |2£ |E.,f|2 £ |f|2.Thus |f|2 = |E.,f|2. Since |(1-E, )f|2 = |f|2- |E,f |2 = 0, we must have (1-E1)f « 0 or f « E.,f € 5ft.,. Thus 5ft2 C 9ft. LEMMA 9. If E^ E2, ... denotes a sequence of pro­ jections with ranges 5ft.,,5ft2, ... respectively and such that E^ Ea+1, then E - lim^^E^ is a projection such that Ea £ E. The range of E is 5ft(5ft.,u 5ft2u ...). £ Ea+1, Lemma 8 implies that Ea*Ea+1« Ea. PROOF: Since Lemma 6 implies that Ea+1~Ea is a projection. Since ~ + (Ecw.1“Ea) is a projection, Lemma 5 implies that E1,

56

VI. PROJECTIONS AND ISOMETRY

E2 ~E1, E^-E2, ... are mutually orthogonal. Furthermore E = lim^^ 15^ = E ^ IJ1.J (Ea+1 ~Ea). Thus Lemma 7 states E is a projection. The expression for EQ in the previous paragraph and the resulting orthogonality relations, shows that if ft^ n, = °* J t followa that EnE = Ei+ zS=i(Ea+rEa)=\ and thus by Lemma 8 that E^ £ E. It follows from Lemmas 7 and 6 that the range of E is 7ft( TT^uTftfTftguTft^Tft^u ...). Corollary 1to Theorem VIof Chap­ ter II, §5 shows that M( W a,Tft^*9fta+1 ) = 7fta+1 • These two results imply that the range of E is 7ft(5711u TftgU ...). LEMMA 10. Let E1, Eg, ... be a sequence of projec­ tions with ranges 9ft.,, 7ft2, ... respectively and such that Ea+1. Then E = linin.-#.00^ is a Prodect;1-on such that E ^ Ea for every a, and the range of E Is •rm2- ... . PROOF: Let £ = 1 -^. Then Ea > Ea+1, implies Fa £ F a+1, since Pa.Fa+1 = (1-Ba)(1-Ea+1) = 1 - V W V o * ! = 1'Ea = V when we use Lemma 8 (b). Lemma 9 tells us that lim Fa is a projection. Hence 1-lim Fa = lim(l-Fa) - lim E^ = E is also a projection. Since lim Fft^ Fn, we must have E £ E^. If f is in ... , then Eaf = f for every a and hence Ef = f. Thus f Is in the range of E and • 7ft2* ... 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 Eya = E we have E^f « E^Ef « Ef « f. Thusf € 7ftn. Since this holds for every n, f € 7ft1•7ft2* ... and this set In­ cludes the range of E. These results together imply the last statement of our lemma. §2

DEFINITION 1. A transformation U from to fig with domain fi1 and range fig and such that (Uf,Ug) * (f,g) for every pair of elements in fi1 is called a unitary transformation from fi1 to fig.

§2. UNITARY AND ISOMETRIC TRANSFBRMATIONS_________ 57 LEMMA 1. A unitary transformation U exists and U* = U _ 1. PROOF:

Given

f 1 and

is linear, U ”1

f2, we have for every

g,

(U(af1+bf2 )-aUfr b«f2,Ug) -blIf2,Ug) - (U(af 1+bf2 ),Ug)-a(Uf,Ug)-b(Uf2,Ug) ( a ^+bf2,g)-a(f-,g)-b(f' + b f ^ g J - a C f ^ g J - b ^ gg)) -« o. o. = (af Since the set of UUgg !s !s fill fill out out 552, 552, wewehave have U(af1+bf2 U(af1+bf2))-allf1+bUf2 and U is additive. Since |f |2 - (f,f) = (Uf,Uf) « |Ufj2 we see that U has bound 1. Thus tl is linear. Since |f |2 - |Uf|2, Uf « 02 e2 implies ~ 11 exists. exists. It It implies ff »» 0011.. Thus Thus UU“ is readily seen to be unitary. Since U is linear, U* exists. For every f € ,, gg € €f>f2>2 we have (Uf,g) « (f,tf 1g). ' 1C CU* former has domain Thus UU” U* and and since the former'has

s, U* == UU_1.. Jfo>2,

LEMMA 2.2. If If U is is a unitary unitary transformation transformationfrom from ^ to fi2 and 1, c|c>|2, >2, ... ... isisa acomplete completeorthonormal orthonormalset set and 1, in , then U ^ , U2, .,. ... is a complete orthonormal set in fi2. fi2.

^

easily seen that Ufy, U2, an or­ or­ PROOF:It It is is easily seen that thetheUfy, U2, ...... isis an in infi2,fi2,g =g Uf thonormal set. set.If Ifg gis is = Uffor foranan f f€ €5o1. 5o1. f f= = Theorem XII XII of of Chapter Chapter II. II. Thus Thus g g- - Uf Uf == ^ - ^ a ^ c c and and 1 s nonmletfl is complete..

^^

“% ThVi3th0 th0set set U1' U1' U2>***2’ ••• **• % '' ThVi3

LEMMA 3. If* > 2> is a complete orthonormal set in fij and ... ... Is is aa complete complete orthonormal orthonormal set in then then the the transformation transformation defined defined by by the the equaequa=1aa^a ^a is is unitary. unitary. tlon U( ,a * ) *»*» ^£""=1aa This is an immediate consequence of Theorem XII of Chapter II. II, DEFINITION 2. An additive transformation V the property that (Vf ,Vg) = (f,g) for every f in its domain is called isometric.

with and g

58

VI. PROJECTIONS AND ISOMSTRY___________________

LEMMA k. An isometric transformation V is bounded and has an isometric inverse. inverse, [V] exists and and is is isometric isometric with domain the closure of the domain domain ofV,V, and and range, range, the closure of the range of V. PROOF: The first statement is proved proved in aa manner manner analogous analogous to the proof of the corresponding statements In in Lemma 1. Theorem II of § 33,, of Chapter II implies that [V] exists and has domain the closure of the domain of V. The continuity of the inner product shows that [V] is isometric. A similar argument will show that [V-1] exists and has domain the closure of the range of V. From Prom graphical considerations we see that [V]~1 ~- [V1]. Hence the range of of [V] [V] is is the the closure closure of of the the range rangeof ofV. V. LEMMA 5. 5. An An additive additive transformation transformation V, V, with with the the property that for every f In in Its its domain |Vf| = |f|, is isometric. This is a consequence of the identity (f»g) (f>g) = -jfC |.f+g|2-|.f-g|2 |f+g|2-|f-gl2 )+5l( )+];i( If+igl2_ |f+ig|2- If-igl2 |f-ig|2 ). )• LEMMA 6. Let Let VVhe be a closed a closed isometric isometric transformation transformation with domain 59ft and range range 9?. 9?. 99fftt and and 9? are are closed. closed. Let S1 be an orthonorcnal set , ,2, 2,... ... such suchthat that 9ft((S1 S ^) = 9ft. Then Vfy, Vfy , V2, ... is an orthonormal set S2 such that 99ft(S2 ft(S2 ) = 9i1. 9ft and 9t are closed since V Is is closed and. bounded. The proof of the remainder of the Lemma is analogous to the proof of Lemma 2 above. Theorem XI of Chapter II, §6 has the consequence: LEMMA 7- Let S1 denote the orthonormal set 1, 2, ... S2 denote the orthonormal set vp1, i|>2, ... and suppose that S1 and S2 have the same number of elements. Then the transformation V defined by the equation V( £ a &aa ) is a closed isometric transformation with domain 9ft(S.j) 771 (S1 ) and range 9ft(S2 ).

§2. UNITARY AND ISOMETRIC TRANSFORMATIONS

59

DEFINITION 3. Let Dft1, 5ftg, ... be a sequence of mutually orthogonal manifolds and V1, V2, ... be a sequence of Isometric transformations such that Va has domain 5fta and range 7la . Let us suppose further that the !Ha *s are mutually orthogonal. It follows from the last paragraph of the proof of Lemma 7 , of the preceding section that Dft( u5ft2u ...) is the set of elements in the form € Dfta« ' We define W

•••

- w « -

LEMMA 8 . ... Is isometric with domain Dft(5ft.ju5ft2u ...) and range Dft(51ju5^u ...). If V 1 and V 2 are closed isometric and V 2 is a proper ex­ tension of V1, the domain and range of V2 include properly the domain and range of V 1 respectively. Thus 0X and HA for V 1 are not J0(. On the other hand, if DA and !RA are not 0 let ! with |4>!| « 1 , be € 3)A and i|>! with |ip*| « 1 , be € IRA. Let VQ be defined by the equation VQ(af) = aip1. Then by Lemma 8 , is a ProPer isometric extension of V 1 . Hence LEMMA 9« A closed Isometric transformation V 1 has a proper isometric extension if and only If both 2)A and DRA are not [0 j. The dimensionality of a closed linear manifold 5ft is the number of elements in an orthonormal set S1 such that 5ft(S1) = Dft. In terms of this definition, we may state LEMMA 1 0 . A closed isometric transformation V 1 has a unitary extension U if and only if the dimensionality of Da is the same as that of DR\ To show that this Is necessary, suppose that V has a unitary extension U. We take a complete orthononnal set ... , ♦{, 2 > ••• with 4>t, 2, ... € D, 4>j, ... € DA. 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 ISOMETRY______________

orthonormal set which consists of a complete orthonormal for 2) and another for DA is complete. Since U is an extension of (>a € *R and since U is unitary U^ € IRA. Since V, Ua = V2 > I determines !R by Lemma 6 above, we must have that U4>*, U^, ... determines !RA. Thus SA BA and !RA must have the same dimensionality. On the other hand, if X)A and D !Ri A have the -same dimensional­ ity, we can find a partially isometric, VQ such that VQDA = 3RA by Lemma 7 above. Lemma 8 can be used to show that V ^ V q is a . unitary extension of V^. §3 DEFINITION 1. An additive transformation W from ^ to ^>2 which is isometric on a linear manifold Tfi and zero on 571A is called a partially isometric transforma­ 5tt is is called called the the initial initial set setof of Wy 5151 the range tion. 5t the range is called calledthe the final final set set of of W. W. of W Wis LEMMA 1. A partial isometric W is linear. The final set of W is a linear manifold. Let V be the contrac­ with domain domain 5715,71, let be the the projection projection of of tion of W with let EE be Let PF be be the on 557711.. Let the projection projection of of JJ5522 on on 5511.. Then « FVE, W* = V_1F = E\T-1F, W = VE = w*w = E, e, m * = F. p. W*W f1 € O JJ fg € TRl, TO1, Wf Is is defined PROOF: Since f = f,+f2, f, Tl,, f2 as its domain. Since and equals Wf1. Hence WW has has ^^ as its domain. Since |Wf1| * Jf-I |f1| =

Af, ip - Ag,

Vcj> = Bf, Vip = Bg and

(^+) = (V,V^) for every and in the domain of V. Furthermore VA* = B! where B T is the contraction of B with domain the domain of B*B. Let f be an element in the domain of [A1]. We can find a sequence such that 1 ^ * ^ ! — ► {f,[A']f!. Hence the sequence fA!fn i converges and owing to the isometry rela­ tion, V, the sequence !B!fni must converge to a g*. Thus ifn,B!fn i converges Iso jf,g*j. Then by the definition of [B* ], [B*]f exists and equals g*. We also have fA,fn>Btfni — ► J[A1 ]f,[BT]f|. Hence this latter pair is in the graph of

§4. C.A.D.D. OPERATORS

63

[V] and [V] [Af]f = [B!]f. This last equation holds for every f in the domain of [Af] and hence [V][Af] C [B13. If we take f in the domain of [B13j. a precisely similar argument shows the reverse inclusion and hence [V][A!] « [B1]. Theorem VIII of §4 of Chapter IV states that [AT] = A, [B’] = B. Thus [V]A = B. It is obvious from the graphical considera­ tions made in the above that A = [V]"1B, 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[Vr1FB = W*B. Now if B = WA, B* = A*W* by the Corollary to Theorem V of §2 , Chapter IV. Similarly A* = B*W. It should be remarked that the partially isometric W is introduced because in general [V]* does not exist.

CHAPTER VII RESOLUTIONS OF THE IDENTITY §1

In this section we will discuss certain properties of selfadjoint transformations whose range is finite dimensional. While these results will he 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 7ttwhich is determined by the orthonormal set ^ , ... , n. Then H is zero in 7711 and there is an n !th order matrix (aa a,p = 1, ... , n with aa>p = ap>a such that Ha - Z(Saoc H ls bounded. Since H = H*, H is zero on 7Jlx byLemma 8 of §3 of Chap­ ter IV. Let E be the projection on 771. 771A C DH implies that for f € U^, (1-E)f € and Ef € Djj. Since is dense, this implies that is dense in 77i. Since 7tt is finite dimensional and 2)^ is additive, must contain 771. Since H € 771, H = Z ^ a ^ ^ by TheoremXI of Chapter II, §6- a«,p - (HW = ^«»HV = = The bound of H is the bound of its contraction defined on 771 and for this we have IHIq^Xq^ I =

I

xa^

=

|

pi 2 5=5 * p I ^c^c^ajpl2

^ ( Ea,p^aa>pl The converse of Lemma 1 holds.

^

LEMMA 2. A finite orthonormal set S, 1, ... , $n, and a symmetric matrix (aaa,3 =1, ... , n (i.e., aa p ~ ^*3,0 ^ determine a self-adjoint transformation by means of the conditions: If f € 77US)1, Hf ® 0; Hp*

6k

§1 . SELF-ADJOINT TRANSFORMATIONS

65

H is readily seen to be symmetric and defined everywhere. The essential result of this section is that , ... > 4>n can be chosen so that aa ^ p **or real Thus we obtain the "diagonal form" for the matrix. LEMMA 3 . If H is a non-zero self-adjoint transfor­ mation whose range is a finite dimensional 7ft, then we can find a in Wland a non-zero A such that if 9ft1 Is the set of iji in W which are orthogonal to then for every f in Hf =* A(f,)+H^f where H 1 is W 1#

is a self-adjoint transformation whose range

PROOF: Since H is not zero, either C+ > 0 or C_ < 0 . (Cf. Definition k of §3 of Chapter IV). We shall suppose C+>0 . (Otherwise our argument would apply to -H) Let E* be the pro­ jection on 7ft. Then EHE = Hsince H is zero on 3ft*. For every f we have (Hf,f) = (EHEf,f) - (HEf,Ef). Now let f1, f2, ... be a sequence of elements with |fn l“1 and such that (Hfn,fn) — ► C+. It follows that |Efn l £ 1 , (HEfn,Efn ) --► C+. All the Efn 1s are in 3ft whose unit sphere is compact. Thus a subsequence of the Efn !s must converge to a g in 3ft such that (Hg,g) = C+, |g| £ 1 . Furthermore |g| « 1 , since if |g| < 1 , (Hg,g)/|g| 2 > C+ a contradiction. We let = g. If ^ € 3ft1, then (+,) = C+ ^ (Hf,f) * cos2 ot(H,)4*2 sinacosaR(H,., )|2+ ••• + P(An)1(f,n)|2. This will imply the desired inequality. We will obtain an infinite analogue of Lemmas b and 5 . How­ ever it must be remembered that in dealing with an infinite di­ mensional space, one must consider not sums but limits of sums. Thus I^==1 (or ) represents that special limiting pro­ cess in which one (or both) limits of summation are permitted to approach oo. The integral ^ in the Rieman-Stieljes sense, is a more general process of taking the limit of a sum, which in­ cludes the preceding method. Thus the generalization of the ex­ pression for Hf in Lemma b need not be an infinite sum I°* ) tt=-ao \Jf, a 9^(x'^cc but a more general method of taking the limit of a sum. §2

DEFINITION 1. A family of projections E( A) defined for -oo < A < +00 is called a resolution of the identity if 1. E(A) ^ E(n) for A > jj. 2. E(A+0) = E(A). * 3-oqEU) = 0 , lim^ ooE(A) = 1 . A resolution of the identity will be said to be finite, if there is a A1 such that E(A1) = 0 and a Ag such that E(Ag) n.i, »*__________ * E(A+0) = lim^ qE(A+€2), E(A-O) « QE(A“€2). **The following are examples of a resolution of the identity. (a) Let ... -1, Q, ^ , ... be a complete orthonormal set

68 _____________ VII. RESOLUTIONS OF THE IDENTITY________________

It follows from Lemma 6 and 8 of Chapter VI, §1, § 1 , that if A A1 1 > Ag,

E(A1 E(A 1 ) - E(Ag)

is a projection.

Lemma 5 of Chapter VI § §1, 1 , that if (E(A1 )-E(A2 )-E(A2 ))(E(m )-E(m2 (E(A1 ))(E(m 11 )-E( m 2 )) - 0 E( A2 A2 )-E( that

It follows from

Aj > Ag ^ ^

> p2, ^2,

then

E(A 1 )-E(m2 1 )-E(A2 )+ E( A 1 )-E(^2 ) - E(A E(A1)-E(A2

since

It Is also a consequence of Lemma 8 8

)+E(|i a, (a,b)

x a < xa+1 xa + 1

x 2, ... , xn = b, with interval

(a,b)

The interval point

x^

into

partition every

If

n

x Q = a,

x 1,

which subdivides the

smaller intervals (xa -i>x a )«

(x .,x ) will be said to be marked if a (xa-(A) 4>(A) a

TTQ

T0 -* TT. TT0 If

such that

is called a finer E( A)

The mesh of is a resol­

a complex valued function

£A£b

and

TT!

a marked sub­

division of this interval we define

IEnn ,«*E(A) Z “_ a )-E(x0 )-E(x0 M ,«*E(A) = - Z “ _ 1* 1* ((x x ;;)(E ) ( E ((x xa M ))). ). LEMMA 1. 1. bound

(a) (a) (b)

Z^,AE( A)

is a bounded transformation with

C == max|(xy max|(xy|.|. WeWealso alsohave have

Zj-jt.j AE(A) + Zn ,2AE(A) (4^ +< (>>2 )AE(A). )AE(A ). AE(A)+ ,2AE(A) = Z j-p ^ f( 4>^ + 1A E ( A ) ) ( Z n t! 4>2AE(A)) « Z Z ^^ ^^ gg AA EEttAA)).. (Zn 4)2AE(A)) =

(c) (E 1(x;)((E(xa )-E(xa_1) ) f ,f ) . (Enn l,*AE(A)f,f) +AE(A)f,f) )-E(xa_1))f,f). (d) |E n ,«|>AE(A)f|2 - r n0(= 1l 1lt-(x t- ( x yyi2*l(B(x0|)-B(3i^.1) i 2*l(B(x0|)-B(3i^.1) ) f | a . whose indices range over the integers from -oo to oo. -0 0 0 0 . If 771 (A) is the manifold determined by the a for which a A, and E( A) is the projection on 9)1(A), then one can easily verify that E(A) is a resolution of the identity. (b) Let ususrealize realize fSfS asas C 0C0 (Cf. (Cf.Chapter Chapter III, III, Defini­ Defini­ tion 1). 1 ). If IfA A 0, 0 , we wedefine define x A(x) x A(x)“ 0“ 0^^ ** 1 > 1> XA (x) = 1when 1 when x x A,x»(x) ^ , x A(x) = 0= 0 if if AA ^^ xx.. IfIf A > A > 1,1 , X^(x) s 11.. It Is readily verified that the transformations defined by the equation E(A)f(x) = x*(x )f(x ) form a resolution of the identity. A

£

£ £

££

§2 . RESOLUTIONS OF THE IDENTITY AMD INTEGRATION

69

These results are Immediate consequences of (E(A1)-E(A2))(E(m 1)-E(m2) = 0 if A, > * 2 ^ M1 > ,i2. Thus |rn AE(A)f|2 = e “ = 1i(A ) be continuous on the interval (a,b), and let TT1 and 17^ denote two marked subdivisions with TT0 a finer subdivision of TT. Given an € > o, there is a number y = |a(c) such that if the mesh of TT, m(TT) is £ y(0, then |lntc(>AE(A)f-rn ,AE(A)f| £ «|f|. PROOF: Since $ is uniformly continuous, we may define M( o, such that when \ x ^ - x 2 \ < (x2)| < €. Let us suppose m(TT) ^ m. We define ^ ( x ) by the equation 4>n,(x) * if xQt_1 £ x < xft. Since m(TT) £ v, |ni (x)-$(x) [ . Furthermore for TT0< TT X^nlAE(A) = ln AE(A). Hence the transformation En ,AE(A)-IntAE(A) — £ ^ r4>^tAE(A )-E niAE(A) Mo has a bound < « by Lemma 1 above. The existence of Ja4>(A)dE( A)f follows from Lemma 2 In a manner entirely analoguouB to the proof of the existence of the

VII. RESOLUTIONS OF THE IDENEITY

70

Rieman-Stieljes integral in the ordinary sense. Thus Lemma 2 implies that every sequence of partitions TTj ,TT2, ... with na < 1 and m(r^) — > 0 as n --► oo will have I^'AE(A)f convergent. The sequences n 1 1 ,T\2 ^ ... and. ^ i yd o,no .. . will have the corresponding I convergent to d9d the same limit as one easily sees if one considers a subdivision FTn which is a finer subdivision of both n, T11- and TTn, 2 0. THEOREM I. If 4>(x) is a continuous function on the interval a £ x £ b, then for every f, Tf = jJ«KA)dE(A)f

exists. Tf is linear with bound £ maxa £ x £ b ^ x ^ (x) is real, T is also self-adjoint. ~

^

The continuity and the bound of T are consequences of Lemma 1 . Suppose (x) is real. If we let for the moment Tn = AE(A) 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 symmetric and since it is linear it is self-adjoint. LEMMA 3 . If H = tinuous then

A)dE(A) for $(A) real and con­

(Hf,f) = jJ(AQ ) =- M. Then there are two numbers A1 and A2 in the interval (a,b) with Aj £ Aq £ Ag, Aj 4 *2 821(1 such that for A1 £ A £ Ag, (A) ^ C++0. For any such pair, we must have E(A2 )-E(A1 ) = 0 , For let f be in the range of E(A2 )-E(A1 ). Then Lemma 8 of Chapter VI, §1 and the preceding results imply (^(A)dE(A)f,f) = (^4>(A)dE(A)(E(A2)-E(A1 ))f,f) = (r%(A)dE(A)f,f) = y%(A)d(E(-A)f,f) A 1

—

1

A

^(A)d|E(A)f|2 ^ (C++tf)^d|E(A)f|2

= (C++tf)}^d(E(A)f,f) = ( C ^ E U ^ - E ^ ))f,f)

= (C++tf)|f|2. The definition of C implies that this is only possible if |f|2 =0. A further consequence of this situation Is ^(A)dE(A) =

LEMMA

5.

«>(A)dE(A)+^(A)dE(A).

For ^(A) and 2 (A) continuous, we have

(a)

jJ^dEW+jJ^dEtA) - ;J(4»1 +2 )dE(A)

(b)

(jJ^(A)dB(A))(^(A)dB(A)) = 5jt,*2 dB(A).

PROOF: The statement (a) is obvious. To show (b) we note first that since ^ is continuous on a closed interval \ 0 such that if Ix^xgl < m(«) |2AE(A)f-In ,4>2AE(A)f| < «|f|.

VII. RESOLUTIONS OP THE IDENTITY

72

Letting mCFT^) --► 0,

we have

|rnl^2AB(A)f-^(A)dB(A)f| < «|f|. Hence by Lemma 3 , since

| < C,

| ^ < | , 1 ( A ) d E ( A ) r rTt(*)dE( A) is a self-adjoint trans­ formation, whose domain consists of all f !s such that l.l2 d|E(A)f I2 < 00 PROOF:

Hf

and

|Hf |2 =

e x i s t s i f and only i f

|4 >12 d| E( A)f | 2. | $| 2 d|E(A)f |2 g) no4>(*)a(E(A)f,g) = llm 11m ^(A)d(E(A)f,g) p( f) - 11m (f,(E(b)-E(a))g*) (f,(E(b)-E(a))g* ) - (f,g*) 4|>»|2d|E(A)g|2 < OO is a bounded linear functional and hence H o l< by Lemma 2 above. Hence if g is in the domain of H* it is in the domain of H. H. This This and and HH CC H* H* imply imply HH == H*. H*. COROLLARY. If If HH -- ^(A)dE(A), ^(*) o, we can find an a and a b such that |Hf-H(E(b)-E(a))f| = and

||2d|E( A)f 12+^’ co ||2d|E( A)f |2 < e2/2

|f-(E(b)-E(a))f|2 < e2 / 2

by Lemma 1 of this section and the preceding Theorem. Now g = (E(b)-E(a))f Is In the domain of HQ. Thus the above inequali­ ties limply ||f,Hf|-fg,H0g!|2 < e2.

Since € is arbitrary, it follows that f is in the domain of [Hq]. HenceH C [HQ]. The inclusions established in thispara­ graph show that H = [HQ]. The proof is similar in the case of an improper integral of the first kind. LESMA 4. Let H, (A)dE(A), H2 - L” 2(A)dE(A) forand 2 continuous. Then H1-H2 is a transforma­ tion with domain, those f !s for which |212d|E(A)f |2 < o o and Ify$212d|E(A)f|2 < o o . For the contraction given in the preceding Lemma, we have H.j(H2)0 = )0#(H2)0 *= Finally [H^ *H2] = H^. 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 H^f and H^Hgf) exist. 1,H2f exists” is equivalent to |2|2d|E( A)f |2 < oo by Lemma 2 above. On the other hand, when Hgf and ^Hgf exist, H^Hgf) - 11m j|^dE(A)(^«>2clE(A)f) - 11m jJ^dBtAMEdO-EteDj^^dEfAJf = 11m (iJ«,«B(A))(^(A)dB(A))f “ 1±m i’ a*l*2dE^A^f = +i*2dE^)f by Lemma k and 5 of the preceding section and Lemma 1 of this lo

§*+. COMMUTATIVITY AND NORMAL OPERATORS__________ section. Thus f is in the domain of H^ and we must have I < 00 oo.• On the other hand, if the f 1s sat­ ^14 1^>22 12d.IE(AJf*|2 12(11 12 K isfy both conditions, a similar argument shows that H^f « H 1« (H2f) and thus f is in the domain of H 1«(H2f). 1•(H^f). We may sum up by stating that the domain of H ^*H2 is the intersection of the domain of H2 and H^. . We note concerning the 3)Q of the preceding Lemma, that if f € 3^, then by Lemma 1 above Hf = ^(A)dE( A)(E(b)-E(a)f) = (E(b)-E(a))y^tJ0 = = (h5)0. The first paragraph of this proof shows that H 1-H2 C H^. H^. On the other hand [H^Hg] Thus [H^Hg] [E^E2] C H3. [H.,*H2] D [(H1)Q-(H^)Q] )Q-(H2)Q] ( ))Q] Q] = * H^ by by preceding lemma.Hence Hence[H1[H1 -H2 = H^. = [ [(H thethe preceding lemma. •Hgl =]H^. One notes that if 2 is bounded the condition I212d[E( A)f |2 < oo is always satisfied. Under such circum­ stances, one has H2*H1 C H1*H2 = H^. LEMMA 5. If a continuous $(x) $(x) is is >> 00 for for aa o,let 4>(x,m,€) = 1 for x £ m, 4>(x,m>0 - 1-(x-m)/€ for m < x £ M+e, (x,ii,€) = 0 for x y jj+€. Now c()(x,}ji,€) is continuous and hence H(n,e) = A,ja,€ )dE(A) commutes with T. Since Ja(A,|u,e)dE( A) = E( |a)-E(a) = E(m ) and j’ ^+€(A,^ji,€)dE(A) * o we have H(|i,0E(m ) - J{f€ 0, wecan find polynomials p(x) and q(x) such that |P(x)-p(x)| < €/2, |Q(x)-q(x)1 < €/2 for -C £ x £ C. Then |R(x)-(p(x)+q(x))| - |P(x)-p(x)-^(x)-q(x) | 0, then there exists a finite resolution of the identity E( A) such that H = ,J?c _6AcLE(A).

We can refine our result somewhat by using the considerations of §2 , Chapter VII, which follow Lemma If C+ for H is < C, we take a 6 so small that C++6 < C. In the discussion referred to, we let A- = C +6, = C. We then obtain that * -Cj.+6 * + 72 J_q_€AdE(A) * I_Q_€AdE(A) for every6> 0 and E(C)-E(C++6)=0. SJnce E(C) = 1, we have E(C++6) « 1. Now IJq ++(A) for which this integral exists. LEMMA k . Let W be partially isometric, W = W* and let E = W*W. Then F^ = -l(E-fW) and F2 = -l(E-W) are projections such that F-j+Fg = E, F^Fg ~ W. F.j and F2 are self-adjoint, since E and W are. Since W = W*, E = W*W = W2 and. gW = W3 = WE = W since E is the initial aet of W. (Cf. Lemma 1 of §3» Chapter VI). Thus F2 = -J;(E-tW)(E-rtlO = ^(E2 -tWE+EM4W2) = ^(2E+2W) = -l(E-tW) = F. Similarly Fg = F2. Lemma 2 of §1 , Chapter VI now implies that F 1 and Fg are projections.

§4. INTEGRAL REPRESENTATION OF A UNITARY OPERATOR

93

For W in Lemma 3 above, E is the projection on [IR^] = ft and thus E = 1. Since W and 1 commute with E(A) for -1 £ A ). If we combine our results, we obtain, u= We may sum up as follows: LEMMA. 5 . Let U be a unitary transformation such that Uf = f or Uf = -f implies f = 0. Then there exists a resolution of the identity F(), with F(o) - 0 , F(2tt) =* 1 , such that

u -

9h

VIII. BOUNDED SELF-ADJOINT TRANSFORMATIONS

To remove the restrictions on U, we recall the situation described, in Lemma 1 above. We apply Lemma 5 to = FUF con­ sidered on the range of F and obtain a family of projections F() of the range of F. Now either Definition 1 or Lemma 2 of §1 , Chapter VI, can be used to show that F()F is a projection of ft. The orthogonality of F, E1 and E_ 1 Insures that E_1 +F(()>)F and E_1 +E1 +F(cj>)F are projections. We define G() = F()F for 0 £ < Tt, G(cj>) ~ E_1 +F()F for it£ < 2tt and G(2 t[) = E1 +E_1+F = 1 . One can easily verify that the G(cj>) form a resolution of the identity with G(0 ) = 0 , G(2 tt) = 1 . Fur­ thermore we have that ^"e^dG^) = (J2lteidP())P+E1+E_1 = PCP+E1-E_1 = U. Thus we have established: THEOREM II. If U is unitary, there exists a resolution of the identity G() with G(o) = 0 , G(2 tt) - 1 , such that 11 _

CHAPTER IX CANONICAL RESOLUTION 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-ad­ joint and. normal operators. The discussion of Chapters VIII, IX and X is essentially based, on the two papers of J. von Neumann to which reference is made at the end of Chapter I. In the present Chapter, however o —i the use of (1 +H ) to obtain the integral representation of an unbounded self-adjoint operator was suggested by the RieszLorch paper also listed in Chapter I. The canonical resolution of a normal operator is used, to obtain the integral representa­ tion 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 §^, Chapter IV tells us that A - (1 +T*T) 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(A) with E(o-o), E( 1 ) = 1 , such that A - £q A&E( A)+0*E( 0) = £qME(A) .

Lemma 6 of §2 , Chapter VII shows that the zeros of A are 1 -E(1 )+E(0 ) = E(o). Since A” 1 exists, 71^ = [0j and hence E( 0 ) = 0 . Lemma 5 of §3, Chapter VII implies that A-1 = 1 /A)dE(A). We make the change of variable p = 1 /A, F( fj) = 1 -E( 1 /p-o) for 1 £ M £ °°' Since E(0 +0 ) = E(0 ) = 0 and l l m ^ 0 ^ y QE ( \ - o ) « l i m ^ a^0E‘ (A) - limM^0F(p) - 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( m) is a resolution of the identity and. that for 1 £ b < oo, 5]/b(l/A)dE(A) = ^ d P ( M)+P(l). * Proc. Imp. Acad., Todyo

15,

pp. 95

2 0 7 -2 1 0 ,

(1939).

9 6 _______________IX. CANONICAL RESOLUTION__________________

When we let b — ► oo, 00, we see that corresponding improper inte­ grals are equal and thus A"1 = j’(l/A)) n'). y J® i M'\i|E(A)f j'1|'d|E( A)f |2 < o o 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, F1 (A) and Fg(A) be as above. Then WF1 (A)W* = F2 (A)-F2 (0 ), W*F2 (A)W F1 (A)-F^ (0 ), WHAT* = C, B = W*CW. Let E1 = W*W, E2 = WW*. These are projections on, respec­ tively, the initial and final sets of W, i.e., [Rg] and C!Rrp]• Now if we use [9^1* (Theorem VI of §2 , Chapter IV) and B = B*, C = C*, 51^ = ftg, 57c = we obtain [Rg] = 5^, [DRrp] = - 3^. Thus E1 is the projection on !>?£, E2 on 7t*. From the expression for B and C given in Lemmas 1 and 2 above, we have that 57-g has the projection F^o) and has the projection F2 (o) (Cf. proof of Lemma 6 of §2 , Chapter VII). Hence E1 = 1 -F^o), E2 = 1 -F2 (0 ). Thus Ea commutes with Fa, a « 1 , 2 . From Lemma 3 of §3, Chapter VI we see that W = WW*W - E2W = WE1. Now WF1 (A)W* is bounded and self-adjoint. Furthermore WP1 (A)W* = WE1P1 (A)W* = WE.,F2(A)W* = WF., (AJE^, (A)W* = WP1 (A)W*WP1 (A)W* = (WP^AJW*)2. Thus P^( A) = WF., (A)W* is a projection. (Cf. Lemma 2 of §1 , Chapter VI). Since E2F^(A) = E2WF.,U)W* - WF^ (A)W* - F^( A), we have F^(A) £ Eg. (Cf. Lemma 8 of §1 , Chapter VI). If we define F*(A) = W*F2 (A)W, a similar argument will show that F* (A) is a projection with P|(A) £ E1. We next observe that C2 = T*T = WB'B/tf* = WB%*. Since is the projection on [!Rg] and on we have W*C% = W*WB%*W - E1B2E1 - B2. Since C2 - WB2W*, we have F^(A)C2 - WF1 (AJW^WB^* WF1 (A)E^B%* =WF-(A)B%* C W ( ^ M2 dF1 (M))W* = WB2F.,(A)W* =* WB2E1F1 (A)W* « W b W ^ A J W * = C2F^(A). (Lemmas 1 and k of §3, Chapter VII aee used here). TJius F|(A)C2 C C2F|(A) and the latter has bound A2. Similarly Fj(A)B2 C B2Fj(A) and the

98

IX. CANONICAL RESOLUTION

O latter has hound A . Lemma 2 above now shows that F^(A) commutes with F2 (|i). We note that F^(A) £ Eg = 1 -F2 (0 ). Consider F^( A)( 1-Fg(A)). Lemma k of §1 , Chapter VI shows that this is a projection. Suppose that an f + 0 is in the range of this projection.For a resolution of the identity, we have lim^^F^jj.) «= 1 and lim€^0P2 (A-re2) = F2 (A). Since f = F£( A)( 1 -F2 (A) )f * 6 , it follows that we can find a A and a AQ with A < AQ < A such that g = F^(A)(F2 (A)-F2 (A1 ))f + 0. Owing to the commutativity of F£(A) and F2 (ji),we also have g = (F2 (A)-F2 (AQ))g = F'(A)g. Hence |C2g| 2 =|C2 (F2(A)-F2(*0))gl2 = ^ Q^d|F2(A)g|2^ ^|(F2 (A)-F2 (A0 ))g| 2 = Aj|g|2. (Cf. Lemma 3 of §2, Chapter VII and Lemma 1 of §3 , Chapter VII). Also |C2 g | 2 = |C2F^(A)g| 2 £ A^|g|2. Since |g| + 0 , AQ ) A ^ 0 , these statements contradict each other and thus f =0. Thus we have shown that f in the range of F^( A)( 1 -F2(A)) implies f = 0. It follows that F£(A)(1 -F2 (A)) = 0 or F»(A) £ F2 (A). (Cf. Lemma 8 of §2 , Chap­ ter VI). Since we also have F2 (A) £ Eg - 1 -F2 (0 ), we have F ’(A)(F2 (A)-F2 (0 )) = f ’(a)(f2 (a)-f2 (a)f2 (o)) - f ’(a)f2 (a)(i-f2 (o), - F’(A)(1 -F2 (0 )) - F£(A), and F ’(A) £ F2 (A)-F2 (0 ). (Cf. Defi­ nition 1 of §2 , Chapter VII and Lemma 8 of §2 , Chapter VI). Now WF1(0)W* - WE1F1(0)W* « W( 1-F1(0) )F1(0)W* = 0. Similar­ ly W*F2(0)W = 0. Thus F^( A) £ F2(A)-F2(0) becomes F2(A)F2(0) ^ WF^ (A)W* » W(F1(A)-F1(o))W*. Multiplying by W* on the left and W on the right, we obtain F.j(A) = W*(Fg( a)-F2(0) )W ^ W*W(F1(A)-F1(0))W*W - E1(F1(A)-F1(0))E1 = (1-F1(0))(F1(A)F1(0)) (1—F^(0 )) - F^AJ-F^O), or F«(A) > F^AJ-F^O). (Cf. Definition 1 of §2 , Chapter VII). But a proof analogous to that of the preceding paragraph will show that F.[(A) £ F^AJ-F^O). Thus we have established that Fj(A) - F^AJ-F^O). This last result may be written W*(F2(A)-F2(0 ))W = F^A)F^O). Multiplying by W on the right, and on the left and proceeding as above we obtain F2 (A)-F2 (0 ) = W(F1 (A)-F1 (0 ))W* = F£(A) - WF^ (A)W*. If we form partial sums, use this last equation and pass to the limit, we obtain J£mcIF2 (m) since either side is defined everywhere. Taking limits, we get

§1 . THE CANONICAL RESOLUTION

99

or C = WEW*. Multiplying on the left by W* and. on the right by W yields W*CW = W*WM*W = E1BE1 =B. Our lemma Is now dem­ onstrated,. 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 [Rg] and final set and such that B is definite self-adjoint and possesses a resolution of the Identity F1 (A) such that B = j‘ ^°ydF1 (y) and furthermore such that T as WB. Suppose that the pair, W, B = ^ydF^y) and the pair, W^, B1 = jjid.G1 (y) both satisfy the given conditions. We can suppose that W and B are as in Lemma 1 above. By the corol­ lary to Theorem V of §2 , Chapter IV, T* = B*W* - B^W* and since W*W is the projection on [!Rg]> we musthave T*T = B2.(Cf. Lemma 1 of §3, Chapter VI). Thus B2 = T*T = B2 and J“ |j2dG1(|i) = j‘* M 2dP1(|4) by Lemma 4 of §3, Chapter VII. Now G.(A) commutes with B? by Lemma 1

of §3 , Chapter VII and thus with B . Lemma 2 above, shows that G-(A) commutes with F^y) and If we consider the bound of B2 G1 (A) we see that precisely the same argument as that used in the proof of Lemma 3 above will show that (A)~G1 (0 ) F1 (A)-F.j (0 ). We also have in, the above that F^y) commutes with G^ (A) and thus we may proceed to obtain the symmetric result F*!(AJF-j(0 ) £ G1 (A)-G,j (0 ). Thus we have shown F^AJ-F^O) = G1 (A)G^o). Since Dig2 1=5 ^ 2 , we also have F^o) = G^O). We may conclude that F^(A) = (A) and B = B1. Since [R-g] = [3^ ], the initial sets of W and W 1 are the same and both W and W 1 are zero on [RgP. (Cf. Defini­ tion 1 of §3, Chapter VI). The equation WB = T = W ^ shows that W = W 1 on Rg.Continuity implies W = W* on [3Rg]. We also have W ~ W 1 = o on and since these transformations are linear we must have W = W 1 . COROLLARY. A similar result holds for T = CW.

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 he c.a.d.d. Then there exists operators W, B, C and resolutions of the identity and F2 (A) such that (a): W is partially isometric with Initial set [Rg] and final set [JRrjJ. (h): B and C are self-adjoint and definite, (c): T - WB = CW. (d): T* = BW* = W*C. (e): C = WBW*, B = W*CM. (f): B = ^"AdP^A), C = J ® A d P 2 (A). (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, ydF^ (y) a definite, self-adjoint C and a partially isometric W such that H = WB = CW. Since H*H = H2 = HH* we see from Lemmas 1 and 2 of the preceding section that B = C and thus H = WB = m .

Thus W commutes with C and Lemma k of §4, Chapter VII shows that W = W*. Let E = W*W. Since W = W*, W = WE = W5= EW. Since W commutes with B, W commutes with B2 and thus Lemma 2 of the preceding section shows that W commutes with F^A). In the proofof the same Lemma 2 , it was also shown that E « i-f (o) and consequently E also commutes with F^A). Thus if P 1 = -|(E+W), P 2 » -|(E-W) then P1 and P2 commute with F1 and F2 are projections by Lemma k of §4 Chapter VIII. F1 (A)F1 and F,j(A)F2 are projections by Lemma k of §1 , Chapter VI. We also have F ^ = -J-(E+W)(E-W) = ^(E2 +WEEW-W2) = -^(E+W-W-E) = 0 . We have then:

101

§2 . SELF-ADJOINT OPERATORS H = EW=

ndF,(n)(Fi—F2) = ^ MdF1 (M)F1 -J“ MdF1 (M)F2.

Let

A = -n, G(A) = (1 -F1 (-A-0 ))F2 for - o o < A < 0 . Then l U n ^ C K A ) = 11m (1 -P1 (m-0 ))F2 - F ^ l ^ . P ^ M J P g - V P2 = o and llm^0 G(-e2) = lim (i-F, (e2 -0 ))P2 = H m ^ d - F , («2) )Fa = (l-P^o+OJPg = (1-F1 (0 ))F2 = EP2 = (F1 +P2 )P2 - F2. Aa in the proof of Lemma 5 of §4, Chapter VIII, we have that for b > e2 >°* 2 -f^MtOFa AdG, (A). Letting

0,

we have

-®*SP1 (M)Pa = lime “ £-b°+

jOjAdG^A) = ^°AdG1(A)+^ _ 0AdG1(A) - -jJfldF,(M)F2+0. Letting b — ► oo, we obtain -^MCHV^F, = ^AdG^A). For 0 < A < cx>, let G1 (A) = F2 +1 -E+F.jF(A). Our previous orthogonality relations and Lemma 5 of §1 , Chapter VI, shows that G1 (A) is a projection. One has also lim^^G^A) = F2+ 1 -E+ lim^F 1F 1 (A) » F2 +1 -E+F1 ~ 1 , since E = Fg+F1 . By a familiar reasoning, we obtain that {“ AdF^AJF -

AdG, (A).

This and our preceding results imply that H - J^MdF1 (M)Fr(fJ,M ® l(»*)F2 “ l!LAdG1 - oo, we have that H = ^+AdG1(A)+C_G1(C_),

while if C__ = - oo, we have H = I-i^G^A). This is shown by a discussion similar to the proof of the Corollary to Theorem I of §3 , Chapter VIII. For a definite operator A, we have C_ ^ 0 . In any case, we may take 0 , oo as our limits of integration and A - J^AdG^A). In Theorem I of§1 above, we now have that (b) implies (f). This demonstrates: COROLLARY 2 . In Theorem I of §1 , above (c) determine W, B and C uniquely.

(a), (b) and

COROLLARY 3 . If T commutes with H, T commutes with G^(A) for - 0 0 = 0 ,) let (P)= P1 (p)G2 (4>). Since P^p) and. G2() com­ mute, one can show that E^P) has the following properties: (a): If P1 and P2 are two points with coordinates (p*l (P2 ) respectively and If Q is the point whose coordinates are (min(p1 ,pg)> min(4>1 ,4>2)) then E1 (P1 )•E1 (Pg) = E1(Q). (b): If we let E1 (p,4>) =E.,(P(p,4>)) then E^p+ 0 ,^+0 ) =

E^p+0,4)) = En (p,+0) = E^ ( p,).

(c): If we now let P^p) = E1 (p,2n-0 ), for p > 0 , F^O) = E^ (0 ,0 ) and F.,(p) = 0 if p .< 0 then F^p) is a resolution of the identity. Similarly if we let G2 () = litn^^ E«j(p,) for 0 £ < (> < 2tt, G2() = 0 for < 0 , G2() = 1 for ^ 2 tt, then G2 () Is a resolution of the identity with G2 (2tt-0 ) = 1 .* DEFINITION 1 . A family of projections E^(P) having the properties (a), (b) and (c) above, will be called a planar resolution of the identity. Let us consider a sector S of a circular ring, S not containing the. polar axis. Thus S is the set ofpoints (p,) with p1 < p £ p2, i 2 *We can associate with it a pro­ jection E1 (S) - (F1 (p2 )-F1 (pl))(G2 (2 )-G2(l)) - E^P^-E^Pg)E1 (P^ )+E1 (P^), where P1, P2, P^ and P^ are the points * These statements are redundant. For instance in (c) it is only necessary to show that linu^F., (p) = 1 and that Gp(2 n-0 ) = 1 , since the other properties are consequences of (a); (d) and the definitions of F-j(p) and G^(). For example a consideration of projections associated witn the areas as in the following dis­ cussion will show that G (+0 ) = G2().

§?. NORMAL OPERATORS

107

(p2 >2)j (P^+g) and (p1,,) respectively. (E^S) is a projection by Lemma 4 of §1 , Chapter VI). Any such S can he expressed as the logical sum of mutually exclusive smaller Sa, i.e. S « S1 u ... u S , and we will call this a partition of S. We can then form for any function the partial sums I i|i(Qa)E(Sa) where Qa € Sa. If is con­ tinuous, it can be shown that for every sequence of partitions such that the maximum diameter of the Sa !s approaches zero, these partial suras approach a limit which we will denote: Jj^KPJdE^P). Furthermore a familiar discussion will show that ^(PJdE^P) = ^ 2 ^ tKP(p,-) = |i(Um (H+i)fn) = ±1 11m (-2 ifn) « 11m f . Simlliarly f* = = 11m Hfn. Thus If we let f = |i(q»-) then Hf exists and = |(+i|i), since H is closed. We also have if= ^(-^)> and thus (H+i)f = ,(H-l)f = Hence i,ip| is in 23. Thus 23 contains its limit points and thus V is closed..It follows by definition that V is closed. Since 13 is a linear manifold, V is additive. Further­ more, if 1 and 2 are in the domain of V then = (H+i)f1, 2 - (H+i)f2, V^ - (H-i)f1 and V2 - (H-i)f2 for some f^ and f2 in the domain of H. Hence Lemma 1 above implies (4>1 f4>2) - ((H+i)f1,(H+i)f2) - ((H-i )f1,(H-i )fg) = (V^ , V2). 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 » 0 implies f = 0 . Lemma k and Definition 2 of §1 , Chapter IV, now show that (H+i) " 1 exists. Now if c(> is in the domain of V, = (H+i )f for f in thedomain of H. Hence (H+i)~1 exists and equals f.Also V = (H-i)f = (H-l)(H+ifV Thus V C (H-i)(H+i)"1. On the other hand, if is in the. domain of (H-i)(H+i)-1, we let f = (H+l)-1 and ip = (H-i)(H+i)-1. Since = (H+i)f, + = (H-i)f, we have V = i|i. This shows that V D (H-i)(H+i) - 1 and with our previous inclusion proves the equality. H1 a proper extension of H2 is equivalent to H.j+1 a proper extension of Hc+i, which in turn is equivalent to (H^i) 1 being a proper extension of (H2 +i) 1 . Now the do­ main 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 0

—

—

112 (H+i)

X. STMMFTRIC OPERATORS _ 1

being a proper extension of

_ i1

(H2+i)

is equivalent

to V or ).). Thus or ff = ^i(V-1 = ^i(V-1 Thus the the range of -i(V-1 ) )includes includesthe the domain domainof of H, H,wh whiicchh is is dense. The statement (g) follows follows easily easily from from this. this. Proof of (h). We first first proveprove that if isif isometric andand We thatV V isisometric &,_y is 1 1exists. 1 1exists if is dense, dense, then then(1-V)” (1-V)” exists.No wN o w(V-1)” (V-1)” exists if and only if of §1, if (V-1 (V-1 )) = = 00 implies implies $$ == 0. 0. ( C f . Lemma ( C f . 4Lemma 4 of §1, Chapter IV). Let us suppose that (V-1 )=© or V = . \\> in the the domain domain V, w e have in of ofV, we have

For

(V,Vip)-(,ip) = (,V^)-(,+) = (>Vip-1|/). 0 = (V,Vip)-(,ip) Thus is is orthogonal orthogonal to to ^^yy-1 -1 and. and. since since this this last lastset is 0. dense, we must have = 0. 0. Hence Hence (V-1 (V-1)== ©©implies implies == 0. Furthermore if f isis in in the the domain domain of of H, H, we we have have for for aa € Dy, -2jf = (V-i )4> and 2Hf -- (V+1 (V+1 ))< It follows €

>.. It follows that that $$ --2i(V-1 )f and HHff = -i(V+1 )(V-1 )“ 1f . Thus HH CC -i(V+1 1 .. )_1f -i(V+1)(V-1 )(V-1))_1 On the other hand, hand, in in the the above, above, we we have have shown shown that that if if gg == (V-1 (V-1 )),, is equivalent g is in the domain domain of of H. H. Thus Thus IRy^ IRy^ CC D H .D HThis . This is equivalent Kv-1 cC Bh D h.. If If TT =- --i(V+1)(V-1 K V + 1 )(V - 1 ff 1, to 2>(v_ i jr- 11 == Ky., \ BDtT == DD (y (y__11)_1 since

V+1

is defined everywhere on the range of

(V-1)- 1 .

811(1 our Thus D t = ®(y-i )~1 c 811(1 our Previous Previous inclusion inclusion HH CC T, T, this shows T = H. If EEVV CCVE, VE, we we see see that that the Proof of (k). If the domain of these transformations include Dy. We also have E(V-1 ) C (V-1 )E. If f is in the range of V-1, i.e. f - (V-1 ), for a $ in the domain of V, then Ef - E(V-1 ) $ - (V-1 )Ef> an d Ef is also

§1 . THE CAYLEY TRANSFORM

112

in the range of (V-1). Furthermore Ef = (V-1 )E= (V-1 )E (V-l)~1 f, or (V-1 )-1Ef = E(V-1 )~1 f. Since this holds for ev­ ery f in we niust have E(V-1 ) _1 C (V-1 )-1E. We also haveE(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 for H obtained in the above now shows that EH C HE. DEFINITION 1. If H is closed symmetric ,and V is as in (a) of Lemma 2 above, then V = is called the Cayley transform of H. In the proof of (h) above, we have shown: LEMMA 3 . If V is isometric and such that dense, then (V-1 ) ” 1 exists.

is

LEMMA 4. Let V be closed and isometric and such that is dense. Lemma 3 above shows that (V-1 ) ~ 1 exists. Let H = -i(V+1 )(V- 1 ) ~ 1 . Then (a): H is closed symmetric, (b): The Cayley transform of H is V. (c): Let 71^ = , then the domain of H and or have only © in common, (d): The domain of H* consists of elements in the form f+g.,+g2 where f € D^, g1 € g2 € \ and H*(f+g.,+g2) - Hf-ig1 +ig2. PROOF OF (a): If is in the domain of V , let f k(V- 1 ). Then = -2 i(V-1 )f, and Hf = (V+1 )(-i(V-1 ) " 1 )f |(V+1). Thus if f1 and fg are in the domain of H and ^ and 2 denote the corresponding we have (Hf.j,f2) = ((V+15^,1 (V-1 )2) - -i( (V+l)1 ,(V-1 )2) - -i[(V1 ,V2)+(1, V2 )-(y1 ,4>2 )-(T,2)] = i[(Vn,2)~(2 ,V1 )] because for an isometric V, (V^ ,V*2) - ( * 1 ,*2). Similarly (f.Hfg) i U V ^ , ^ ) - ^ / ^ ) ] . Thus for every f1 and. fg in DH, (f ,Hf2) - (Hf\j,f2). Furthermore Dg « D^v _ 1 is by hypothe­ sis dense. Thus Definition 1 of §3, Chapter IV shows that H is symmetric. The proof that H is closed is analgous to the proof of •the closure of V, in (b) of Lemma 2 above.

X. Sm/JETRIC OPERATORS PROOF OF Cb): If 4> is in the domain of V, f = ~i(V-l) is in the domain of H with Hf = ^(V+l)4>. Thus = g(V+l) -1 (Y-1 ) = (H+i)f and V = ^(V+1 )+l(V-1 )$ =(H-i)f. Thus V is included in the Cayley transform of H. If, however, were a proper extension of V, (vh~ 1 w011^ ^e a proper ex­ tension of (V-1 ) ~ 1 since ^ y _ 1 j-i ~ 2y. Hence j-1 is included in but not equal to ^(vh-1 ) - 1 * However we see from our hypotheses and (j^) of Lemma 2 above that these sets are both 2^. This is a contradiction and we have V = V^. PROOF OF (c). Let us suppose that g is inWj/2^ and g^ 9. Then g= i(V-l ) for € 2y. Since 5^ = 2y, we must have 0 = (g,) = i(V-,). This Implies (V,) = (,) = || 2 = |V|.||. Thisis only possible if V * k for a constant k. If g $ 0, $^ 0 and (V,«|>)= (,) impliesk =» 1 . Thus V == or (V-1 ) = 0 . This implies g = I(V-1 )cj> =© contrary to our supposition. Thus g = © and ^*2^ - !©!. The proof of is similar. PROOF OF (d). Let 6 denote the graph of H and 6 * de­ note the set of pairs jf,H*fi, i.e. the graph of H*. Since H is symmetric, we have C 6 *. Consider *6 * and let us suppose that Jh,H*hj is in We have for every f in the domain of H, (f,H*h) - (Hf,h), 0 = (if,Hfi,|h,H*h|) - (f,h)+(Hf,H*f). This implies that for every in the domain of V, (ii(V-l),H*h-ih)-(c|>,H*h+ih) « 0 , (V,H*h-ih)+(4>,H*h+ih) - 0 , and these equations are equivalent to (V,H*h-ih) = o, (, H*h+ih) = o. Thus if -2 igl = H*h-ih, 2 ig2 = H*h+ih, g., is in Jl_± = Jty , g2 is in ^ » h = g.,+g2, H*h = -igl+ig2. Thus if |h,H*hj € h = g-i+gg, g, € g2 € 5^, H*h = -ig^igg. On the other hand if H = g.,+g2, h* = -ig+ig2, reversing

§1 . THE CAYLEY TRANSFORM

115

the above discussion, will show that g1 € 93 ^ and g2 € 9^ implies (f,h*) - (Hf,h), o - (|f,HfI,|h,h*|), for every f in the domain of H. Theorem II of §2 , Chapter IV, shows that H*h exists and equals h*. Thus we may con­ clude that fh,h*j € If k is in the-domain of H*, Jk,H*kl is in R* and Jk,H*k j - [f,Hf! + lh,H^h! where ff,Hf! € «, !h,H*hi € «*.«*, by Corollary 1 to Theorem VI of §5, Chapter II. From ^he above, we obtain k = f+h - f+g-j+gg where f € 3^, g1 € 91^, g2 € 9^, and H*k = Hf+H*h » Hf-ig1 +ig2. Thus every element k in is in the desired form and the converse is also readily shown when our previous results are used. Furthermore, the given for­ mula for H*k holds. This completes the proof of the Lemma. We may now state: THEOREM I. If H is closedsymmetric, there exists a closed isometric VH called the Cayley Transform, having the properties (a) to (i) of Lemma 2 above. If V is closed isometric and such that 9RV- 1 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 VR is unitary, i.e. B*. = = |0 |. If VH is unitary and B^ = }©}, !R^ « }©}, (d) of Lemma 3 shows that the domain of H* is simply that of H. Since H € H*, we must have H = H*. If VH is not unitary, either B^. or !R^. ^ fei. Let us suppose that g-j ^ 0 is in B^.. By (c) of Lemma 2 above, g1 is not in B^. However (d) of Lemma 2 above shows that g1 is in BH#. Thus H* f H. COROLLARY 2 . If H is closed symmetric, H has a maximal symmetric extension. (Cf. Definition 3 of §3, Chapter IV). H has a closed self-adjoint extension and only if B^. has the same dimensionality as 9*y.

if

116

______________X. SYMMETRIC OPERATORS__________________

PROOF: If VH is such that either !R^ or 2^. = [0j, (©}, then VH V jj has no isometric extension and it follows from (d) of Lemma 2 above that H is maximal symmetric. Thus we may consider consider the case where both !R^ and and 2^ 2^ are arenot not{©}. {©}. For convenience convenience let us assume that 2) ^^.. has dimensionality less than or equal to !R^. By using Lemma 77 of of §2, §2, Chapter Chapter VI, VI, we we can find find an an iso­ iso­ metric V* with domain 2^ 2^ and and range range included included in !!RR^^.. Lemma Lemma 88 of §2, Chapter VI shows that VV^ = V ® V 1 is an isometric trans­ formation such that V 1 3 V, 2y^ = fi. Since V^ 3 V, IRy^ 1133 V r Since the latter is dense, Kyl-1 is also and. Lemma k above shows that there is a symmetric H1, whose Cayley trans­ form is V^. V^ • Since 2y = f),H1 must be maximal symmetric as we remarked above. Since V 1 3 V, Lemma 2 (d.) above shows that H1 is a proper symmetric extension of H. A similar argument holds if the dimensionality of 2^ is greater than that of with however the result that I V1 V has a unitary extension V^, if and only if the dimension­ ality of !Ry is the same as that of 2y. (Cf. Lemma 10 of §2, Chapter VI). Since Ry_., is dense, we see from Corollary 1 above, that H1 can be taken as self-adjoint if and only if the dimensionality of !Ry equals that of 2y. We have also shown: COROLLARY 3. H is maximal symmetric if and only if at least one of the 2^.or ! or R^. consists consistsofof 0 0 alone. §2

In this section, we present an analysis of maximal symmetric operators, obtaining both structural and existential results. DEFINITION 1. Let 0, , 2,.... be a complete orthonormal set in fi. (Cf. the end of §6, Chapter II.) Let VQ be the transformation defined by the equation, V X~=oa« V = Za=oacA:+1 Ia=oaaa+1 * • Let E denote the projection on W«*= OT( | i . i.e. e . the range of VQ. (Cf. §6, D?( | !1,2,... ...}), Chapter II, Theorem XI). LEMMA 1 (a): VQ is isometric with domain fS and range (b): VQ is partially isometric with initial set £

§2 . STRUCTURE OP MAXIMAL SYMMETRIC OPERATORS

117

and final set m a . VQ* - V0‘1E, VQ*V = V0"1Ve= 1, V QV Q* = E: (c): Ry-1 and !Ry_-]_1 are dense: (d): (VQ-1 )“1 and (V„~1-1 )-1 exist and (V0"1+1)(V0"1-1)"1 = -(VQ+1)(V0-1 -1 )-?. (a): is a consequence of Lemma 7 of Chapter VI §2 . (b): follows from Definition 1 and Lemma 1 of §3 , Chapter VI. (c): We first show that Ry is dense. By Theorem VIof §2 , Chapter TV, we have Wy0_i )A = fty . Thus if f € (HVcr1 )S (VQ*-1 )f « 0 or VQ*f -°f. Since° f € 8, f - aQ0+ a ^ ^ ... (Cf. Theorem XII of §6 , Chapter II). We have VQ*f =* V0 lE0^a0^0+ai1+ #** ^ “ V0 1 +a22+**• ) = ai^o+a2^1 +-Thus VQ*f = f implies a = a1, a1 = a2 ,....etc. Since laal2< 00 , lima >aoaa = 0, and hence aa = o for «= o, 1 Hence f = ©. Thus f € (5?y _1)* implies f = e and hence 3*y j is dense. ^VcT1 -1 is also dense* Consider !Ry0*„g and suppose g € !RVo*_e - Then g » (VQ*-E)f = (V0_1E-E)f°= (V0_1-l)Ef. Thus g € %o*-E m v l l e 3 g € ^ Vo"1-i 0I> RV*-EC5V Q*1-r Hence c (RVo*_b> )* = 71 v0-e by Theorem VI of §2, Chapter IV.° Thus f € (RVq_i_1 )* implies f € 9?Vo_E or (VQ-E)f = 6 or VQf = Ef. Lettingf = aQ0+a1^ +.... as before we obtain V0f 88 aQct>r+a12+.... while Ef = a ^ +a22+.... Since V f = Ef we must have a = a1, a1 = ag,.... etc. Thus ijaj2 < oo again implies a = 0 , a = 0 ,1 ,... and f - ©. Thus f €(^yQ-l_ 1 )l implies f = 0 and ^yQ-i_ 1 is dense. (d): Lemma 3 of §1 above shows that (VQ-1 ) ~ 1 and (VQ1-1 ) ~ 1 exist. Now V *~1-1 is defined only on 5ft and thus V 1-! = V0-1-V0V0-? = (1-V )V -1 since T V ’1 = 1 on m . Thus (V0_1-1 f 1 = ((1-V0)V0“T)"T - V0(1-V0)"T, and (V0-1 +1)(V0-1-1)“1 = (V0_1+1)V0(1-V0)-1 = ( U V ^ d - V ^ ' 1 = -(V0+l). (VQ-1 ) ~ 1 and this completes the proof of the Lemma. LEMMA 2 . Let HQ correspond to of the precedingsection. Then H _1 o and -Hq corresponds to V .

VQ as in Theorem I is maximal symmetric

Since ^y0««| is dense, there is a symmetric H whose Cayley

X. SYMMETRIC OPERATORS

118

transform is V0. Corollary 3 of §1, above shows that HQ is maximal symmetric. Lemma 1(d) and Lemma 4 of §1 show that -H0 -1 corresponds to VQ LEMMA 3 . Let V be maximal isometric, i.e. either 3Xy= ft or !Ry= ft. Then we can find a finite or Infinite set of manifolds 77lQ, 7711, 77l2 , ... with corresponding projections E0, E1 , Ep, ... having the properties: 1 . 'E Y » VEa = EaVEa 1= W v v = Za=oEaVE«- 2- If m 0 * !0!’ then BoVEo when considered on DW0 alone is unitary. 3 . If 77l0 + ft and (a): Dy = ft, then for 307^ a^ 1 , we can find an orthonormal set * 0>♦«, 1 ' ' * * i )

•£,01, every

a

119

3UC]tl that 'I'a

is defined. Prom the above, it is readily seen that the S^'s are m utually orthogonal. Since V( = r p=oapa,p+l ’ VEa = EaVEa . Furthermore V * ( I ^ _ 0a ^ p ) - V 1 (1-F)< “

V

^ i y « (p = r p = i y « , p - r

®ms

V*Ea = E aV*Ea.

Taking

adjoints, we obtain EaV = EaVEa = VEa. By Lemmas 5 and. 7 of §1 , Chapter VI, Ea=s-|Ea ls a Projection with range 771{711 771 ...) - 77l( ia ). Since E^V = VE^, forming sums and if necessary taking limits, we obtain ( Ia=1E )V = V ( I « .lV and

L rt

1 - I o,=0Ea .

Eq = l - Z a= 1Ea. Furthermore

Then

VE0 = EQV = EQVE0,

V = 1-V = ( I a=0Ea )V

-

^a=0EaV

« Z a==0EaVEa. The properties listed in (1 ) have now been estab­ lished. The range of 1 -EQ is |a ) and hence includes 771 (|a 0 1) = 77l{ |a 1, ... Thus we may state THEOREM II. Suppose H is maximal symmetric. Then there exist^ mutually orthogonal manifolds, W]0, 57^, 9tt2, with projections, EQ, E1, ... respectively, such that Z a==0Ea = 1 and such that E ^ C HEa for every Ea. Let H ^ denote the contraction of H with domain ^ and which is regarded as a transformation on Then for every f € Hf = I a=sQH^a^Eaf. Furthermore if 5W0 ^ {0}, H ^ is self-adjoint. If !ffi0 ^ ft, we have at least one for an a ^ 1 andtwo cases are pos­ sible: (a) If V Is the Cayley transformof H,then Dy - ft and H^for the ot^ 1 , is such that there Is an orthonormal sequence a Q,a 1,... for which H^ is analogous to the HQ of*Lemma 2 or (b) Ry = ft and H ^ for the a ^ 1 Is such that there Is an orthonormal se( oc) quence, a 0 >a ••• for which Hv J is analogous to -Hq of Lemma 2 . The converse to this resultcan also be given. Let a number of Hilbert spaces, rt, ftI ,, ftr be given, /0\ ftU tzt,... and consider a self-adjoint Hv ' in ft~ and realize Hn //y\ ^ ^ as a Hv } in each fta for a - 1, 2 , ... . We may form ft = ftQ © ftf «. .as in Definition 1 or 2 of §3, Chapter III. Let HffQ, f,j,...| = {H^0 ^fQ,H^1 ^f1,...! when both sequences exist and are in ft. Now if V ^ is the Cayley transform of H ^ , we know from Lemma 8 of §2 , Chapter VI that V = V0 « V 1 *.... is Isometric with Dy = ft. Onecan also readily show that V | f 0, f

= |V ^ °'f0, V

^

f

t

h

a

t

5lv _1

i s dense,

and that H corresponds to V as in Theorem I of §1 above. Thus H is maximal symmetric. If we take H^0^ as a realization of -HQ, we have !Ry = ft and H Is again maximal symmetric. Thus we have: COROLLARY 1. H

If

H

is maximal symmetric.

Is constructed as in the above,

CHAPTER XI REFERENCES TO FURTHER DEVELOPMENTS Our main purpose in this Chapter is to give references in a number of topics for further reading. We will also give a brief heuristic Introduction to each topic. Our references will be numbered asthey are introduced. Two essential references are the following: (1 ) M. H. Stone. "Linear Transformations in Hilbert Space". Amer. Math. Soc. Colloquium Publications. Vol. XV, New York, N.Y. (1932). (2 ) J. v. Neumann. Princeton Lecture Notes, for the years, 1933 -3^, 193^ - 35. §1 In the footnote to Definition 1 of Chapter VII, §2 , we In­ dicated two different kinds of resolutions of the Identity. Other types are possible; for instance: Let E2 (A)denote the example (b) in this footnote and let (A) denote a monotonically increasing continuous function with (A) = 0 for A