A Mathematical Introduction to Dirac's Formalism (North-Holland Mathematical Library)

A MATHEMATICAL INTRODUCTION TO DIRAC’S FORMALISM

North-Holland Mathematical Library Board ofAdvisory Editors: M. Artin, H. Bass, J. Eells, W. Feit, P. J . Freyd, F. W. Gehring, H. Halberstam, L. V. Hormander, J. H . B. Kemperman, H. A. Lauwerier, W. A . J . Luxemburg, F. P. Peterson, I. M. Singer and A . C. Zaanen

VOLUME 36

NORTH-HOLLAND AMSTERDAM * NEW YORK * OXFORD *TOKYO

A Mathematical Introduction to Dirac’sFormalism S. J. L. VAN EIJNDHOVEN J. DE GRAAF Eindhoven University of Technology Eindhoven, The Netherlands

1986

NORTH-HOLLAND AMSTERDAM * NEW YORK * OXFORD *TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V., 1986 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 70127 3 Published by: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 1991 1000 BZ Amsterdam The Netherlands

Sole distributors for the U.S. A . and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52Vanderbilt Avenue NewYork, N.Y. 10017 U.S.A.

Library of Congress Catalogingin-PublicationData

Eijndhoven, S. J. L. v8n (Stephanus J. L.), 1956A mathematical introduction t o D i r a c ' s formalism. (North-Holland mathematical l i b r a r y ; vol. 36) Bibliography: p. Includes index. 1. Quantum theory. 2. Dirac, P. A. M. (Paul Adrien Maurice 1, 1902I. G r a a f , J. de (Johannes) 11. T i t l e . 111. Series: North-Holland mathematical l i b r a r y ; v. 36. QC174.12 .E33 1986 530.1'2 86-19885 ISBN 0-444-70127-3

.

PRINTED IN T H E NETHERLANDS

A MATHEMATICAL INTRODUCTION TO

DIRAG'S FORMALISM S.I.1. VAN EllNOHOVEN

I. DE GRAAF

This Page Intentionally Left Blank

AAN ONZE OUDERS

This Page Intentionally Left Blank

ix

PREFACE

From the first years of its founding Quantum Mechanics has had a tremendous influence on the development of Functional Analysis. Application of functional analytic methods in theoretical physics has become very popular, since. The famous physicist Paul Dirac has had essential part in the introduction and description of the underlaying principles of quantum mechanics. He has put his mark upon the development of quantum mechanics and his FundamentaZ

principles of quantwn mechanics is the standard work for all research workers in this area. Dirac approaches quantum mechanics by means of a

so-

called symbolic method, the bracket formalism. Although this formalism has a mathematical flavour, in fact, it is based upon bold claims which lack mathematical foundation. Because of these shortcomings in Dirac's approach, certain mathematicians rejected Dirac's bracket formalism and searched for another more rigid mathematical approach. Others did not reject Dirac's ideas and tried to develop a sound mathematical basis for the bracket formalism. In this context we mention John von Neumann, the founder of Hilbert space theory and Laurent Schwartz, the founder of distribution theory. However, Hilbert space theory seems too limited to provide a tractable description of quantum mechanics. Also, distribution theory does not provide all satisfactory solutions to get rid of the weak spots in Dirac's formalism, although many people ungroundedly believe so. Anyhow, for a remarkably great number of mathematicians the mathematical aspects of quantum mechanics and of Dirac's formalism, in particular, are a major source of inspiration. Yet, one did not really succeed in developing a theory which contains the mathematical devices that can cope with all aspects of the bracket formalism. Probably nobody ever will. In this monograph we present a mathematical interpretation of the bracket formalism that is more in line with Dirac's original ideas than any interpretation we know of. We illustrate this with the following example. In Dirac's formalism two kinds of vector spaces appear: the space B of bra vectors and the space K of ket vectors. Dirac assumes that there exists a sesquilinear form, the bracket, on the space K x B and a l s o a one-to-one

Preface

X

correspondence between t h e k e t space and t h e b r a space. Mathematically t h i s means t h a t Dirac assumes t h a t it i s p o s s i b l e t o e s t a b l i s h a p a i r i n g between generalized functions. In almost a l l i n t e r p r e t a t i o n s one f e i g n s t h a t t h e s e claims of Dirac a r e n o t t h e r e and one r e p r e s e n t s t h e b r a space by a t e s t function space and t h e k e t space by t h e corresponding space of generalized functions. Then t h e b r a c k e t i s i n t e r p r e t e d by t h e u s u a l p a i r i n g of t e s t functions and generalized f u n c t i o n s . I n our i n t e r p r e t a t i o n no d i s t i n c t i o n

i s made between k e t and b r a space, what we consider t h e main p r i n c i p l e of t h e bracket formalism. So both t h e b r a space and t h e k e t space are represented by spaces of generalized f u n c t i o n s . But o u r b r a c k e t i s no longer a complex number but a complex valued f u n c t i o n on ( 0 , m ) .

It leads t o a

mathematical j u s t i f i c a t i o n of v a r i o u s h e u r i s t i c formulae of Dirac, f o r i n stance ( 6 , 6 X

) Y

= dX(y).

The m a t e r i a l i s presented i n t h r e e p a r t s . In P a r t A we introduce t h e mathematical concepts, which l i e a t t h e b a s i s of D i r a c ' s formalism. They a r e described i n t h e s e t t i n g of Sobolev t r i p l e s of H i l b e r t spaces. We mention t h e concept of Dirac b a s i s and of Federer measure space. The concept of Dirac b a s i s i s t h e measure t h e o r e t i c a l generali z a t i o n of t h e concept of orthonormal b a s i s i n H i l b e r t space theory. A Dirac b a s i s i s l a b e l e d by a measure space. In o r d e r t o o b t a i n a

canonicaZ l a b e l i n g

we introduce Federer measure spaces, measure spaces which admit a r e l a t i v e d i f f e r e n t i a t i o n theorem. Related t o t h e s e concepts w e prove a measure theor e t i c a l Sobolev lemma. Thus we o b t a i n a s o l u t i o n of t h e g e n e r a l i z e d eigenvalue problem f o r a r b i t r a r y s e l f - a d j o i n t o p e r a t o r s which i s c l o s e l y a l l i e d t o t h e s o l u t i o n of t h e c l a s s i c a l eigenvalue problem f o r Hermitean o p e r a t o r s on f i n i t e dimensional v e c t o r spaces. It t u r n s o u t t h a t t h e generalized eigenvectors of any s e l f - a d j o i n t o p e r a t o r l i e a t t h e p e r i p h e r y of H i l b e r t space. This r e s u l t looks new. I n many o t h e r approaches people s t i c k t o o much t o an a p r i o r i chosen d i s t r i b u t i o n theory f o r s o l v i n g t h e g e n e r a l i z e d eigenvalue problem. In P a r t B we introduce t h e t r i p l e of t o p o l o g i c a l v e c t o r spaces

'X,A

c

x

c

TX,A and give i t s f u n c t i o n a l a n a l y t i c d e s c r i p t i o n . Here

sX,A

de-

notes t h e s e t of a n a l y t i c v e c t o r s of t h e nonnegative s e l f - a d j o i n t o p e r a t o r

A i n t h e H i l b e r t space The space

x.

On

SX,A an i n d u c t i v e l i m i t topology i s imposed.

T X , A i s a r e p r e s e n t a t i o n of t h e s t r o n g dual of SX,A;

Frechet space. I f

x

it i s a

i s an L2 -space over a t o p o l o g i c a l measure space

(M,u),

Preface

then SX,A is a space of test functions and T

xi

X,A

a space of generalized

functions on M. We lift up the concepts introduced in Part A to the triple

SX,A c X c Part

C

TX , A

is entirely devoted to our mathematical interpretation of Dirac's

formalism and of the related free field formalism. Starting from Dirac's own description of his own formalism we interpret its various features. Herein both the bra space and the ket space are represented by spaces of type

T X , A . We try to follow Dirac's original ideas. In particular, our mathematical theory involves an interpretation of the pairing between kets and bras, of expansions of kets with respect to continuum sets of eigenkets, of orthogonality in complete sets of eigenkets and of matrices with respect to these complete sets. The book is intended for mathematicians and theoretical physicists both students and research workers, who are familiar with the fundamentals of functional analysis, primarily Hilbert space theory and the theory of unbounded linear operators in Hilbert space. With respect to the involved topological vector space theory the book is self-contained. Also certain aspects of the theory of measure and integration are included. We give a summary of the fundamental concepts of measure theory. Besides we employ a deep result on relative differentiation of measures on certain topological measure spaces. We prove this result for Borei measures on the real line. Its general proof is omitted. The references to literature and additional comments are appended at the end of each chapter. Finally we want to thank everyone who has contributed in one way or another in the realization of this monograph. Mrs. Anita Klooster took care of the typing of the manuscript. The pages which follow are a testimonial for her proficiency and dedicatioh. Eindhoven, June 1986. S.J.L.

van Eijndhoven

J. de Graaf

This Page Intentionally Left Blank

xiii

CONTENTS A. Some basic concepts and developments in Sobolev triples

I. CARLEMAN OPERATORS 1.1.

1

3

Some measure theoretical prerequisites

3 8

1.2.

Carleman operators

1.3.

Operators of Carleman type

13

1.4.

Strong Carleman operators

19

Some bibliographical notes and comments

22

11. A MEASURE THEORETICAL SOBOLEV LEMMA

23

11.1.

An elementary illustration

24

11.2.

Sobolev triples of Hilbert spaces

28

11.3.

Federer measure spaces

30

11.4.

A measure theoretical generalization of the Sobolev

11.5.

embedding theorem

38

Some applications

44

Some bibliographical notes and comments

50

111. DIRAC BASES

53

111.1. The concept of Dirac basis

55

111.2. Canonical Dirac bases

64

111.3. Dirac-Riesz bases

69

111.4. Canonical Dirac-Riesz bases

74

Some bibliographical notes and comments

79

IV. THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

81

IV.l.

Commutative multiplicity theory

82

IV.2.

An application of the measure theoretical Sobolev lemma

90

IV.3.

A solution

IV.4.

Some illustrations

95 101

Some bibliographical notes and comments

107

V. DIRECT RESOLUTIONS IN SOBOLEV TRIPLES

109

V.l.

Preliminaries and summary

109

V.2.

Direct resolutions of the identity

113

V.3.

A general construction of a direct resolution

119

xiv

V.4.

Contents

Equivalence classes of direct resolutions, 125

canonical direct resolutions V.5.

Generalized eigenprojections related to commutative von 132

Neumann algebras

145

Some bibliographical notes and comments

147

B. A theory o f generalized functions I. ANALYTICITY SPACES, TRAJECTORY SPACES AND THEIR DUALITY

sX , A

The analyticity space

1.2.

The trajectory space

1.3.

Pairing and duality of

1.4.

Sequence space representation

T

X,A

sX , A

149 150

1.1.

163 and

T

X,A

171 178

Some bibliographical notes and comments

181

11. LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

183

11.1.

Continuous linear mappings between analyticity and trajectory spaces

11.2.

183

Topological tensor products of analyticity and trajectory spaces

193

11.3.

Kernel theorems

202

11.4.

Matrix representations

208

Some bibliographical notes and comments

2 16

111. ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

2 17

111.1. Analyticity spaces based on the Laplacian operator

217

111.2. The spaces

sB of

Gelfand and Shilov

111.3. Analyticity spaces based on classical polynomials

229 234

111.4. Analyticity spaces related to unitary representations of Lie groups

245

Some bibliographical notes and comments

250

IV. THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

253

IV.l.

A measure theoretical Sobolev lemma for analyticity spaces

253

IV.2.

Diric bases in trajectory spaces

258

IV.3.

Canonical Dirac bases in trajectory spaces

267

IV.4.

The generalized eigenvalue problem for self-adjoint operators solved in the setting of trajectory spaces

271

Contents

Some bibliographical notes and comments

C. A mathematical interpretation

o f Dirac's formalism

xv

280 281

I. DIRAC'S FORMALISM ACCORDING TO DIRAC AND ITS RELATIONS WITH LINEAR ALGEBRA 1.1.

Some elementary concepts of linear algebra

1.2.

Dirac's formalism according to Dirac

285 285 29 1

Some bibliographical notes and comments

308

11. A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

309

i.

Kets

310

ii.

Bras

313

iii.

Brackets

316

.

Linear operators

321

V.

Dirac bases

325

vi .

Representatives of kets and bras

332

vii.

Representatives of linear operators

338

viii.

Observables

351

ix.

The harmonic oscillator

353

X.

Angular momenta

359

iv

Some bibliographical notes and comments

365

111. THE FREE FIELD OPERATOR FORMALISM

367

111.1. The Fock space

316

111.2. Required mathematical tools

389

111.3. Annihilation and creation operators

400

111.4. The free field formalism

406

Some bibliographical notes and comments

413

REFERENCES

417

INDEX

425

LIST OF SYMBOLS

429

This Page Intentionally Left Blank

1

SOME BASIC CONCEPTS AND DEVELOPMENTS

IN SOBOlEV TRIPLES In this part of the monograph we introduce the mathematical concepts, which appear in our mathematical interpretation of Dirac's formalism, in the bare setting of Sobolev triples of Hilbert spaces. The main theme of Part A is the concept of Dirac basis, which is a measure theoretical generalization of the concept of orthonormal basis in Hilbert space. A Dirac basis is

labeled by a measure space. Only recently, we have noticed a one-one relation between our notion of Dirac basis and the notion of Carleman operator. Besides, we introduce the concept of canonical Dirac basis. A canonical Dirac basis is "canonically" labeled by a measure space which obeys a relative differentiation theorem. We call such a measure space a Federer measure space. In terms of canonical Dirac bases we present a complete solution of the generalized eigenvalue problem for any finite number of commuting self-adjoint operators in a Hilbert space. Finally, we introduce the concept of (canonical)direct resolution of the identity in the setting of direct integrals of Hilbert spaces. This concept is irrelevant for our mathematical interpretation of Dirac's formalism, but is affiliated to the topics of Part A in a natural way. The present part is organized as follows. In Chapter I and I1 we discuss the notion of Carleman operator (Carleman kernel) and the notion of Federer measure space, respectively. Also, we prove a measure theoretical generalization of Sobolev's embedding theorem based on geometric measure theory. The first two chapters provide the mathematical tools needed in the remaining chapters.

2

Chapter I11 is devoted to (canonical) Dirac bases. Each positive bounded Carleman operator, in particular each positive Hilbert-Schmidt operator, generates a Dirac basis and, conversely, each Dirac basis defines a positive bounded Carleman operator. The theory of canonical Dirac bases, presented in Chapter 111, and the well-known commutative multiplicity theory for self-adjoint operators yield a complete solution of the generalized eigenvalue problem for any self-adjoint operator. This solution is described in Chapter IV. Chapter V contains the concepts of direct resolution of the identity and of generalized eigenprojection. It may be skipped in first reading.

3

CHAPTER I

CARLEMAN OPERATORS

This chapter is partially based on Weidmann's study of so-called Carleman operators. Originally Stone introduced a Carleman operator as follows. Let a < b <

5

-m

all x

OJ.

-

Define a function K on (a,b) X (a,b) such that for almost

(a,b) the function y

E

K(x,y) is square integrable with respect to

the Lebesgue measure. Then, following Stone, K is called a Carleman kernel.

a

and

1, is an integral operator in L2(a,b). In 1923 Carleman has studied this kind of integral operators. He took the kernel real and symmetric. His investigations were carried through by Stone (1932) and von Neumann (1935).

1.1. Some measure theoretical prerequisites

Let

M denote a non-empty set.

A

class

B

of subsets of M is called a a-

algebra if B satisfies the following conditions:

- @ E B . - If U E B then - If U . E B for 3

If

r

also its complement M\U j E N, then

m

n U.

j=1

'

E

B

E

B

. m

and U U . j=1 J

E

B .

is a collection of subsets of M then there exists a smallest o-algebra

B ( r ) which contains r ; B ( r ) is called the a-algebra generated by I'. A

set

M

which carries a o-algebra B is called a Bore1 space; the elements

4

of

CARLEMAN OPERATORS

B

are called Borel sets. (In our terminology, topology on

necessarily involved.)

mapping p from B into I R '

A

u {-I

M

is not

is called a measure

if p posesses the following properties:

-

p(lb) = 0 .

-

If Un, n

m,

E

are pairwise disjoint elements of B, then

The triple ( M , B , p ) is called a measure space, if (M,B) is a Borel space and p a measure on (M,B). If

M

is the countable union of elements of

B

each of

which has finite p-measure, then the measure space (M,B,p) is called ufinite and p a o-finite measure. We express this feature as follows

"M

is

a measure space with u-finite measure p " . Often we deal with a topological space

M.

we take

In this case, there is a natural choice of a u-algebra

r

the collection of all open subsets of

generated by l'.

M

and

B

B.

Indeed,

the o-algebra

Throughout this monograph a topological space M is always

considered as a Borel space in this way. If p is a measure on the topological space M then (M,p) is called a topological measure space. Let

(M,B) denote

a Borel space.

Borel function, if for each a

B. c p

A

real valued function cp on M is called a

IR the set {x

E

M I cp(x)

< a) is a member of

Icpl

+cp

,

cp

-

u,

1 cpI , cp+ = 2 + 6$ ( a , @ E IR) are Borel functions. A complex on M is called a Borel function, if cp = cpl + icp2, where

If cp and JI are Borel functions, then also the functions =

valued function cp cpl

E

$ and acp

and cp2 are real valued Borel functions. The set of Borel functions is

closed with respect to pointwise sequential convergence. So if (cpn)ncm is a sequence of Borel functions which converges pointwise with limit cp,

then

cp is a Borel function.

Let p denote a o-finite measure on the Borel space (M,B). We are going to introduce the class S ( M , p ) of p-integrable functions. First, we define the concept of simple function.

A

simple function s on

M

is a complex function

on !b which can be written as

Here the

u

denote mutually disjoint elements of B,

teristic function of

u

and the o

x

denotes the characun are complex numbers. The integral of s

SOME MEASURE THEORETICAL PREREQUISITES

5

denoted by

is defined as

Each Borel function cp on

M

can be expressed as the pointwise limit of a

sequence of simple functions. Suppose that cp is a positive Borel function. Let (sn)ncm denote a monotone increasing sequence of simple functions, which converges to cp pointwise. We define the integral

1

cp dp = lim

M

sndp

MJ

cpdp of cp by

.

M

This definition does not depend on the particular choice of the sequence ( S n ) n cN *

If the integral

MI

rpdp is finite, then cp is called p-integrable.

A real valued Borel function cp is called p-integrable, if cp = cp

cpl

-

(p2

where

and cp2 are positive p-integrable functions. We define

The class of real valued p-integrable functions is denoted by observe that for cp

E

S,(M,p)

also ( c p ( , cp

+ , cp -

s IR ( M , p ) .

belong to S,(M,v).

We

Moreover

the set SIR(M,p) is linear. The following theorems are very fundamental. They express that

s IR (M,p)

is a large set.

The monotone convergence theorem Let (cpn)ncm be a monotone sequence of functions in S,(M,p)

the integrals

of which are bounded. Then there is a Borel set N of p-measure zero such that the sequence ( c p ) converges to a function cp pointwise o n M\N, n n6lN where cp belongs to s I R ( M , p ) . Further,

6

CARLEMAN OPERATORS

Remark. Pointwise convergence except on a set of measure zero is also called convergence a h o s t everywhere. The dominated convergence theorem Let (cpn)nEm be a sequence of functions in

s w (M,u) which

converges point-

wise to a function cp. Suppose (cpn)nEm is dominated by a positive function

1

J, in SIR(M,p), i.e. 1cp

5 J,

for all n

6

hl.

Then cp

E

S

IR (M,p) and

Finally, the class S(M,p) consists of all Borel functions c p , which can be written as cp

=

cpl

+

icp2 where cpl

and cp2 belong to

Sw(M,u).

We put

Mj'TJd!J = M I ' P +~ iMj'P2dp~ ~ A

Borel function cp on M is said to be square p-integrable if the function

I cp 1

2

belongs to S ( M , p ) . The linear space of square integrable functions is

denoted by S2(M,p). In

s2 (M,p)

cpi'dw

s(cp,J,) =

we define the sesquilinear form

.

M This definition makes sense because lcpJ,l form

s

s(cp,cp)

5

2

+

'"I2

is degenerate; put differently, there exists cp = 0 and cp

. The sesquilinear E

S2(M,p) such that

# 0. To overcome this problem of degeneracy we introduce

an equivalence relation

- in S2(M,p):

It follows that two functions cp and J, in S (M,p) are equivalent if there 2

exists a set of p-measure zero such that cp quotient space

S, (M.u)/- by L, (M,p).

= J,

on M \ N . We denote the

So L, (M,p) consists of equivalence

classes of square integrable functions. It is proper to denote the elements of L (M,p) by Ccpl, where cp denotes any representative of this equivalence 2 class. For a,B E C and Ccpl,[J,l E L2(M,p) we put

aCcpl + acq1

= Cacp

+B$3

.

SOME MEASURE THEORETICAL PREREQUISITES

Thus

L 2 ( M , ~ ) becomes a (complex) vector space. The s e s q u i l i n e a r form

s,(M,1.1)induces

kere

7

(p and

on

an i n n e r product i n L 2 ( M , p ) .

4 denote

v e c t o r space

s

L2(M,u)

any r e p r e s e n t a t i v e of with i n n e r product

[TI

and

[$I,

r e s p e c t i v e l y . The

i s a H i l b e r t space.

( a , . )

L2

( M 1' B 1) and ( M 2 ' B 2 ) denote two Borel spaces. On t h e product space MI x M, we consider t h e s m a l l e s t a-algebra B generated by t h e c o l l e c t i o n B, x B, = {U, x U, I U, E B,, U, E B2}. Thus (M, xM2,B) i s a Borel space. I f p 1 and 1 . 1 ~ a r e a - f i n i t e measures on M, and M,, r e s p e c t i v e l y , then t h e aa d d i t i v e s e t f u n c t i o n 1.1 = 1 . 1 ~ x 1 . 1 ~ on B, x B2 i s d e f i n e d by Let

M,. (M2,p2) a r e a - f i n i t e measure spaces, then a l s o (M,

I n a n a t u r a l way p extends t o a measure on

M,

I f both (M,,p,)

x

xM2,p1 x u 2 ) i s a

and IJ-

f i n i t e measure space. A n important r e s u l t f o r product spaces i s t h e theorem

of Fubini-Tonelli

.

Fiiini-Tone 1l i Let 8 E S ( M 1 ~ M 2 , p 1 ~ p 2 )Then . there e x i s t s a pl-null s e t N s e t N such t h a t f o r a l l x1 2

and, s i m i l a r l y , f o r a l l x

The functions

(p,

and

(p2

2

E

E

M 1\N 1 t h e f u n c t i o n

M 2 \N 2

the function

defined by

,

i f x,

E

N,

,

E

M,\Nl

,

(P1(xl) = @(x1,x2)dp2, if x1

1

and a 1.1 - n u l l 2

CARLEMAN OPERATORS

8

(P2(x2)=

ToLMIJ

,

@(x1,x2)dul I

belong to s(Ml,ul)and S(M2,v2),

,

if x2

E

N,

if x2

E

M2\N2

respectively. We have

put differently,

An

application of product measure spaces is the following: Consider the

Hilbert spaces

L2 (Ml,ul) and L2 (M2,v2).

Then a topological tensor product

of L2(Mlrwl)and L2(M2,u2) is given by L2(Ml X M 2 , u 1 xu2).

It is well known

that the Hilbert space of Hilbert-Schmidt operators from L2(Ml,ul) into

L2 (M2,l! 21 is a topological tensor product of these Hilbert spaces, also. It follows that any Hilbert-Schmidt operator is represented by a "kernel" out of

L2 (M1 xM2,p1xp2).

1.2. Carleman operators

Throughout the present monograph by

x we denote a

separable Hilbert space

and by (M,u) a o-finite measure space (with respect to a suitable Bore1 oalgebra

B).

We always assume that the Hilbert space

This is the case if

M

L2 (M,u)

is separable.

is a separable topological space.

I.2.1. D e f i n i t i o n .

x -+ L2 (M,p) be a linear operator with domain D ( T ) . Then T is said to be a Carleman operator, if there exists an X-valued function h on M with

Let T :

the following property: For each f

E

D ( T ) the function

CARLEMAN OPERATORS

is a r e p r e s e n t a t i v e of t h e c l a s s Tf the function (BY (

T

i s s a i d t o be induced by

h.

, ) X or

,

(

) w e denote t h e i n n e r product

in

x.)

.

Exmp l e s

-

L2(M,p).

E

9

The H i l b e r t space

e2

of square summable sequences can be w r i t t e n as

L 2 ( m , ~ ) . Here T denotes t h e counting measure s a t i s f y i n g T ( n ) = 1 f o r all n from

E

x

N. into

Let ( k n ) n E m denote any sequence i n

e2 i s

x.

Then t h e o p e r a t o r

defined by

The o p e r a t o r T i s a Carleman o p e r a t o r induced by t h e X-valued f u n c t i o n

n n kn on W.

-

Let (M1,pl) and (M2,v2) denote two a - f i n i t e measure spaces and suppose

L2(Ml,pl) and L2(M2,p2) Bore1 f u n c t i o n on

M1

x

be square p2-integrable

a r e s e p a r a b l e H i l b e r t spaces. Let K denote a

M2

and l e t t h e f u n c t i o n

for a l l x

E

i s a Carleman o p e r a t o r from L2(M2,p2)

M,

x

1.2.2.

M,

Mi.

Then t h e o p e r a t o r T defined by

i n t o L2(Ml,pl). The f u n c t i o n K on

i s c a l l e d a Carlernan k e r n e l .

&.

a. Every Hilbert-Schmidt o p e r a t o r T b. A h u n d e d Carleman o p e r a t o r T :

x

:

x

+

+

L,(M,!.i)

i s a Carleman o p e r a t o r .

L2(M,p) i s Hilbert-Schmidt i f f i t s

10

CARLEMAN OPERATORS

inducing function

h

has t h e property t h a t x

I+

11 k ( x ) 1 1 , x

E

M

i s square

u-integrable.

Proof. a . Suppose T

x

L (M,p)

i s a Hilbert-Schmidt o p e r a t o r . Let ( 2 note an orthonormal b a s i s i n Then t h e s e r i e s :

E

v

~ de-)

~

x.

m

1

IITvkII

2

k= 1 i s convergent. For each k

Tvk m

c

k= 1

E

L2(M,p).

1 (pk 12

Define

k

E

IX

i s pointwise convergent except on a s e t of measure zero N . :

M

+

0 k(x) =

m

c

b . Let

T

,

i f x c N

-k=l Then

he take a r e p r e s e n t a t i v e 'pk o f t h e c l a s s

Then by t h e monotone convergence theorem t h e s e r i e s

cp,(X)

vk

'

if x

E

.

M\N

b has t h e r e q u i r e d p r o p e r t i e s . :

x

+

L 2 ( M , p ) be a Hilbert-Schmidt o p e r a t o r . Then from ( a ) it

follows t h a t i t s inducing f u n c t i o n

k

satisfies

On t h e o t h e r hand, l e t t h e inducing f u n c t i o n

k

of a bounded Carleman

operator T s a t i s f y

Then f o r any orthonormal b a s i s (u )

n nElN

Thus t h e s t a t e d equivalence follows.

in

x

we have

0

~

~

CARLEMAN OPERATORS

11

The following characterization of Carleman operators is useful

I . 2.3.

Theorem.

Let T be a bounded linear operator. Then T is a Carleman operator iff for

any orthonormal basis (vnInEm in

n

E

x

there are representatives ( T v

E

IN, such that the series

is convergent for all x

E

M'bN,

v

~ denote ) ~ an~orthonormal ~ basis in

( T vn)- : x n (~,,k(x))~, n

E

Tv n'

where N is a set of p-measure zero.

w. Let T be a Carleman operator with inducing function k (

)I

x.

:

M + X and let

Then we put

IN.

Conversely, for the orthonormal basis (v ) let there be given repren new sentatives (Tvn)- and a null set N as indicated. We define

1

k(x) =

o

Then the function

ifxEN 1

k

1

:

x w k(x), x

E

.

M, induces the operator T as a

0

Carleman operator.

Remark. Let T be a bounded Carleman operator, and let ( denote two orthonormal bases in

x.

Then for almost all x

v E

~ and) ( u~ ) ~ ~ m mEIN

M

1

where the (Tvn)

and the (Turn) are representatives of the classes T v

and T u , respectively. m The family of bounded Carleman operators possesses the following properties. I. 2.4. Theorem. Let T

1

and

T2

be bounded Carleman operators from

x

into L (M,LI) 2

and let B

CARLEMAN OPERATORS

12

x.

be a bounded l i n e a r o p e r a t o r from a separable H i l b e r t space Y i n t o

+ T,

T,

Proof.

and T B a r e Carleman o p e r a t o r s . 1 Let T, and T,

kl and b,. Then it induced by h, + h, a s a

denote t h e inducing f u n c t i o n s of

follows simply t h a t t h e o p e r a t o r T

1

+ T2

is

Carleman o p e r a t o r and t h e o p e r a t o r T B by t h e f u n c t i o n x 1

x

E

Then

I+

B * ( h1 ( x ) ) ,

M.

0

Let T be a bounded Carleman o p e r a t o r from

x

i n t o L2(M,p) and l e t U be a

u n i t a r y o p e r a t o r from L (M,p) onto X. Then TU : L2(M,p) 2

+ L2(M,p)

is a

Carleman o p e r a t o r . So i n studying Carleman o p e r a t o r s we have t o consider only Carleman o p e r a t o r s on L, (M,p).

Therefore, l e t T

:

L2 (M,u)

denote a bounded Carleman o p e r a t o r induced by t h e f u n c t i o n Fixing f o r each k ( x )

E

L2 (M,p), x

E

h

:

+

L,

M

+ L2(M,p)

(M,p)

M, a r e p r e s e n t a t i v e KX' we o b t a i n a

f u n c t i o n K on M x M,

with t h e property t h a t

1

where cp denotes any r e p r e s e n t a t i v e o f t h e c l a s s [ q ] . We a r r i v e a t t h e following d e f i n i t i o n .

I. 2.5.

Definition.

Let K be a Bore1 function on M x M f o r which t h e r e e x i s t s a p-null s e t N such t h a t f o r a l l x

E

M\N t h e f u n c t i o n y

I+

K(x,y) i s square p-integrable.

Then K i s c a l l e d a Carleman k e r n e l .

1.2.6.

Theorem.

L e t K denote a Carleman k e r n e l . W e d e f i n e t h e l i n e a r o p e r a t o r

bY

L on L

2

(M,p)

OPERATORS OF CARLEMAN TYPE

13

Then L is a (possibly unbounded) Carleman operator in L (M,p). The linear 2

operator L is closed.

Proof. There exists a null set N such that y * K(x,y) is square integrable. We define

Then

k

k(x)

=

o

k(x)

=

Cy

H

K(x,y)l

for all x

E

the mapping

ifxeN

,

if x c M\N

.

M\N the function

k

:

M

-+

L2(M,p) by

induces 1 as a Carleman operator.

Next we prove that L is closed. Let (Cvnl)nEN denote a sequence in D ( L ) with limit [ q l in L (M,p) and let the sequence (LCvnl)nEhTconverge to

CJ11 in

2

L2(M,p).

Then for all x

t

M

and lim

JI

$

+(x)l2dp(x)

=

o

L2

n” M for any

-

(Cvnlrk(x))

[$I. It follows that the function x

E

presentative of

[$I.

Hence

[TI

E

D(L)

([ql,h(x))

and LCqI = [$I.

L2

is a re-

0

By the closed graph theorem we get the following consequence of the preceding theorem.

I.2.7.

Corollary.

Let K denote a Carleman kernel and operator L is continuous iff D ( L )

L be defined as in Theorem 1.2.5. The 0

= L2(M,p).

Remark. It can be proved that each Carleman operator L from

L2(M,p)

into

L2(M,p) arises from a Carleman kernel.

1.3. Operators of Carleman type If

R

:

x

-t

x

is a Hilbert-Schmidt operator and U

:

x

operator, then U R is a Hilbert-Schmidt operator from

-+ L2(M,p)

x

a unitary

into L2(M,p),

and

14

CARLEMAN OPERATORS

hence a Carleman operator. In general a bounded operator does not possess this property. This phenomenon leads to the definition of Carleman type operator.

I.3.1. D e f i n i t i o n . Let

R

:

x

+

x

be a linear operator. Then

R

is said to be of Carleman type

with respect to the measure space (M,p) if there exists a unitary operator U from X onto

L 2 (M,p)

such that

UR is a Carleman operator.

Theorem.

1.3.2.

Let the linear operator

R

in

x

be of Carleman type with respect to (M,p).

Let V denote a unitary operator on Then the operators

ProqS. Let

UR :

:

x

X

-+

-t

VR

and

RB

x

and

a bounded linear operator on

denote the unitary operator from X onto L,(M,p)

L2(M,p) is a Carleman operator. Since UR

L2(M,p) is unitary, the operator

Theorem 1.2.4 the operator

x.

are of Carleman type with respect to (M,p).

URB

VR

such that

UV*VR and UV*

=

:

is of Carleman type. A l s o , by

is a Carleman operator and hence

RB

is an

0

operator of Carleman type. 1.3.3. Let

R

COPO~k!P~.

denote a bounded linear operator, which is of Carleman type with

respect to ( M , p ) . Then

IRI

=

(R*R)' is of Carleman type with respect to

(M,v), also.

Proof.

IR/Jwhere J denotes isometry which maps ran(R) onto ran( I RI 1 . So we have IR/ = RJ*, ( R / is of Carleman type with respect to (M,p). JJ

The operator

a partial and hence

R

admits a polar decomposition

R

=

In this section we present a characterization of the class of bounded normal operators of Carleman type. 1.3.4.

-.

Let R be a bounded self-adjoint operator with pure point spectrum such that 0 belongs to its essential spectrum oe(R). Then

R

is of Carleman type with

respect to any o-finite measure space (M,u) with the property that

L 2 (M,u)

is a separable Hilbert space.

Proof.

Since

R

has pure point spectrum, there exists an orthonormal basis

15

OPERATORS O F CAFGEMAN TYPE

(vnLntN o f e i g e n v e c t o r s o f =

C

X (w,vn)vn, w

n=l n

x.

E

If

R s u c h t h a t R can b e r e p r e s e n t e d a s Rw = m m t h e n R i s a Hilbert-Schmidt IXnI2

.gl

o p e r a t o r and hence an o p e r a t o r of Carleman t y p e w i t h r e s p e c t t o any measure space (M,p)

.

m

p o i n t of t h e sequence sequence ( A n )

nE

lhn12 =

w e suppose t h a t

SO

such t h a t

~

s e q u e n c e by (p )

(An ) n6mm Z

s i n c e 0 i s an accumulation

n= 1 I n a s t a n d a r d manner we t u r n t h e s e q u e n c e ( p

nEm.

m

i n t o a d o u b l e s e q u e n c e (pkL)k,LEIN.

Now w e d e f i n e sequences ( A

f o r each k

m.

t h e r e e x i s t s a subsequence (p ) of the n ntB lpnI2 <-. Denote t h e r e m a i n i n g p a r t o f t h e

So f o r e a c h k

k,L)kIN

E

IN,

Lgl

lpkLl

)

n n t IN < m.

as f o l l o w s

IN.

E

L e t (M,p) b e a a - f i n i t e measure s p a c e . Then t h e r e e x i s t s a c o u n t a b l e number of d i s j o i n t Bore1 sets

Mk

such t h a t

m

M

=

U

k= 1

M,

and

p(Mk) <

m

2nd such t h a t t h e H i l b e r t s p a c e L2(Mk,!.i) i s i n f i n i t e d i m e n s i o n a l ( a n d s e p a r a b l e ) . I t i s clear t h a t

I n e a c h H i l b e r t s p a c e L2(Mk,p) w e f i x an o r t h o n o r m a l b a s i s ( [ q k,l’)LelN. Next w e d e f i n e the H i l b e r t - S c h m i d t o p e r a t o r Tk, k E IN, b y

and t h e n t h e bounded s e l f - a d j o i n t o p e r a t o r T by m

TCqI

=

1

k= 1

m

o

TkCql, ,

Then T i s a Carleman o p e r a t o r on

Iql

=

L 2 (M,p)

1

k=l

@

Cql,

E

L,(M,p)

.

i n d u c e d by t h e f u n c t i o n

CARLEMAN OPERATORS

16

k

=

1

k,

M,

-f

:

M

+. L 2 ( M , u )

k=1

k,

Here each

:

denotes an inducing f u n c t i o n of t h e H i l b e r t -

L2(Mk,p)

Schmidt o p e r a t o r T,,

k

E

.

IN. Since T and R both possess a pure p o i n t

spectrum (with t h e same m u l t i p l i c i t y of t h e e i g e n v a l u e s ) , t h e s e o p e r a t o r s a r e u n i t a r i l y e q u i v a l e n t . I t means t h a t t h e r e e x i s t s a u n i t a r y o p e r a t o r

x

from

u

onto L 2 ( M , p ) such t h a t UR = TU. Thus it follows t h a t R i s of Carleman

type with r e s p e c t t o t h e measure space (M,u).

e.

1.3.5.

R be a p o s i t i v e bounded o p e r a t o r with

Let

0

-

0 i n i t s e s s e n t i a l spectrum. Then

t h e r e e x i s t s a p o s i t i v e bounded o p e r a t o r R with pure p o i n t spectrum and 0 1 i n i t s e s s e n t i a l spectrum, such t h a t R R and a r e bounded o p e r a t o r s .

--

Proof.

Replacing R by

(EA)AECO,l, denote

IIRII-lR

we may a s

R-'Z

well assume t h a t IIRII

= 1. L e t

t h e s p e c t r a l r e s o l u t i o n of t h e i d e n t i t y corresponding t o

on X by

R. We d e f i n e t h e p r o j e c t i o n s P 1

Pn =

,

j'dEA

n

E

rm

1 -

n+ 1 Y

and t h e o p e r a t o r R by

Then R has pure p o i n t spectrum and 0 belongs t o i t s e s s e n t i a l spectrum. Further

be

have

--1

R

R =

1

n=l

--1 so R R 1.3.6. Let

N

Then

1

m

n+ 1 and R

I"

2

nA dEX ,

m

R-'Z

1 n=l

-1"

R a r e bounded o p e r a t o r s with

-- 1 IIR RII

In

2

&dEA

.

n+ 1 = 1,

IIR

-1-

RII

= 2.

0

Theorem. denote a bounded normal o p e r a t o r on

N

=

1 -

x

with 0 i n i t s e s s e n t i a l spectrum.

i s of Carleman type with r e s p e c t t o any o - f i n i t e measure space (M,p)

OPERATORS OF CARLEMAN TYPE

with the property t h a t

Proof.

17

L2 (M,p) i s a s e p a r a b l e H i l b e r t space.

The o p e r a t o r N admits a p o l a r decomposition

isometry. By Theorem 1 . 3 . 2

N

l N ] J with J a p a r t i a l

=

t h e proof i s complete i f it can be shown t h a t

I N 1 i s of Carleman type with r e s p e c t t o any a - f i n i t e measure space we put R = I N 1 and observe t h a t 0

(M,p).

o (R).

E

(R)

Following Lemma 1.3.5 t h e r e e x i s t s a p o s i t i v e s e l f - a d j o i n t o p e r a t o r

-R 'R

-R with

E u such t h a t i s a bounded l i n e a r e w t h e o p e r a t o r R i s of Carleman type with r e s p e c t t o ---1 , any admissible measure space (M,!-I). Hence R = RR R 1s of Carleman type

a pure p o i n t spectrum and with 0

o p e r a t o r . By Lemma 1 . 3 . 4

0

with r e s p e c t t o any admissible measure space ( M , p ) .

Under c e r t a i n c o n d i t i o n s on t h e o - f i n i t e measure space ( M , p ) , t h e c o n d i t i o n that 0

E

o

(N) i s

a l s o a necessary c o n d i t i o n . The notion of p u r e l y atomic

measure space e n t e r s t h e d i s c u s s i o n .

I. 3.7.

Definition.

A Borel s u b s e t

u

of M i s c a l l e d a p-atom i f f o r each Borel s u b s e t

ul

C U

e i t h e r p ( u ) = 0 o r p ( u ) = p(u). A o - f i n i t e measure space ( M , p ) i s c a l l e d 1 1 m p u r e l y atomic i f M can be w r i t t e n a s t h e union u M of mutually d i s j o i n t k=l k p-atoms

Mk .

L e t (M,p) denote a purely atomic measure space with a r e mutually d i s j o i n t p-atoms.

Let

[cpl

E

L 2 ( M , p ) . Then

a r e p r e s e n t a t i v e which takes a c o n s t a n t value p ( c p )

k

l i n e a r o p e r a t o r U from

M

m

=

kgl b$ i n [TI

where t h e

on each atom M.,

1.3.8.

L2 (M,p) i n t o l2defined by

Theorem.

N denote a bounded normal o p e r a t o r on

x.

a. I f ( M , p ) i s a purely atomic measure space, then N i s of Carleman type with r e s p e c t t o ( M , p ) . b.

If

(M,u)

i s n o t a purely atomic measure s p a c e , then

type with r e s p e c t t o (M,p) i f f 0

E

ue(N).

N

k

So t h e

i s a unitary operator.

Let

M

there exists

i s of Carleman

18

CARLEMAN OPERATORS

m. a. Let

M

m

Mk

u

=

k=l f u n c t i o n s $,

k

'

M

by

([Jlk])kEIN

The family

M mutually d i s j o i n t p-atoms. Then d e f i n e t h e

m t h the

on

i s an orthonormal b a s i s i n L 2 ( M , p ) . Choose any

orthonormal b a s i s ( v ) in k kern U : X L,(M,p) by

x

and d e f i n e t h e u n i t a r y o p e r a t o r

-+

m

For each x

Then

E

M ,

a.

IN,

E

we put

k i s a well-defined X-valued f u n c t i o n on M.

For each w

E

x

and a l l x

E

Me

we have m

Hence x I+ b.

If 0

E

( ~ , k ( x ) i)s~a r e p r e s e n t a t i v e of UNw. N i s of Carleman type with r e s p e c t t o (M,p) following

o e ( N ) , then

Theorem I. 3 . 7 . Conversely, suppose t h e r e e x i s t s a u n i t a r y o p e r a t o r U :

x

-t

L,(M,p)

such

U l N l and UINIU* are Carleman L2(M,u) denote an inducing f u n c t i o n o f t h e opera-

t h a t UN i s a Carleman o p e r a t o r . Hence a l s o o p e r a t o r s . Let t o r U/N)U*. Let

k

:

M0

M

+

denote a Borel s u b s e t of

M

which does n o t contain any

Since t h e measure space ( M o , p ) i s o - f i n i t e t h e r e e x i s t s a count-

p-atoms.

a b l e number of mutually d i s j o i n t Borel sets such t h a t m

Y

Introduce

Mo

u

the s e t

Mo

,n

In

=

Ix

E

Mo

,n

1

Ilk(x) II < n ) . Then f o r n s u f f i c i e n t l y l a r g e Let P denote t h e canonical projec-

has p o s i t i v e p-measure.

19

STRONG CARLEMAN OPERATORS

w

t i o n of L,(M,p)

onto L 2 ( M 0 , n , p ) .

Then UlNlU*P : L,(M,p)

Hilbert-Schmidt o p e r a t o r , induced by

h,(x)

,

= 0

?.

Since L 2 ( M o , n , p )

h,

:

M

-f

is a

else

i s i n f i n i t e dimensional, f o r a n y orthonormal sequence N

E

L,(M,p)

L,(M,p)

i n L 2 ( M 0 , n , p ) we have UINIU*[qnI + 0 i n

([qn’) nE follows t h a t 0

+

oe(UINIU*)

=

o e ( ( N / ) .Hence 0

E

L 2 (M,p)

as n

-t m .

It

o ( N ) and t h e proof i s

0

complete.

1.3.9.

CoroZlary.

The c l a s s of bounded normal o p e r a t o r s on

x

which a r e of Carleman type with

r e s p e c t t o ( M , p ) i s t h e same f o r a l l admissible o - f i n i t e measure spaces (M,p)

which a r e n o t purely atomic.

1.4. Strong Carleman o p e r a t o r s I n the previous s e c t i o n w e have introduced t h e notion o f Carleman type o p e r a t o r . A Hilbert-Schmidt o p e r a t o r i s o f Carleman type with r e s p e c t t o any o - f i n i t e measure space (M,p). In f a c t , f o r any o - f i n i t e measure space from x onto L (M,p) and any Hilbert-Schmidt 2 R on X t h e o p e r a t o r UR i s a Carleman type o p e r a t o r from X i n t o

( M , u ) , any u n i t a r y o p e r a t o r operator

L 2 ( M , p ) . Thus i n a n a t u r a l way t h e notion of s t r o n g Carleman o p e r a t o r comes UP.

I. 4.1. Let

z

Definition. be a l i n e a r o p e r a t o r i n

x.

Then

z

i s c a l l e d a s t r o n g Carleman o p e r a t o r

with r e s p e c t t o a o - f i n i t e measure space ( M , p ) , i f f o r each u n i t a r y o p e r a t o r

U from X onto L

2

(M,l.i)

t h e o p e r a t o r UZ i s a Carleman o p e r a t o r .

I n t h i s s e c t i o n we c h a r a c t e r i z e t h e c o l l e c t i o n of s t r o n g Carleman o p e r a t o r s . W e observe t h a t each Hilbert-Schmidt o p e r a t o r on

x

i s a s t r o n g Carleman

o p e r a t o r f o r any ( M , p ) .

1.4.2.

Theorem.

L e t t h e o - f i n i t e measure space ( M , p ) be n o t purely atomic. A bounded l i n e a r

20

CAFLEMAN OPERATORS

operator

Z on X i s a s t r o n g Carleman o p e r a t o r with r e s p e c t t o

(M,p)

iff

2

i s a Hilbert-Schmidt o p e r a t o r .

Proof.

p a r t of t h e a s s e r t i o n . The proof i s divided

W e prove t h e only-if

i n t o four steps.

S t e p 1 . Suppose f i r s t t h a t

XI

=

+

U(AI -z)u*

E

u (XI

that

XI

-z);

i s self-adjoint.

If A

u e ( z ) , then

E

:

x

-f

L2(M,u)

(Observe t h a t i s a Carleman o p e r a t o r on L (M,I.I). 2 fJ e x i s t s . ) We conclude

such t h a t U ( h I - Z ) U *

0

z

UzU* where w e t a k e t h e u n i t a r y o p e r a t o r

s o Theorem 1.3.6 s t a t e s t h a t such

i s a Carleman o p e r a t o r on L2(M,p). Since (M,p)

X

i t follows t h a t We d e r i v e t h a t

z

=

i s n o t p u r e l y atomic,

0.

possesses a d i s c r e t e spectrum with only p o s s i b l e accumula-

t i o n p o i n t 0 . So

z

can be w r i t t e n a s

m

where t h e h

a r e t h e eigenvalues counted t o m u l t i p l i c i t y and where t h e e

c o n s t i t u t e an orthonormal b a s i s i n

x.

S t e p 2. For any u n i t a r y o p e r a t o r V

:

([vn])n6N

Carleman o p e r a t o r . So l e t For each n

.

CL

N l e t cp

E

[rp

x

-t

L2(M,p) t h e o p e r a t o r VZV* i s a

denote an orthonormal b a s i s i n L2(M,p).

].Then f o r almost a l l x

p u t d i f f e r e n t l y , f o r almost a l l x

E

M

E

M

S t e p 3. W e c o n s t r u c t a s p e c i a l orthonormal b a s i s i n L (M,p) i n o r d e r t o m

a r r i v e a t ,fl Since

lhnj2 <

-.

2

(M,u) i s n o t p u r e l y atomic, t h e r e e x i s t s a Borel s u b s e t M1 of M which 1 and, which i s of f i n i t e ( p o s i t i v e ) y-measure. 2 2 i n t o two d i s j o i n t Borel s u b s e t s M I and M2 such t h a t

does n o t c o n t a i n any u-atoms, Now we s p l i t

Mi

STRONG CARLEMAN OPERATORS

21

Repeating t h i s process, it follows by i n d u c t i o n , t h a t f o r each j E IN t h e r e j+l , 1 5 k 5 2 1 , with t h e following propere x i s t 2' d i s j o i n t Bore1 s u b s e t s b$ ties I+' = $ , " M2k

Mj+l 2k-1

M,,-,n M,, @ , j+l

j+l

=

Now we d e f i n e t h e orthonormal system ([$,I)

nc IN

i n L 2 ( M , p ) a s follows

W e extend t h i s orthonormal system t o an orthonormal b a s i s ( c q n l ) with nc IN 1 [q2,1 = [qnl, n E IN. Then w e have f o r c e r t a i n x E M 1

S i m i l a r l y , we can take [ q

m

Hence

,El / ' , I 2

Step 4. If

Z

=

z

<

z

3

= [$,I,

which y i e l d s

i s a Hilbert-Schmidt o p e r a t o r .

is not self-adjoint,

then t h e p o l a r decomposition y i e l d s

IZIJ h e r e J denotes a p a r t i a l isometry.

a djoi nt operator So

and

m

2n-1

IzI

and hence

IzI z

I t follows t h a t t h e s e l f -

i s a s t r o n g Carleman o p e r a t o r with r e s p e c t t o ( M , p ) .

0

i s Hilbert-Schmidt.

I t follows that t h e c l a s s of s t r o n g Carleman o p e r a t o r s i s independent of

t h e choice of t h e measure space

(M,u),

i f ( M , p ) i s not p u r e l y atomic. A

s i m i l a r r e s u l t h a s been proved f o r Carleman type o p e r a t o r s .

22

CARLEMAN OPERATORS

1.4.3.

Theorem.

L e t (M,p) be a purely atomic measure space then any normal bounded o p e r a t o r

on X i s a s t r o n g Carleman operator with r e s p e c t t o (M,u).

Proof. Every to

(M,u).

bounded normal o p e r a t o r on

x

i s of Carleman type with r e s p e c t

Also f o r every u n i t a r y o p e r a t o r U from

X onto L 2 ( M , u ) , t h e

o p e r a t o r UNU* i s a bounded normal o p e r a t o r on L 2 ( M , p ) . Hence UKU* i s a Carleman o p e r a t o r on L (M,p), 2

and UN = UNU*U i s a Carleman o p e r a t o r on

X. 0

Some b i b l i o g r a p h i c a l notes For a profound treatment of t h e theory of measure and i n t e g r a t i o n w e r e f e r t h e r e a d e r t o t h e book [Weir] of Weir. For a g r e a t e r p a r t t h e l a s t t h r e e s e c t i o n s of t h i s c h a p t e r a r e based on t h e paper [ W e i 2 1 of Weidmann. In t h i s paper Weidmann e l a b o r a t e s r e s u l t s of Korotkof [Kor], [Schrl.

of Misra, Schreiber and Targonski [MST] and of Schreiber

I n the papers of Carleman [Gal, Stone [ S t o l and von Neumann [Neu],

Carleman o p e r a t o r s were s t u d i e d a s a p a r t i c u l a r kind of i n t e g r a l o p e r a t o r s . Another e x c e l l e n t r e f e r e n c e i n t h i s r e s p e c t i s Weidmann's monograph [Wei 11, Chapter 6 , and t h e monograph [HSl of Halmos and Sunder.

23

CHAPTER II A MEASURE THEORETICAL SOBOLEV LEMMA

L e t (M,p) denote a o - f i n i t e measure space. The l i n e a r space S 2 ( M , p ) con-

s i s t s of a l l square i n t e g r a b l e Bore1 f u n c t i o n s . I n S,(M,p)

we define the

s e s q u i l i n e a r form s by

The s e s q u i l i n e a r form s i s degenerate. To g e t r i d of t h i s degeneracy, an

- i s introduced i n s,

equivalence r e l a t i o n

(M,p),

M So 'p and

J1 i n S,(M,p) a r e e q u i v a l e n t i f ~ ( x =) $(x) f o r almost a l l

The q u o t i e n t space S,(M,p)/-

x

E

M.

i s denoted by L 2 ( M , p ) . I t follows t h a t L 2 ( M , p )

c o n s i s t s of equivalence c l a s s e s of square i n t e g r a b l e f u n c t i o n s . Therefore it i s proper t o denote t h e elements of L 2 ( M , p ) by

[cpl, where

[TI

denotes

t h e equivalence c l a s s with r e s p e c t t o ,., of t h e square i n t e g r a b l e f u n c t i o n cp.

The s e s q u i l i n e a r form s induces an i n n e r product i n L 2 ( M , p )

(C'pl,C61)

=

S ( t 1 3 )

L2

here

and

3

denote a r b i t r a r y r e p r e s e n t a t i v e s of

ccp1 and [$I. The space

L, ( M , p ) i s a H i l b e r t space. Having solved t h e problem of degeneracy and gained t h e H i l b e r t space L,(M,p),

w introduce a new problem: t h e space

L,(M,u)

functions b u t of equivalence classes of f u n c t i o n s . no meaning i n L 2 ( M , p ) . I n s t e a d of L,(M,p) dense subspaces

D

o f L2(M,p)

does n o t c o n s i s t of

So p o i n t e v a l u a t i o n has

a s a whole, we consider c e r t a i n

h i c h c a r r y a s t r o n g e r topology than t h e norm

24

A MEASURE

topology of L 2 ( M , p ) . I n such each c l a s s

[PI

THEORETICAL SOBOLEV

LEMMA

D w e can s e l e c t a r e p r e s e n t a t i v e

D such t h a t t h e l i n e a r f u n c t i o n a l

E

i s l i n e a r and continuous on D f o r each x

M.

E

E

:

[PI

+

o u t of

;(XI,

[PI

E

D

The question i s how t o

generate t h e s e subspaces 0. In t h e p r e s e n t chapter we study t h i s problem i n a more g e n e r a l c o n t e x t . In p a r t i c u l a r we o b t a i n a c h a r a c t e r i z a t i o n of t h e t r i p l e s of H i l b e r t spaces

i n which p o i n t e v a l u a t i o n i s a continuous l i n e a r o p e r a t i o n .

11.1. A n elementary i l l u s t r a t i o n On t h e i n t e r v a l C - r r , ~ ] w e consider t h e usual Lebesgue measure dx. The corresponding H i l b e r t space functions ek, k

E

Z,

L 2 ([-IT,T])

=

L2([-rr,n],dx)

i s s e p a r a b l e . The

defined by

satisfy

e

e k ( x ) e ( x ) dx = 6 ke

,

k,L

5

.

z?,

-11

So t h e family ice,]

1k

E

Z}

i s an orthonormal s e t i n L 2 ( [ - ~ , a l ) . As a

consequence of weierstrass'approximation theorem t h e s e t (Ce

k

an orthonormal b a s i s i n

L 2 ([-IT,IT]).

So each

E

L2 ( [ - n , n l )

I)

k c Z is i s represented

by an L2-convergent s e r i e s

Next we introduce t h e p o s i t i v e bounde

OF

r a t o r R by

W e observe t h a t R i s a p o s i t i v e Hilbert-Schmidt o p e r a t o r . The space

25

AN ELEMENTARY ILLUSTRATION

H m ( C - n , n I ) , m > 0 , i s d e f i n e d a s follows

For each m > 0 , t h e space s t r u c t u r e : L e t cwl,Cvl R-mCvl

E

L, ( [ - n , n I ) .

E

H m ( [ - T , T I ] ) can be given i t s own H i l b e r t space H m ( C - n , n l ) . Then we have R-mCwl E L 2 ( [ - n , a l ) and

Now p u t

( ~ w l , c v l ) H= m

(R"Cw1

.

,R-mCvl) L2

Hm ( C - n , n l ) becomes a H i l b e r t space with i n n e r product ( * , * ) Hm proof of t h i s statement i s s t r a i g h t f o r w a r d . We observe t h a t II[wlII

Thus

9

-

5

L2 SO t h e topology generated by

f o r a l l Cwl E H m ( [ - n , n l ) . m i s s t r o n g e r than t h e topology generated by t h e norm 11 * 11 L

II R~ II 11 C W I I I H

II II H

. The

.

2 m Since R" i s a p o s i t i v e bounded operator with an unbounded i n v e r s e R-m, t h e norm II-II

H-,

on

IICcplllH

L2 ( [ - n , n l ) =

defined by

,

IIRmCqlII

-m

[cpl

E

L,(C-a,sl)

L2

I I L of L 2 ( [ - n , n l ) . The 2 . By t h e usual space L ([-n,a]) i s n o t complete with r e s p e c t t o 11 * I I H i s weaker and n o t e q u i v a l e n t t o t h e usual norm 11.

2 -m procedure of completing we o b t a i n a Banach completion H - m ( [ - n , n l )

of

L2(C-n,n]).

I~*IIH

The space H-m([-n,al)

is even a H i l b e r t space, s i n c e

-m

s a t i s f i e s t h e p a r a l l e l l o g r a m law. A r e p r e s e n t a t i o n of H-m(C-n,al) can be given by means of formal s e r i e s

2

where ( ( l + k )

0

E

H

-'s ak)kEz i s an 1 -sequence. H e u r i s t i c a l l y speaking,

([-n,~rI) i f f -m

2

=

R-mCrpl f o r c e r t a i n [rpl

m t h e l i n e a r o p e r a t o r R from L 2 ( C - n , n ] )

E

L 2 ( C - r , n 1 ) . We can extend

t o H - m ( C - n , n l ) . Denoting t h i s exm R maps H - m ( C - n , n l ) i s o -

t e n s i o n by Rm a l s o , it i s n o t hard t o s e e t h a t m e t r i c a l l y onto L 2 ( C - n , n l ) . There e x i s t s a p a i r i n g

<-,a>

m

onHm([-n,n]) x H-,([-n,n]),

l i n e a r f u n c t i o n from Hm(C-n,n])

is defined a s follows,

x H-m([-n,n])

i . e . a sesqui-

i n t o C. T h i s p a i r i n g

m

A MEASURE THEORETICAL SOBOLEV LEMMA

26

Hm ([-n,n]) and H -m ( [ + , T I ) a r e i n d u a l i t y ; soeach c o n t i n u o u s l i n e a r f u n c t i o n a l o n Hm (C-n,nI)

By some s t r a i g h t f o r w a r d arguments i t can be shown t h a t

i s of t h e form

Cwl

I-+

,

CWIE Hm(C-n,nI) ,

r

and each continuous l i n e a r f u n c t i o n a l on H-m(C-n,n])

NOW

suppose t h e r e e x i s t continuous l i n e a r f u n c t i o n a l s

Hm(C-rr,nl),

Cwl

E

Hm([-n,nl).

E

eX '

x

E

C-n,nl,

on

m 2 0, such t h a t xk+ ex(Lwl) i s a r e p r e s e n t a t i v e of Then t h e s e r i e s

has t o be convergent i n all x

is o f t h e form

H -m ( C - n , n l )

for a l l x

[-n,nl.

E

I t means t h a t f o r

I-n,nl

])I

Since IL (Ce = 1 f o r almost a l l x E [-n,n] and a l l k E 2 2 , t h e l a t t e r x k condition implies t h a t m has t o be g r e a t e r than 5 . For 0 5 m 5 5 t h e r e do n o t e x i s t c o n t i n u o u s l i n e a r e v a l u a t i o n f u n c t i o n a l s on H (C-n,nI).

m

Let m > 5 . For each k

[e,].

E

.

23, l e t us f i x a r e p r e s e n t a t i v e ek i n t h e c l a s s

Then f o r almost a l l x

E

C-n,nl

E

Z we have I z k ( x ) I = 1.

[-n,nl\N

where N denotes a n u l l

and a l l k

So t h e s e r i e s

i s convergent i n H-m(C-n,nl)

s e t . Putting

zx

= 0

for x

function x

I+

<[Wl,ZX>

E

for a l l x

N

E

it follows t h a t f o r a l l

Cwl

E

Hm(C-n,nl) t h e

AN ELEMENTARY ILLUSTRATION

i s a r e p r e s e n t a t i v e of t h e c l a s s [ w ] .

21

So t h e choice of r e p r e s e n t a t i v e s

i n [ e 1 induces a family of e v a l u a t i o n f u n c t i o n a l s k r e p r e s e n t a t i v e s Ox. 1

2

k

with H ( [ - n , n ] ) -m

C l e a r l y , vu m u s t b r i n g i n more uniqueness i n our d e f i n i t i o n of e v a l u a t i o n f u n c t i o n a l s . We must e v a l u a t e "canonical" r e p r e s e n t a t i v e s of t h e elements of H m ( [ - n , n ] ) .

I n the p r e s e n t example t h i s i s done a s follows. In each 1 ikx c l a s s [ e 1 w e t a k e t h e canonical r e p r e s e n t a t i v e ek : x t+ - e , k

x

6

[-n,n],

E

Ji;;

k

E

a.

6x

=

1 fi

Then we d e f i n e f o r each x

eLikX[ekl

-m

( C - n , a l ) . A l s o , f o r each Cwl

1 1 keZ



L2.

t h e function x t+

[-n,n],

keZ

i s an element of H

with ( a k ) k E ZE

E

( C U I , C I~)

e

ikx

E

Hm (C-n,al)

-

-

L2

Because of t h e e s t i m a t i o n

([wl,6

> i s t h e continuous r e p r e s e n t a t i v e of t h e c l a s s

Cwl. In t h e remaining s e c t i o n s of t h i s c h a p t e r we g e n e r a l i z e t h e s e concepts t o g e n e r a l measure spaces (M,p),

M

i n p a r t i c u l a r t o measure spaces (M,p) with

a metric space and p a r e g u l a r Bore1 measure. This g e n e r a l i z a t i o n can be

regarded as a measure t h e o r e t i c a l Sobolev lemma.

A MEASURE THEORETICAL SOBOLEV LEMMA

28

11.2. Let

Sobolev t r i p l e s of H i l b e r t spaces

R denote a p o s i t i v e bounded o p e r a t o r on t h e s e p a r a b l e H i l b e r t space

with p o s s i b l y unbounded i n v e r s e R

-1

. In

introduce t h e s e s q u i l i n e a r form ( * , - )

(w,v)l = ( R - ' ~ , R - ' ~ ) ~

t h e subspace R ( X ) = { R f

x we

. i s non-degenerate.

Lemma.

The space R ( X ) with i n n e r product ( - , - )

Proof.

XI

E

1

Because R i s i n j e c t i v e , t h e s e s q u i l i n e a r form ( . , . ) ,

11.2.1.

If

Observe t h a t Ilwll, = I I R - l w l l

Next i n t h e H i l b e r t space

x

1

. So

i s a H i l b e r t space.

R-'

: R ( X ) -+

X i s an isometry.

we consider t h e s e s q u i l i n e a r form ( * , * )

1

given

by (f,g)-l

The v e c t o r space

= (Rf,Rg)X

x

,

f,g

E

x . i s a pre-Hilbert space. Let

with i n n e r product

R-l ( x ) denote a completion. Then we extend t h e s e s q u i l i n e a r form ( - , * )-1 -1 t o an i n n e r product i n R ( X ) and t h e o p e r a t o r R t o an isometry from

R-l(X)

onto

x.

For F , G

E

R-l(X)

W e observe t h a t formally each F c e r t a i n unique f

E

we have

E

R-'(X)

can be w r i t t e n as F =

R -1 f

for

X.

Thus we o b t a i n t h e t r i p l e of H i l b e r t spaces

W e c a l l such a t r i p l e a Soboleo t r i p l e .

Remark. I f R i s a p o s i t i v e bounded continuously i n v e r t i b l e o p e r a t o r then R(X) =

X

= R-l(X)

a s t o p o l o g i c a l v e c t o r spaces. W e note t h a t

R is c o n t i -

nuously i n v e r t i b l e i f 0 i s n o t contained i n i t s e s s e n t i a l spectrum.

0

SOBOLEV TRIPLES O F HILBERT SPACES

On t h e product space R ( X ) x R - l ( X )

29

w introduce the p a i r i n g <*,-> a s

follows, = (R

-1

u , R F ) ~ ,'

For each u E R ( X ) and f E

= ( u , f )

x

u

,

R(X)

E

F E

.

R-'(X)

we have

X '

Theorem.

11.2.2.

The spaces R ( X ) and R-'(X)

e s t a b l i s h a dual p a i r with r e s p e c t t o t h e

p a i r i n g <',.>.

roof.

The o p e r a t o r R~ maps R - ' ( x )

i s o m e t r i c a l l y onto R ( x ) . Since R ( X ) i s

a H i l b e r t space any continuous l i n e a r f u n c t i o n a l on R ( X ) can be w r i t t e n a s

L(w)

=

,

( W , U ) ~ = <W,D

w

E

R(X)

f o r c e r t a i n u E R ( X ) where F i s uniquely determined by RF = R

-1

u. Hence

any continuous l i n e a r f u n c t i o n a l on R ( X ) can be w r i t t e n a s w ++ < w , D , -1 w E R(X1, for certain F E R Conversely, t h e l i n e a r f u n c t i o n a l

(x).

w

t+

< w , O i s continuous on

R ( X ) f o r each w

Similarly, a l i n e a r functional

for certain u

e on

R

-1

E

R ( X ) because

( X ) i s continuous i f 1 can be

0

R(X).

E

L e t (M,p) denote a o - f i n i t e measure space, and l e t

D

:

X

-t

L2(M,p)

denote

a densely defined l i n e a r o p e r a t o r with R ( X ) contained i n i t s domain. The o p e r a t o r I%? i s everywhere defined. On DR we impose t h e following c o n d i t i o n .

DR

11.2.3.

Let

b

:

M

+

:

X

-+

L 2 (M,p) i s a (bounded) Carleman o p e r a t o r .

X be an inducing function f o r t h e Carleman o p e r a t o r described

i n D e f i n i t i o n 1.2.1. Then f o r each f

E

x

t h e function x

++

( f , h ( x ) ) X is n

30

A MEASURE THEORETICAL SOBOLEV LEMMA

r e p r e s e n t a t i v e of t h e c l a s s PRf. For each x tionals

eX

M w e d e f i n e t h e l i n e a r func-

on R ( X )

ex(w) Since k ( x )

for a l l w

E

( R - ' ~ , k ( x ) ),~

X for a l l x

E

E

=

R(X).

E

R(X)

M we o b t a i n t h e e s t i m a t i o n

E

ex, x E M, a r e continuous on t h e R ( X ) t h e f u n c t i o n x t+ e x ( w ) , x E M, i s

t h e functionals

SO

K i l b e r t space R ( X ) . For each w a r e p r e s e n t a t i v e of

w

ow

E

L

2

E

(M,p).

Thus t h e l i n e a r f u n c t i o n a l s

C?

can be

regarded as a g e n e r a l type of e v a l u a t i o n f u n c t i o n a l s . In p a r t i c u l a r , t a k e

x

=

L2(M,p), R a p o s i t i v e bounded Carleman o p e r a t o r on

L2(M,p) and D t h e i d e n t i t y o p e r a t o r . Then i n t h e above d e s c r i b e d way we o b t a i n continuous l i n e a r f u n c t i o n a l s A

Ax([wl)

such t h a t x

,

x

E

i s a r e p r e s e n t a t i v e of

M

[wl.

Here

d

:

M

-t

L2(M,u)

denotes an inducing f u n c t i o n of R.

Although t h e above s e t up i s extremely simple, mathematically it i s n o t very s a t i s f a c t o r y , because t h e i n t r o d u c t i o n of t h e l i n e a r f u n c t i o n a l s

e

i s f a r from canonical. Indeed, i n s t e a d of t h e f u n c t i o n k : M + X ( d : M + L2(M,p)) we can take any o t h e r f u n c t i o n k ( d ) which d i f f e r s from h (d) only on a s e t of measure zero. I n t h e next s e c t i o n s w e prove t h a t

- -

(Ax)

f o r c e r t a i n t o p o l o g i c a l measure spaces (M,p) functionals

C?

we can i n t r o d u c e t h e l i n e a r

( A ) i n a canonical way. x x

11.3. Federer measure spaces Let

M

denote a t o p o l o g i c a l space. Then it i s n a t u r a l t o regard M a s a Borel

space with t h e o-algebra

B

M.

of

As usual the elements

a r e c a l l e d Borel measures.

generated by t h e c o l l e c t i o n o f open s u b s e t s o f

B

are c a l l e d Borel s e t s and measures on (M,B)

31

FEDERER MEASURE SPACES

11.3.1.

Definition.

A Borel measure p on a t o p o l o g i c a l Borel space

measure i f f o r each an open s e t

E

> 0 and each Borel s e t

u

M i s c a l l e d a r e g u l a r Borel with p ( U ) <

m

there exists

il and a closed s e t X i n M such t h a t 1 c U c il and p ( i l \ X ) <

E.

Example. Any Borel measure p on IR” (where IRn regular i f

A!

c a r r i e s i t s Euclidean topology) i s

has t h e property t h a t bounded Borel s e t s i n lRn have f i n i t e

p-measure.

T h i s r e s u l t i s contained i n t h e following theorem.

11.3.2. Let

Theorem.

M be a l o c a l l y compact m e t r i c space and l e t

such t h a t bounded s u b s e t s of

p be a Borel measure on

M have f i n i t e p-measure.

If

M

M i s o-compact,

i . e . M can be w r i t t e n a s t h e countable union of d i s j o i n t compact s e t s , then

0

p i s a r e g u l a r Borel measure.

We omit t h e proof of t h i s theorem. This theorem i n d i c a t e s t h a t under c e r t a i n t o p o l o g i c a l c o n d i t i o n s on a metric space r e g u l a r i t y of a Borel measure

i s n o t a very r e s t r i c t i v e c o n d i t i o n .

Next we introduce t h e notion of Federer measure space. This i s nology t o i n d i c a t e c e r t a i n t o p o l o g i c a l measure spaces (M,p)

our termi-

of a very

s p e c i a l kind.

11.3.3.

Definition.

A o - f i n i t e measure space ( M , p )

i s s a i d t o be a Federer measure s p a c e , i f i t

possesses t h e following p r o p e r t i e s .

-

M i s a s e p a r a b l e t o p o l o g i c a l space metrized by t h e m e t r i c d. p i s a r e g u l a r Borel measure on

M such t h a t bounded Borel s u b s e t s of M

have f i n i t e p-measure.

-

The measure space (M,p) admits a r e l a t i v e d i f f e r e n t i a t i o n theorem: Let cp :

M+

C denote a B o r e l function which i s i n t e g r a b l e on bounded

Borel s e t s . Then t h e r e e x i s t s a n u l l s e t all x

E

M\N

the

N

cp

such t h a t f o r a l l r z 0 and

closed b a l l B ( x , r ) with r a d i u s r and c e n t r e x has

cp p o s i t i v e measure. Moreover, t h e l i m i t

A MEASURE THEORETICAL SOBOLEV LEMMA

32

exists for all x E

M\N

u

The f u n c t i o n x

9’ with cp = 9 p-almost everywhere.

9 ( x ) d e f i n e s a Borel f u n c t i o n

N

In l i t e r a t u r e ( c f . r e f e r e n c e s a t t h e end of t h i s c h a p t e r ) t h e r e a r e given c o n d i t i o n s on t h e metric d of a s e p a r a b l e metric space

M

such t h a t t h e

measure space (M,p) i s a Federer measure space for any r e g u l a r a - f i n i t e Borel measure p on M with t h e property t h a t bounded Borel s e t s of f i n i t e u-measures.

M

have

These c o n d i t i o n s are of a geometric n a t u r e . Examples

of such metric spaces (M,d) a r e t h e following:

-

F i n i t e dimensional v e c t o r spaces with d ( x , y ) = v ( x - y ) where v i s any norm on M.

-

A Riemannian manifold (of c l a s s 2 2 ) with i t s Riemannian m e t r i c .

The above conditions p u t r e s t r i c t i o n s on t h e geometry o f t h e m e t r i c space

M. Other c o n d i t i o n s l a y r e s t r i c t i o n s on t h e r e g u l a r Borel measure p . In t h a t case i t i s assumed t h a t

f o r almost a l l x

E

M.

The measure theory which i s r e q u i r e d t o prove t h e above s t a t e m e n t s , i s t o o involved f o r t h e a i m s o f t h i s monograph. However, t o g i v e an i d e a of t h e c o n s t r u c t i o n of such p r o o f s , w e show t h a t ( I R n , p ) space f o r each n

E

i s a Federer measure

JN and each a - f i n i t e Borel measure p on IRn f o r which

bounded Borel s u b s e t s of IRn have f i n i t e p-measure.

Let n

E

IN be f i x e d . In IRn we consider t h e maximum norm

Then t h e metric topology i s f i x e d by cubes whose edges a r e p a r a l l e l t o t h e coordinate axes. By Qx(hl we denote t h e closed cube with c e n t r e x and edge

FEDERER MEASURE SPACES

Q,

l e n g t h h, by

we denote any cube with c e n t r e

We say t h a t a family

c

where

B

E

IRn

t h e c h a r a c t e r i s t i c f u n c t i o n of t h e cube

Put

9.

Theorem (Besicovitch covering lemma).

be a bounded s u b s e t of IRn and l e t

B such t h a t f o r each x

E

B

c

be a family of cubes covering

t h e r e e x i s t s a cube

Q

c

E

t h e r e e x i s t s a countable number of p o i n t s (x 1

in

k krlN

-

c.

belongs t o a t most c cubes of

h a s bounded o v e r l a p s , i f

xQ denotes

11.3.4. Let

c

X.

of cubes has bounded overlaps, i f t h e r e i s a fixed

c o n s t a n t c such t h a t every x differently,

33

with c e n t r e x. Then

B

such t h a t

1 k E IN] has bounded o v e r l a p s , where t h e o v e r l a p c o n s t a n t xk c can be chosen such t h a t it depends on n, only.

The s e t

Proof.

{Q

We prove t h i s theorem f o r n = 2. The proof f o r n > 2 runs s i m i l a r l y .

F i r s t we show an a u x i l i a r y r e s u l t .

m. Let

(QkIkelN

denote a sequence of cubes with c e n t r e ( \ )

j < k , then xk

p!

Qj

and

1 Qk I

t i v e l y ) . Then t h e family {Qk

i 2

Ik

1 Q 3,I E

( i.e.

kc IN

such t h a t i f

t h e a r e a s of Qk and Q ,

respec-

3'

IN) h a s hounded o v e r l a p s and t h e o v e r l a p

c o n s t a n t c can he chosen t o depend only on n.

Proof.

.

, m = 1 , 2 , . . , denote those Qk which c o n t a i n t h e o r i g i n with m c e n t r e s i n the f i r s t quadrant. L e t h denote t h e edge l e n g t h of Qk Let &I

m

m

.

81

covers a t l e a s t t h e region

{(x,y)

E

2

IR 1 0 i x 2 $ h l , 0 2 y i $hl]

.

N o Qkm can have i t s c e n t r e o u t s i d e the s e t { ( x , y ) 10

s i n c e otherwise we would have h

> 2hl and hence

m d i c t i o n . Therefore t h e c e n t r e of each regions I -1V.

Qk , m m

L

5

I km

IQ

x 5 h l , 0 i y 5 h1} > 41Qk,I, a contra-

2 must l i e i n one of t h e

A MEASURE THEORETICAL SOBOLEV LEMMA

34

hl

3h 1

Since xk

%,

p

m

f o r m > 1 t h e c e n t r e xk

c e n t r e l i e s i n 11, then, s i n c e 0 B most one

Qk

I11 and I V .

of

Qk , m

kmdoes

not l i e i n I. I f the

i t covers 11, and s o t h e r e i s a t

with c e n t r e i n 11. The same arguments apply f o r t h e regions

m

T r e a t i n g t h e o t h e r quadrants i n t h e same way we d e r i v e t h a t a t

most 16 cubes 2 p o i n t of IR

Qk

contain t h e o r i g i n . By t r a n s l a t i o n t h e same holds a t any

.

Proof of Besicovitch l e m u . Let a = sup {lQxl I x E B}. I f a l = -, t h e r e 1 are a r b i t r a r i l y large Q , and s i n c e B i s a bounded s e t we simply choose one which c o n t a i n s 6. I f al < -, w r i t e 6, = B and f i x x 1 E 6, with 1Q I > $ a , . X1

Put

6, If

B, # 8,

=

B,\Qx

a2 =

I

SUP

1

tIQxI 1

x E

,

with

and hence a2 > 0, choose x

B,

E

B21

.

lQx2[ >

%a2. Proceeding i n

t h i s way, w e o b t a i n subsequently

Bk

=

B

\Q k-l

=

k- 1 B \ U Qx,

Xk-l

The process can be continued a s long a s

xk

E

Bj

,

J

j=1

B k # 8.

Since x

E

6, w e have

f o r a l l j 5 k. Therefore

lax 1 k

ak 5 aj

5

21Qx,I 3

j 5 k

,

.

From t h e preceding lemma it follows t h a t t h e sequence overlaps. I t remains t o show t h a t B c ( u

kc lN

Q,).

(QXk 1 kcIN

If 6 ,

0

has bounded

i s empty f o r cer-

35

FEDERER MEASURE SPACES

t a i n ko

E

ko-1 then 6 i s contained i n k = Ul

IN,

i n f i n i t e l y many

$ak <

lQxkl

Q

xk'

I f n o t , then t h e r e a r e

Since t h e sequence ( a

f o r some 6 > 0 and a l l k

E

IN.

k ~ r ni s d e c r e a s i n g and s i n c e

I

IQ

-+ 0 a s k + m o r l Q x k l t 6 Xk The second p o s s i b i l i t y cannot a r i s e because

5 ak it follows t h a t e i t h e r

f o r each k

Q,,.

Q,,

IN and j 5 k, xk does not belong t o

E

be bounded. Hence

x

E

19

Bk

1

0 and ak + 0 as k xk f o r a l l k e IN and hence -f

+ m.

IQ

I

For x

E

6\

,U

11.3.5.

6 \jErnQxj

B would not

Q x , it would mean t h a t J

5 ak f o r a l l k E 7N.

c o n t r a d i c t i o n with t h e assumption t h a t f o r a l l x Thus w e have proved t h a t

and hence

3

E

This l e a d s t o a

6 there exists Q

c.

E

X

0

i s empty.

a.

L e t 1.1 and V denote r e g u l a r Borel measures which a r e f i n i t e on bounded Borel s e t s . W e make t h e redundant assumption t h a t u ( Q x ( h ) )7 0 f o r a l l x and a l l h > 0 ( c f . Lemma IV.l.l).

IRI'

E

Let c denote t h e o v e r l a p c o n s t a n t i n t r o -

duced i n Besicovitch covering lemma and l e t 6 denote a bounded Borel s e t o f lRn.

Proof. Because

Then f o r each a > 0 t h e r e e x i s t s a Borel s e t A

Fix a > 0 and l e t

s

> a}.

:=

v i s r e g u l a r w e can take an open s e t

For each x

E

s

t h e r e e x i s t s a cube

G

QE such t h a t

Using Besicovitch covering lemma we can s e l e c t

and

a

B QE

3

QL

Let

E

with V ( G ) < V ( B ) c G and c G, k

E

> 0.

+

E.

v(Qx) > a p(Qx). IN,

such t h a t

36

A MEASURE THEORETICAL SOBOLEV LEMMA

k!NQ:

I t follows t h a t f o r A" =

p(A")

Put A =

v(G)

5

Then

fl nE i N

s

we have

s

( ~ ( 6 +) E )

.

A and p(A)

c

c A" and

U(6)

S

.

0

Before e s t a t e t h e main r e s u l t we prove f i r s t t h a t each p-integrable function can be approximated by a sequence of continuous f u n c t i o n s .

11.3.6. Let

$3

*.

be a p-integrable

function on IR. Then t h e r e e x i s t s a sequence

( g k ) k E m o f continuous f u n c t i o n s such t h a t

Proof. Since t h e r e e x i s t s t h a t mj I'p-'pkl dp * 0 as functions xu where u i s a

IRn

u

c $? and p ( n \ C )

I q - g k l dp

0 as k

-f

+ m.

a sequence ( q k ) k E m of simple f u n c t i o n s such

k + =, w e can r e s t r i c t o u r s e l v e s t o c h a r a c t e r i s t i c Borel set with f i n i t e p-measure.

Since p i s r e g u l a r , t h e r e e x i s t s an open s e t

Z c

I

L e t L > 0.

and a c l o s e d s e t C such t h a t

< E. Now use Urysohn's lemma. I t y i e l d s a continuous

function g such t h a t 0 5 g

5 1

with g

Z

*

1 on C and g Z 0 on

.

Hence

0 11.3.7. Theorem ( R e l a t i v e d i f f e r e n t i a t i o n theorem). Let p s a t i s f y t h e c o n d i t i o n s of Lemma 11.3.5 and l e t cp be a Borel function which i s p-integrable on bounded Borel sets i n lRn.

mn

for a l l x c

Prqof. Assume have

p

except on a s e t of measure zero.

f i r s t t h a t cp i s p-integrable.

I (9,

Then we have

( h ))

-'

cpdu

-

Then f o r any p-integrable

g we

q(x)l 5

Qx ( h ) 5

p ( Q x ( h ) -1 )

1 Qx(h)

Icp -91

dv + / P ( Q x ( h ) ) - '

j

gdu

-

Blx)/

.

Qx( h )

If g i s a l s o continuous t h e l a s t term on t h e r i g h t converges t o / g ( x )- q ( x )

1

37

FEDERER MEASURE SPACES

a s h J. 0. Hence with L(x) denoting t h e limsup of t h e term on t h e l e f t hand hJ.0 s i d e of t h e previous i n e q u a l i t y , we o b t a i n

Let

6 denote a bounded Borel set. The s e t

is contained i n t h e union of t h e two s e t s

and

1

1

By Lemma 1 1 . 3 . 5 t h e r e e x i s t s a Borel s e t AE(B,g) such t h a t sE(B,g) and

-' J

1

u(AE(B1g)) 5 ( 4 ~ ) c

IV

C

1 A (6,g)

-gI dv

B and, f u r t h e r

v

and

N o w l e t (qk)kcm be a sequence of continuous f u n c t i o n s such t h a t

mn

f Icp -gkl

du

+

0 as k +.

m.

We p u t N E ( B ) = k:mAE(B,gk). Then we have

38

A MEASURE THEORETICAL SOBOLEV LEMMA

N

U

=

N1(BK)

U

j < I N KCIN

N is

Then

.

7

7

a n u l l s e t and

u

{x

E

I L(X)

IRn

€1

c

N

.

E>O

d \ N we

SO f o r a l l x E

get

I f cp i s n o t p-integrable,

then employing t h e o - f i n i t e n e s s of

(lRn,u)

we

m

can w r i t e cp = cp,

3

.E cp ]=I j '

( x ) = 0 o r cp. (x) I

=

where each c p . i s p-integrable

I

cp(x).

and where e i t h e r

0

11.4. A measure t h e o r e t i c a l g e n e r a l i z a t i o n of t h e Sobolev embedding theorem

R > 0 denote a bounded o p e r a t o r on

Let

measure space and l e t o p e r a t o r such t h a t

D

:

X

-P

x,

let

(M,u)

denote a Federer

L 2 ( M , p ) denote a densely defined l i n e a r

R ( X ) i s contained i n i t s domain. On t h e everywhere DR w e impose t h e following conditions.

de-

fined l i n e a r operator

11.4.1. a . The l i n e a r o p e r a t o r

k

Let

:

M

.+

X

DR i s a bounded Carleman o p e r a t o r .

denote an inducing f u n c t i o n of t h e o p e r a t o r

b. The Borel f u n c t i o n x ++

of

VR.

Ilb(x) ( I 2 i s p-integrable on bounded

B o r e l sets

M.

Remarks. - I f DR i s a Hilbert-Schmidt o p e r a t o r , then t h e corresponding f u n c t i o n x

I+

. Ilk(x) ll 2 LS p-integrable.

So i n t h i s case t h e Condition (b) i s re-

AN EMBEDDING THEOREM

39

dundant ( c f . Corollary ( 1 . 2 . 4 ) ) .

-

In general t h e second condition i s not redundant a s t h e following example shows. Define

k

: IR

*

L 2 ( R ) by

,

L O

x = o .

f o r x # 0 and hence x on t h e bounded i n t e r v a l ( - a , a ) , a

t

t+

Ilk(x) I1

2 .

1s not integrable

0.

The aim of t h i s s e c t i o n i s t o c o n s t r u c t c a n o n i c a ~e v a l u a t i o n f u n c t i o n a l s

ex,

x

E

M ( c f . Section 1 1 . 2 ) .

O u r method i s c o n s t r u c t i v e and t h e r e f o r e

rather t e c h n i c a l . I n t h e appendix t o t h i s chapter w e p r e s e n t an e l e g a n t b u t non-constructive method. Let (

v

~ denote ) ~ an ~orthonormal ~ basis i n

longs t o L 2 ( M , p ) .

x.

For each k

E

IN,

DRv, be-

Since bounded Borel s e t s of M have f i n i t e p-measure,

each

r e p r e s e n t a t i v e (DRvk)- of DRvk i s i n t e g r a b l e on bounded Borel s e t s . So there e x i s t representatives that for all k

Since I U R V , ~ all k

E

E

E

L,

(M,p),

IN and a l l x

E

k

E

k

E

and a s e t of p-measure zero

N,

such

M\N,

E

IN, t h e r e e x i s t s a p-null s e t N 2 such t h a t f o r

H

II k(x) II

M\N2

Also, t h e Borel function x

and f o r almost a l l x

of URv

'pk

IN and a l l x

M

* is i n t e g r a b l e o n bounded Borel s e t s

we have

I t follows t h a t t h e r e e x i s t s a p-null set N, such t h a t f o r a l l x

E

M\N,

A MEASURE THEORETICAL SOBOLEV LEMMA

40

In t h e remaining p a r t o f t h i s s e c t i o n m e n the Relations II.4.2.a-b

are

N

i n d i c a t e s t h e null s e t

valid for a l l x

E

N, u N, u N,.

M\N.

Remark. I n p r a c t i c e , t h e aim i s t o choose t h e r e p r e s e n t a t i v e s

(pk

of DRv,,

k

E 34, such t h a t t h e n u l l s e t N is as small a s p o s s i b l e . I f a continuous 1 r e p r e s e n t a t i v e can be taken then N, = N, = 0.

Definition.

11.4.3. Let k

E

IN and l e t t h e functions (pk be defined a s i n I I . 4 . 2 . a .

We introduce

Y

t h e r e p r e s e n t a t i v e (URv,)

of URv,

11.4.4. a g . a. L e t x

E

M

b. Let x

E

M\N.

belongs t o

and d e f i n e

ex

=

as follows

kil ( 2 % ~ ~ )(x) " Rv,.

Then

ex

Then f o r a l l r > 0

R ( X ) . Further,

Proof. a. From t h e d e f i n i t i o n of t h e functions (DRv )- we d e r i v e k

Hence

e

E

b. For each k

R(X), E

IN t h e following i n e q u a l i t y i s v a l i d

E

R(X).

AN EMBEDDING THEOREM

5 p(B(x,r))-'(

B (x, r )

I

I (DRvk)"12 d u ) .

as a consequence o f t h e B o l d e r i n e q u a l i t y . Thus we o b t a i n

Because of C o n d i t i o n II.4.1.b

ex(')

E

R(X).

Next VE prove t h a t l i m

r+O Let

E

t h e l a t t e r e x p r e s s i o n i s f i n i t e , whence

IIeX - ex(') 11

> 0. Take a f i x e d ko

E

=

0.

IN so l a r g e t h a t

m

(*)

k=k 4-1

I (DRvk)" ( x ) l 2

< ~

~

./

4

0

Then b e c a u s e of II.4.2.a-b

w e c a n t a k e ro > 0 so s m a l l t h a t €or a l l r ,

O < r < r 0

(**)

1 (DRv,

for a l l k =

I,

Now t h e f o l l o w i n g e s t i m a t i o n can be c a r r i e d o u t :

41

42

A MEASURE THEORETICAL SOBOLEV LEMMA

BY ( * * ) we o b t a i n

and by (*) and (***)

< 2 E 2+ 2 E 2 < 3 E

2

.

4

Thus we have proved t h a t f o r a l l r , 0 < r < r 0

Theorem (Measure t h e o r e t i c a l Sobolev lemma).

11.4.5.

L e t t h e pair

D,R s a t i s f y C o n d i t i o n I I . 4 . 1 . a - b .

For each w

R(X) there e x i s t s a representative

N

-

a.

E m

( h )= kgl ( R - ' W , V , ) ~ ( D R V ~ )

( h )o f Dw.

w

w i t h p o i n t w i s e convergence.

b . L e t x E M. Then t h e l i n e a r f u n c t i o n a l w

N

i+

(%I)

(x) i s c o n t i n u o u s on

R ( X ) ; i t s Riesz r e p r e s e n t a t i v e i n R ( X ) e q u a l s ex. c. L e t x

E

M\N. m e n

2 d . Suppose i n a d d i t i o n t h a t t h e f u n c t i o n x t+ Ilh(x) [ I x i s e s s e n t i a l l y bounded

on M . Then t h e convergence i n (i) i s uniform o u t s i d e a s e t o f p-measure z e r o No. Moreover

43

AN EMBEDDING THEOREM

- i s c o n t i n uMo u s x.

e . Suppose i n a d d i t i o n t h a t t h e f u n c t i o n k each w

Proof.

R(X)

E

Let w E N

R ( X ) . Then w

=

kil (R-'w,vk)

:

-+

i s c o n t i n u o u s . Then f o r

(Dw)

kzl ( R

(DRvk)-.

-1

W , V ~ Rvk ) ~

with convergence i n

R(X).

N

Then o b v i o u s l y

(Dw)

i s a representative

Dw.

kEl

a. S i n c e ( W , C ~ )= ~

(R-'w,v

c o n v e r g e s p o i n t w i s e on N

b. T r i v i a l because C.

=

m

P u t (Dw)

of

the fuhction

Let x

E

M\N

(Dw)

and w

E

k

)

(DRv k )

N

( x ) and s i n c e

e

E

R ( X ) t h e series

M. (x)

= (w,C

1

x 1'

R ( X ) . Then

Since f o r a l l r > 0 m

w e can i n t e r c h a n g e summation and i n t e g r a t i o n i n t h e l a s t e x p r e s s i o n . It yields

d. By assumption t h e r e e x i s t s a n u l l s e t No s u c h t h a t

A MEASURE THEORETICAL SOBOLEV LEMMA

44

So f o r each w

E

R(X)

f o r a l l K > L , K,L

and x

M\No we o b t a i n

E

IN.

E

N

e . One r e a d i l y checks t h a t we can t a k e t h e f u n c t i o n (DRvk ) that for all x E M

This y i e l d s f o r each w

(Dw)

-

(x) =

E

R(X)

and a l l x

6

continuous, and

M

( w , ~ , ) ~= ( R - l w , k ( x ) ) X N

and t h u s the c o n t i n u i t y of (Dw)

.

-

Remark. The t h i r d statement of t h e above theorem i n d i c a e s t h t t h e repreN

s e n t a t i v e (Dw) and a l l x

E

i s canonical. Indeed, f o r each r e p r e s e n t a t i v e

M\N (Uw)

N

(XI

= l i m u(B(x,r))-'(

rJO

B(x,r)

I

(Dw)

of

Dw

.

(Dw)^d~)

11.5. Some a p p l i c a t i o n s

AppZication I : 6-functions on Federer measure spaces. Let

(M,u) denote a Federer measure space and l e t R > 0 denote a bounded i n t o L2(M,u). W e consider t h e Sobolev t r i p l e

Carleman o p e r a t o r from L2(M,u)

a s introduced i n Section 1 1 . 2 f o r a general s e p a r a b l e H i l b e r t space

h

:

M

I+

L,(M,p)

denote an inducing f u n c t i o n of

measurable function x t+

Ilh(x) I1

x.

Let

R. W e assume t h a t t h e

i s u-integrable on bounded Bore1 s e t s of M. L2

45

SOME APPLICATIONS

Following Theorem 11.4.5 t h e r e a r e continuous l i n e a r f u n c t i o n a l s and a n u l l set

-

u

q : x

I+

k!

N

such t h a t f o r each [ q l

([TI)

E

k!

X'

x

E

M,

R(L,(M,p))

i s a r e p r e s e n t a t i v e of [ q ]

;

I n Section 1 1 . 2 we have shown t h a t t h e space R - l ( L z ( M , p ) ) i s a representat i o n of t h e dual of

R(L,(M,u))

with r e s p e c t t o t h e n a t u r a l p a i r i n g <.,.>

on R ( L , ( M , p ) ) x R - l ( L , ( M , p ) ) . So f o r each x E M , t h e r e e x i s t s an element -1 6x E R ( L 2 ( M r u ) ) such t h a t k!x(Cq31) = < C q l , G x > , Cql E R ( L 2 ( M , p ) ) . Since f o r each element C q l E

with

c [q],

L 2 (M,p)

we have

we can w r i t e formally

H e u r i s t i c a l l y speaking, R - l

(L, (M,p) ) c o n t a i n s "a complete s e t of d e l t a

functions", c f . Chapter 111.

Application 2: The c l a s s i c a l Sobolev embedding theorem on C-7r,nln. On t h e n-dimensional

dx = dxldx 2 . . . d x

cube

c

=

. I n L 2 (cn ,dx) +

A = l - ( $

... +

[-n,7rIn

we t a k e t h e u s u a l Lebesgue measure

we consider t h e o p e r a t o r A ,

7) a2

ax

1

where we impose p e r i o d i c boundary conditions. normal basis of eigenvectors

where k

E

Zn, k = ( k l ,

l ek I ,

...,k n ) .

2 Acek] = (1 + k l

W e have

+ . . . + k n2 ) [ekl

.

The o p e r a t o r A has an ortho-

46

A MEASURE THEORETICAL SOBOLEV LEMMA

Next we introduce t h e p o s i t i v e bounded o p e r a t o r

Rm,

Rm

m > 0 , by

=

A-m/2

(c

Then R ( L , d x ) ) i s t h e Sobolev space of 2n-periodic " f u n c t i o n s " H m ( C n ) m 2 n -1 of p o s i t i v e o r d e r m. Correspondingly t h e space Rm (L2(Cn,dx)) i s denoted by H-,(C

).

The well-known Sobolev embedding theorems can be formulated a s

follows.

Theorem.

11.5.1.

e

Let m > n/2 and l e t 0 5 such t h a t f o r each [ u l

E Z.

-u

Then t h e r e i s a n u l l set N

Hm(cn ) t h e r e e x i s t s a r e p r e s e n t a t i v e

E

t h e following property: For a l l u exists y

e

< m-n/2,

(IN u {O))",

E

u1

+... + u

5

ce,

of [u] with

k?, t h e r e

> 0 independent of [ u ] such t h a t

Here 'D

denotes t h e d i f f e r e n t i a l o p e r a t o r

norm of

H m( Cn ) . That i s IIC~III m

Praof.

Let o

each y

E

(IN

E

[-n,nIn

=

($-yl ... (
and 11 *

I I m the

I I A ~ / ~ C ~ I .I I L2

u {O})". The o p e r a t o r D'Rm

i s a Carleman o p e r a t o r i f f o r

the s e r i e s

il'l

I: kEZn

kyl

2

...

(1 + k l +

i s convergent i n L

2

(cn , d x ) . kf"1

1 kcZn

... k?

-

2 m/2 +kn)

e k ( Y ) lekl

I t means that t h e s e r i e s

... k2', ... + k n2 ) m

2 (1 + k l +

has t o be convergent. Comparison with t h e i n t e g r a l

... x20n (1 + x l +... )

mn

shows t h a t f o r 2 ( m Hence f o r l o

I

dxl

... dx

+X

5

e

- la I ) C

- (n - 1 ) > 1 t h i s i s indeed t h e case.

m -n/2

inducing function on

c

the operator f A - m / 2

i s given by

i s a Carleman o p e r a t o r . I t s

SOME APPLICATIONS

47

W e have

.. . k Z u n

k1 :'

k e d

for a l l y

E

2 (l+kl+

... + k 2n )m

Cn.

Now t h e s t a t e d r e s u l t s follow from Theorem 11.4.5.

AppZication 3: The c l a s s i c a l Sobolev embedding theorems on mn. L e t dx = dx l...dx

denote t h e Lebesgue measure on IRn.

In L2(mn,dx) we

i n t r o d u c e t h e p o s i t i v e s e l f - a d j o i n t o p e r a t o r A by

a2

A=l-(-

L e t En denote t h e Fourier transform on

AFn

Then t h e o p e r a t o r IF

Put M

Rm

rl

=

= A-m'2

I F :

+

+

...

+

y;

.

i s bounded f o r each m > 0 . So a l s o A IFn. The o p e r a t o r (M rl i s a bounded o p e r a t o r f o r each m > 0 . The space Rm(L2(IF?,dx))

is t h e Sobolev space

R m-1

yf

i s t h e p o s i t i v e s e l f - a d j o i n t o p e r a t o r of m u l t i -

,

p l i c a t i o n by the function

q(y) = 1

L 2 (IRn,dx) formally given by

(L,(d,dx))

Hm (lRn)

by H-m(IRn).

of o r d e r m. Correspondingly, w e denote Similar t o Theorem 11.5.1 we have t h e following

formulation of t h e Sobolev embedding theorems on lRn.

11.5.2. Let

)'(N n

Theorem.

m > n/2 and l e t 0

5

L < m -n/2, 1 E IN u

such t h a t f o r each [ul e H m ( I R n )

Then t h e r e i s a n u l l set N

t h e r e e x i s t s a r e p r e s e n t a t i v e u of

[ul with t h e following property: For a l l u

exists y

(0).

> 0 independent of [ul such t h a t

E

(IN u {O})",

101 5

L , there

A MEASURE THEORETICAL SOBOLEV LEMMA

48

I (uaL) ( x ) I

vxEIRn\N(L) :

n

Da

=

I1 [ u l I I m

=

Here

Proof.

(ey' (ern ...

11 Am/2[ul II

Let a

2 /(1 + y l

E

(IN u

and

IICUI

11 * [ I m d e n o t e s t h e norm o f H m ( I R n ) .

and p u t I-

Then rl

a,m

: y

a,m

*

i l a Jy;'

That i s

.. ./ :y

i s s q u a r e i n t e g r a b l e on lRn i f

With t h e a i d o f t h e F o u r i e r t r a n s f o r m IF

la

1

< m

- n/2.

it c a n b e s e e n t h a t t h e o p e r a t o r

i s a Carleman o p e r a t o r inducednby t h e f u n c t i o n

2)'A-m/2

: IRn

'a,m

I I .~

.

L2 {O])"

+ .. . + y f ) m / 2 .

5 y,

+

L2 ( R n , d x ) g i v e n by

where w e t a k e a w i t h la1 2

l l C k a , m ( x ; * )I

II

e. =

W e have for a l l x

IICI-a,mlII

L*

E

IRn

.

L2

The s t a t e d r e s u l t s f o l l o w from Theorem 1 1 . 4 . 5 .

Appendix

I n t h i s a p p e n d i x w e p r e s e n t an e l e g a n t b u t n o t v e r y c o n s t r u c t i v e way t o o b t a i n the f a m i l y Let

h

(ex)xEM.We

d e n o t e an i n d u c i n g f u n c t i o n o f t h e Carleman o p e r a t o r

11.a. I.

DR.

Proposition.

There e x i s t s a null s e t

Proof.

u s e t h e n o t a t i o n s o f S e c t i o n 11.4.

Let f

E

x.

N'

such t h a t f o r a l l x

The f u n c t i o n y

H

E

M\N'

I l h ( y ) - fll is i n t e g r a b l e on bounded

Bore1 s e t s . So t h e r e e x i s t s a n u l l s e t

Nf

such t h a t f o r a l l x

E

M\Nf

49

Since A c

x

i s a separable H i l b e r t space, t h e r e e x i s t s a dense countable s e t

X. P u t N ' =

Nf.

f;A

Then

N'

i s a n u l l s e t . Let x

sequence ( f n ) n E N i n A such t h a t l i m n+m

o

11 f

- k ( x ) IIx

E

M\N'.

Choose a

= 0 . Then we o b t a i n

Ilk(x) - fnIIX =

= l i m

n+-

Now we d e f i n e t h e family

11.a. 2.

Definition.

For x

N1

E

we p u t C?'

a s follows.

0 and f o r x

=

E

M\N'

we p u t t i = R ( k ( x ))

.

Then a s i n Lemma 11.4.4 we prove

II.d.3.

a.

a. L e t x

E

M.

b. Let x

E

M\N'.

Then C?;

E

R(X).

Then f o r a l l r > 0

Proof. a. T r i v i a l . b. Since t h e f u n c t i o n y

I+

IIk(y) IIx i s i n t e g r a b l e on bounded Bore1 s e t s , t h e

expression k(y) d u ( ~ )

B (x,r )

A MEASURE THEORETICAL SOBOLEV LEMMA

50

denotes a well-defined element of

Remark.

x.

Hence

e'(r)

L e t ( v ~ ) denote , ~ ~ an orthonormal b a s i s i n

E

R ( X ) . F u r t h e r we have

x.

Then w e have

m

I t follows t h a t t h e d e f i n i t i o n of

d e f i n i t i o n of

e

e'(r)

i n Lemma 11.4.4

( r ) i n t h e corresponding Lemma I I . a . 3 .

s e t N' depends on the choice of

a g r e e s with t h e However, t h e n u l l

k and on t h e countable dense s e t

A in

x,

whereas t h e n u l l s e t N i s p r e c i s e l y d e s c r i b e d by II.4.2.a-c.

Some comments and b i b l i o g r a p h i c a l n o t e s Our d e s c r i p t i o n o f o - f i n i t e measure spaces (M,p) i s taken from t h e Prel i m i n a r i e s of Yosida's book

w e r e f e r t o Dunford-Schwartz,

[ Y I . For much more d e t a i l s on L 2 (M,p)-spaces [DS] V o l .

I. The measure spaces t h a t w e c a l l

Federer measure spaces have t h e d i f f e r e n t i a t i o n p r o p e r t y s t u d i e d by Federer i n Section 2 . 9 of h i s monograph [Fe]. The ad hoc proof f o r lRn presented i n Section 1 1 . 2 i s an a d a p t a t i o n of Theorem 10.49 i n Wheeden-Zygmund's [WZ] General c o n s i d e r a t i o n s on Sobolev t r i p l e s can be found i n Gel'fand-Shilov

[ G S Z I , Lions-Magenes [LMl and P a l a i s [ P a l ] . A thorough c l a s s i c a l introduct i o n t o t h e c l a s s i c a l Sobolev spaces can be found i n Smirnov, [ S m ] V o l . A more r e c e n t r e f e r e n c e f o r Sobolev spaces

5.

is a book by Kufner e t a l . [KJF].

For a simple statement and proof of t h e Sobolev lemma s e e Yosida [ Y ] .

For

i t s measure t h e o r e t i c a l g e n e r a l i z a t i o n s e e [EG7]. We f i n d i t remarkable t h a t i n t h e c l a s s i c a l Sobolev lemma t h e " open set i n mn ,, can be replaced by "Federer measure space" and t h e o p e r a t o r

51

Schmidt i n v e r s e o r a n o p e r a t o r A with Carleman i n v e r s e .

This Page Intentionally Left Blank

53

CHAPTER I l l

D l R A C BASES

Only few mathematicians a r e n o t acquainted with t h e concept of orthonormal b a s i s , i t s elegance and p r a c t i c a l use. I n a c e r t a i n s e n s e , t h e concept of Dirac b a s i s i s a s powerful a t o o l . Any orthonormal b a s i s i s a Dirac b a s i s . Let

x

denote an orthonormal be a separable H i l b e r t space and l e t ( v n nEIN We d e f i n e t h e l i n e a r o p e r a t o r 4 from onto 2 by

basis i n

x.

e

x

By t h e theorem of Riesz-Fischer t h e o p e r a t o r 4 i s u n i t a r y . However, s t a r t i n g with a u n i t a r y o p e r a t o r Y from u

x

onto

e2

= Y* en, i s an orthonormal b a s i s i n

x,

t h e s e t (unInCIN given by where e

E

l 2 i s given by

(e ) , = 6 n j , j c IN. Thus t h e choice o f an orthonormal b a s i s i n n i onto L, and v i c e versa. u n i t a r y o p e r a t o r from

x

fixes a

x

From a measure t h e o r e t i c a l p o i n t of view t h e H i l b e r t space

l 2 c o n s i s t s of

square i n t e g r a b l e functions from IN i n t o 6 where t h e Borel a-algebra f o r IN i s brought about by t h e d i s c r e t e topology and where t h e a - f i n i t e measure i s

t h e counting measure denoted by T . I t implies t h a t any s u b s e t Borel s e t and

r(U)

=

#u.

An orthonormal b a s i s ( n

H

u

Of IN i s a

x

such t h a t

So @ i s t h e only n u l l s e t f o r 'r.

v

~ i n)

x

i~s a ~f u n~c t i o n from IN i n t o

(w,v ) i s a Borel function f o r each w

E

x

and

Conceptually seen, Dirac bases come up i f t h e measure space ( N , T ) is replaced by a general o - f i n i t e measure space (M,u). H e u r i s t i c a l l y speaking, t h e index s e t IN i s replaced by a more g e n e r a l index s e t M. Thus t h e con-

DIRAC BASES

54

c e p t of Dirac b a s i s i s a continuum s u b s t i t u t e of t h e d i s c r e t e concept of orthonormal b a s i s . L e t (M,p)

denote a a - f i n i t e measure space which i s n o t p u r e l y atomic. For a

X onto L 2 (M,p) t h e r e d o n o t e x i s t u E X , x E M, such t h a t x ++ ( f , u x ) X , x E M, i s a r e p r e s e n t a t i v e of 8 f . This would mean t h a t t h e r e e x i s t s a continuous e v a l u a t i o n f u n c t i o n a l on L (M,p) a t any p o i n t 2

u n i t a r y mapping 8 from

x

E

M, which is impossible because

(M,p)

i s not p u r e l y atomic. However,

posed i n t h e s e t t i n g of a Sobolev t r i p l e

it i s an a p p r o p r i a t e q u e s t i o n whether t h e r e e x i s t elements G

x

E

M, such t h a t f o r each w

i s a r e p r e s e n t a t i v e of Ow

E

in

(x) ,

R ( X ) t h e function

L

( M , p ) . In our terminology t h e family ( G ) 2 x XEM i s a r e p r e s e n t a t i v e of a Dirac b a s i s . E

W e conclude this i n t r o d u c t o r y sketch with an elementary example. Consider t h e Lebesgue measure on IR and denote t h e corresponding H i l b e r t space by L2(IR).

On

L 2 (IRR) w e d e f i n e t h e bounded m u l t i p l i c a t i o n o p e r a t o r Ma, a >

+,

by

MaCqI = Cy The o p e r a t o r M

M

(L2(IR))

*

(1+y2)””

q(y)l

.

i s p o s i t i v e and has an unbounded i n v e r s e . The subspace

c o n s i s t s of a l l

[$I

E

L2(M,p)

f o r which

The space Ma1 (L, (IR)1 i s represented by equivalence c l a s s e s of Bore1 functions

e

on IR with the property

Let IF denote t h e Fourier transformation on L 2 ( I R ) ;

it i s a u n i t a r y opera-

55

THE CONCEPT OF DIRAC BASIS

tor. For each x

m whence

[ex]

j

E

IR t h e f u n c t i o n

: y H

2 ( l + y 2 r a lCX(y)I dy <

M a 1 (L, (IR)1.

E

e

-.

1 iyx e

Jzn

satisfies

I t follows t h a t t h e function

is a r e p r e s e n t a t i v e of t h e c l a s s FCqI f o r a l l [ q l hard t o check t h a t t h e f u n c t i o n x

t+

<[pl,[C

E

M

(L2(IR)).

I t i s not

I> i s continuous. The family

s e l e c t s t h e canonical r e p r e s e n t a t i v e o u t o f t h e c l a s s IFCpI, [ql

E

Ma(L2(IR)). In t h i s c h a p t e r w e give meaning t o strong/weak expansion

results like

me

s e t ([Cx])xEM

i s an example of a so-called canonicai! Dirac b a s i s .

111.1. The concept of Dirac b a s i s Let

X.

R

denote a p o s i t i v e and bounded o p e r a t o r on t h e s e p a r a b l e H i l b e r t space -1 Let R ( X ) 5 5 R be t h e Sobolev t r i p l e a s introduced i n Section

x

(x)

1 1 . 2 and l e t (M,p) i n d i c a t e a o - f i n i t e measure space. A function

8

:

M + R-l(X) i s c a l l e d a (weak) Borel f u n c t i o n i f f o r each w E R ( X ) t h e H < w , ~ ( x ) > ,x E M, i s a Borel f u n c t i o n from M i n t o C . In t h e

function x

l i n e a r space of Borel f u n c t i o n s from equivalence r e l a t i o n

el

-

-

€I2 :CI

M t o R-l(X)

B1(x) = 0 2 ( x )

we introduce the n a t u r a l

p-almost everywhere

.

By [€I] we denote t h e equivalence c l a s s of 8. The Borel f u n c t i o n 8 is s a i d

t o be weakly p-integrable i f f o r each w p-integrable.

E

R ( X ) the function x

H

<w,O(x)> i s

The Borel f u n c t i o n 8 i s s a i d t o be s t r o n g l y p-integrable

t h e function x

H

IIB(x) 11

-1

i s p-integrable.

a Borel function, s i n c e R - l ( X )

(We note t h a t x

H

i s a separable H i l b e r t space.)

118 ( x ) 11

-1

if is

56

DIFlAC BASES

Proposition.

111.1.1. Let

8

M

:

R-l(X)

+

u-integrable.

d e n o t e a Bore1 f u n c t i o n and l e t 8 be s t r o n g l y

Then 8 i s weakly u - i n t e g r a b l e and f o r a l l w E

R(X)

the

following i n e q u a l i t y is v a l i d

So t h e l i n e a r f u n c t i o n a l

i s continuous.

Proof.

Let w

R(X).

Then S c h w a r t z ' i n e q u a l i t y y i e l d s

Thus we g e t

III. I . 2. Let

8

:

Definition.

M

-t

R-'(X)

be s t r o n g l y u - i n t e g r a b l e .

denotes t h e element I

M,P

(8)

E

I

< w , ~ M , u ( e )=>

R-l(X)

Then t h e i n t e g r a l e x p r e s s i o n

d e f i n e d by

<w,e(x)>d p ( x ) ,

w

E

R(X)

.

M The e x i s t e n c e of I

(8)

Mr

t h e d u a l i t y between

u R(X)

E

R-l(X) f o l l o w s from t h e p r e v i o u s R-'(X). Cf. Theorem 1 1 . 2 . 2 . W e

and

p r o p o s i t i o n and observe t h a t

57

THE CONCEPT OF DIRAC B A S I S

In t h e sequel we w r i t e B(x) d u ( x ) i n s t e a d of I (€4) and we say t h a t t h e MI M, lJ l a t t e r i n t e g r a l converges i n s t r o n g R - l ( X ) sense. So f a r t h e needed i n t e -1 g r a t i o n theory f o r R (X)-valued Bore1 f u n c t i o n s on M. In o r d e r t o a r r i v e

a t an e l e g a n t d e f i n i t i o n of t h e notion of Dirac b a s i s t h e following a u x i l i a r y r e s u l t i s needed.

*.

111.1.3.

There e x i s t s an orthonormal b a s i s (

~ i n)

u

x

which ~ ~ i s~ e n t i r e l y contained

i n R ( X ) and which i s moreover a Schauder b a s i s i n all w

R(X) the series of R ( X ) . 1

E

nEl

R(X).

I t means t h a t f o r

(w,un)u a r e convergent with r e s p e c t t o t h e

norm l I * I I

Proof.

The proof c o n s i s t s of two s t e p s .

S t e p 1 : Let u s assume t h a t R has pure p o i n t spectrum. Then t h e r e e x i s t s an orthonormal b a s i s (

u

~ and ) p o~ s i t i~v e ~numbers p

such t h a t f o r a l l f

E

x

m

The orthonormal b a s i s ( each w

E

R ( X ) and a l l M >

i

II

u

~ i s ) a Schauder ~ ~ ~basis i n

N,

M,N

( w , u ~ ) ~ R - 'U n l 1 2

E

R ( X ) , because f o r

IN

Ip,

2

n=N+ 1

1

(w,un)x12

=

n=N+1

1 I ( R - ' W , U ~ ) ~ 2I . m

=

n=N+1 So i f

R has pure p o i n t spectrum, then each orthonormal b a s i s of eigenvectors

of R i s a Schauder b a s i s i n R ( X ) .

S t e p 2: L e t R > 0 be a bounded o p e r a t o r on 0 < R < 7. Let

x.

We may a s w e l l assume t h a t

denote t h e s p e c t r a l r e s o l u t i o n of t h e i d e n t i t y of R.

W e d e f i n e t h e mutually orthogonal p r o j e c t i o n s

Pk ,

k

E

IN,

by

58

DIRAC BASES

N

and the o p e r a t o r

R by

m

l P 1k + l k

N

R =

k=l

'

s

It is clear that

Moreover, both

IIR-l?II

R i s a p o s i t i v e bounded o p e r a t o r with pure p o i n t spectrum. 1 and R-'? a r e bounded o p e r a t o r s with 1 I R R II 5 2 and

?-' R

N-

5 1. N

Now l e t ( u n ) n E m denote an orthonormal b a s i s of e i g e n v e c t o r s o f

un

=

1

R(R-'%?un and so un E R ( X ) , n - 1 R ( R R)R-lw E and

z(X)

N

have w =

f o r all M > N, M,N

IN. Also, f o r each w

E

R(X) we

IN.

E

I t follows t h a t (

c

R. Then

u

~ i s) a Schauder ~ ~ ~basis i n R ( X ) .

0

Remark. I f R i s a p o s i t i v e bounded o p e r a t o r which i s continuously i n v e r t i b l e then t h e spaces

R ( X ) and

x

are equal and t h e norms II II

e q u i v a l e n t . In t h i s case each orthonormal b a s i s i n in

x

and I1

Itx

are

i s a Schauder b a s i s

R ( X ) ( i n f a c t a so-called Riesz-Schauder b a s i s ) .

111.1.4. Let (M,p)

Definition. denote a o - f i n i t e measure space and R a p o s i t i v e bounded o p e r a t o r

i n the s e p a r a b l e H i l b e r t space Bore1 f u n c t i o n s G from

x.

Let [ G I denote an equivalence c l a s s of

M t o R-'(X).

Then ( C G I , M , p , R , X )

b a s i s i f t h e r e e x i s t s an orthonormal b a s i s (u ) k Schauder b a s i s i n

R ( X ) such t h a t f o r ^G

E

E

i~n

is called a D i r a c

x,

which is a

[GI t h e following r e l a t i o n s a r e

valid.

( I m p l i c i t l y we assume t h a t t h e f u n c t i o n s x

H

, x

integrable. 111.1.6.

Corollary.

Each orthonormal b a s i s ( v k I k c N i n

x

i s a Dirac b a s i s .

E

M, a r e square

THE CONCEPT OF DIRAC BASIS

-.

For

M

w e take t h e space IN with i t s d i s c r e t e topology and c o r r e -

sponding Bore1 o-algebra. measure T , f o r

k E IN.

59

For t h e a - f i n i t e measure we t a k e t h e counting

R t h e i d e n t i t y o p e r a t o r and f o r

e2

Observe t h a t L2(IN,r) =

G the function k

1

(

~ an) orthonormal ~ ~ ~ basis i n

u

v

k'

n

[ G I and R ( X ) , such

denote a Dirac b a s i s , G a r e p r e s e n t a t i v e of

Let (CGI,M,p,R,X)

H

c o n s i s t s of genuine functions.

x,

which i s a Schauder b a s i s i n E

IN w e l e t qk denote belongs t o L 2 ( M f p )

f o r each k E IN.

. Then Cq,l k' I t follows from 111.1.5 t h a t (Cqkl,Cq

So t h e s e t {Cq,l

Ik

t h a t t h e R e l a t i o n s 111.1.5 a r e s a t i s f i e d . For each k t h e square i n t e g r a b l e f u n c t i o n x

E

H

IN) i s orthonormal i n L 2 ( M , p ) .

Next, we d e f i n e t h e o p e r a t o r V from

x

into

L 2 (M,v)

1) L2

= 6ke, k,t

E

IN.

by

m

v

It is clear that m

series

E

k= 1

i s an isometry from

x

i n t o L2(M,p).

( w , ~ ' )3 ~~a r e convergent i n R ( X ) . So f o r each x

m

So t h e f u n c t i o n x

~w

=

Let w

kil( w , \ ) x

E

M

we o b t a i n

<w,G(x)>, x

C ~ , Ic L , ( M , u ) .

E

M,

i s a r e p r e s e n t a t i v e of m

The s e r i e s

kgl

(w,uk)x qk converges point-

wise.

111.1.7.

Theorem.

L e t (CGI,M,p,R,X)

be a Dirac b a s i s . 1

a. For each r e p r e s e n t a t i v e G Plancherel formula

The

m

.

I+

R(X).

E

E

[GI and a l l w1,w2

E

R(X)

we have t h e

DIRAC BASES

60

properties. N

b. For each r e p r e s e n t a t i v e G

[ G I and each w

E

E

R ( X ) t h e function

N

x

I-+

(The d e f i n i t i o n of vw does not

<w,G(x)> i s a r e p r e s e n t a t i v e o f Vw.

depend on t h e choice of

^G

[GI.)

E

c. The bounded o p e r a t o r VR i s a Carleman o p e r a t o r .

Proof.

.

a. Fix G

[ G I and c o n s t r u c t t h e isometry

E

theorem. Then t h e f u n c t i o n

vw f o r each w

R(X).

E

(w1'w2)x

=

X H

<w,;(x)>,

E

x

I-+

(VW1,VW

1

=

)

[ G I . Then f o r each w <w,;(x)>

i n t h e way p r e s e n t e d above t h i s x

E

M , i s a r e p r e s e n t a t i v e of

I t follows t h a t

2L2 b. A l s o f i x

v

and

x

I+

E

< w l , t ( x ) > <w2,G(x)> d u ( x )

.

R ( X ) the functions

<w,E(x)>

a r e e q u i v a l e n t . I t follows from a . t h a t

Hence

the

operator

i s an isometry on

so v

-v

defined by

R(X) 5

x

N

and does n o t depend on t h e choice o f G

E

N

=

v.

c. Take any

^G

E

[ G I . Define t h e f u n c t i o n k

Then f o r a l l f

Hence x

I+

E

:

M

+

x

by

x,

( f , k ( x ) ) i s a Bore1 f u n c t i o n which i s a r e p r e s e n t a t i v e of

[GI.

61

THE CONCEPT O F DIRAC BASIS

VRf

0

L2 ( M , p ) . By D e f i n i t i o n 1.2.1, VR i s a Carleman o p e r a t o r .

E

There e x i s t s a one-to-one and i s o m e t r i e s

V

:

x

-C

correspondence between Dirac bases ( [ G ] , M , u , R , X )

L 2( M ,uI

with t h e p r o p e r t y t h a t VR i s a Carleman

operator.

111.1.8.

Let

V

Theorem.

denote an isometry from

X into

L2(M,p) f o r which t h e o p e r a t o r

VR i s a

( C G I , M , u , R , X ) such t h a t R ( X ) t h e Borel f u n c t i o n x H <w,G(x)> i s a

Carleman o p e r a t o r . Then t h e r e e x i s t s a D i r a c b a s i s I

f o r each G

A

E

[ G I and f o r a l l w

E

r e p r e s e n t a t i v e of Vw.

Proof.

Let

k

:

M

+

X denote an inducing f u n c t i o n

( u n ) n c m denote an orthonormal b a s i s i n

We d e f i n e t h e Borel f u n c t i o n G

:

M

+.

x

of t h e o p e r a t o r

UR. Let R(X).

which i s a Schauder b a s i s i n

R-'(X)

by

m

where t h e s e r i e s converges i n t h e norm 11. I1 Let w E

-1

of R-l

(x) .

R ( X ) . Then w e have

Hence x n <w,G(x)> i s a r e p r e s e n t a t i v e of Vw.

So f o r a l l w1,w2

E

R(X)

M' I f [Gldenotes t h e equivalence class of G, then ( C G I , M , p , R , X )

i s t h e wanted

0

Dirac b a s i s .

Corollary 111.1.6 and t h e next r e s u l t c h a r a c t e r i z e t h e Dirac b a s e s which a r e orthogonal bases i n t h e usual sense.

111.1.9.

Proposition.

L e t t h e measure p be p u r e l y atomic. Then each Dirac b a s i s ( [ G I , M , v , R , X ) an orthogonal b a s i s i n

x.

is

62

DIRAC BASES

-.

M

u-atoms, N

Let G

E

Mj, j

E

can be w r i t t e n a s t h e d i s j o i n t union of a countable number of a

.u M . where p ( M . ) < a and M is a p-atom. 3=1 3 3 1 [ G I be t h e canonical r e p r e s e n t a t i v e which i s c o n s t a n t on each atom i.e.

M

=

IN. We p u t e .

where w e t a k e an a r b i t r a r y x . 3

= p(Mj)-’Z(x.) 3 3 I t follows t h a t f o r a l l w E R ( X )

E

M.. 3

m N

N

<w,G(x)> <w,G(x)> d p ( x ) =

(w,w)x =

M.

]=I

3

a

=

1

j=1

The l i n e a r o p e r a t o r U :

Uw : j

H

’

<w,e.> <w,e.>

R(X)

+.

1

e2

defined by

<w,e > j

extends t o an isometry on t h e whole of

x

f H ( u f ) ( j ) on (fj)jEmi n

x

Hence t h e l i n e a r f u n c t i o n a l

i s continuous. So there e x i s t s an orthonormal b a s i s

(w,f.)

3

Let

x.

such t h a t

In p a r t i c u l a r f o r a l l j

Hence e . = f . 3 3

.

E

([Gl,M,u,R,X)

x

E

IN and a l l w

= c w , f . > = <w,e.>

3

3

for all j

E

E

R(X)

we have

. n

lN.

denote a Dirac b a s i s and l e t G

E

[GI. From t h e Plancherel-

type r e s u l t s t a t e d i n Theorem 111.1.8 we o b t a i n t h e expansion

with convergence i n t h e weak sense, i . e .

for a l l v E

R(X)

THE CONCEPT OF DIRAC BASIS

A

sharper result is valid if

R

63

is a Hilbert-Schmidt operator. In this case R-1 (X)-sense. Hence it is fully

the expansion converges in the strong

justified to call (CGI,M,p,R,X) a basis. 111.1.10. Theorem. Let

R

be a positive Hilbert-Schmidt operator and let (CGI,M,p,R,X)be a L

Dirac basis. Let G

E

[GI. Then for all w

E

R(X)

the function

is p-integrable. So we get the strong expansion

it means that the convergence is in strong

Proof.

Let

denote an orthonormal basis of eigenvectors of

corresponding eigenvalues pk, k

R(X).

R-1 (X)-sense.

We have for all x

E

E

R

with

lN. Then ( u k ) k c m is a Schauder basis in

M

Hence

So we obtain

n In Chapter 1 we have characterized the operators on

x which are of Carleman

type. With the aid of this characterization the problem of existence of Dirac bases in any Sobolev triple

R(X) c X

5 R-'(X)

is completely solved.

64

DIRAC BASES

Theorem. R denote a p o s i t i v e and bounded o p e r a t o r i n

111.1.11. Let

x,

l e t (M,p) be a a - f i n i t e

measure space. a . Suppose R - l b. Suppose

R

-1

i s unbounded. Then t h e r e e x i s t s a Dirac b a s i s ( [ G I , M , p , R , X ) .

(CGI,M,u,R,X)

i s bounded. Then t h e r e e x i s t s a Dirac b a s i s

iff

the measure space (M,p) i s purely atomic. c . L e t t h e measure space (M,p) k e n o t p u r e l y atomic. Then any isometry from

X into

L 2 ( M , p ) g i v e s r i s e t o a Dirac b a s i s ( C G I , M , p , R , X )

v

i n the

described way, i f f R i s a Hilbert-Schmidt o p e r a t o r .

Proof. i s unbounded, then 0 belongs t o t h e e s s e n t i a l spectrum of R. So

a. I f R - l

following Theorem 1.3.6 t h e r e e x i s t s a u n i t a r y mapping L 2 ( M , p ) such t h a t

u

from

x

onto

UR i s a Carleman o p e r a t o r . Then t h e r e s u l t follows from

Theorem 111.1.8. b. Observe t h a t

V

:

X

+

R(X)

=

x

=

R- 1

(x).

By Theorem 1.3.8 an isometry

(M,u)

L2(M,p) i s a Carleman o p e r a t o r i f f

i s a p u r e l y atomic

measure space. Cf. P r o p o s i t i o n 111.1.9. From Theorem 1 1 1 . 4 . 2

C.

t h e o p e r a t o r VR : operators V :

X

X

it follows t h a t -+

R i s a Hilbert-Schmidt o p e r a t o r i f f

L 2 ( M , p ) i s a Carleman o p e r a t o r f o r a l l u n i t a r y

+. L 2 ( M , p ) .

I f we p u t geometrical r e s t r i c t i o n s on t h e measure space

(CGI,M,u,R,X)

Dirac b a s i s class

[GI.

(M,u),

then f o r each

u

a canonical choice G can be made i n t h e equivalence

It l e a d s to t h e notion of canonical Dirac b a s i s .

1 1 1 . 2 . Canonical Dirac bases A

Dirac b a s i s (CGl,M,p,R,X)

M t o R-1 ( X I .

i s an equivalence class of Bore1 f u n c t i o n s from

N o r e s t r i c t i o n s a r e imposed on t h e o - f i n i t e measure space

(M,u).

However, if we r e s t r i c t t o Federer measure spaces introduced i n D e f i n i t i o n N

11.3.3, then f o r c e r t a i n R > 0 we can make a canonical choice G i n [ G I .

Such

a choice i s c a l l e d a c a n o n i c a l Dirac b a s i s .

111.2.

Let

1.

Definition.

(M,u) be

a Federer measure space and l e t ( [ G I , M , p , R , X )

be a Dirac b a s i s .

65

CANONICAL DIRAC BASES

N

representative G

A

a null set

isuch

E

[ G I i s c a l l e d a canonical Dirac b a s i s i f t h e r e e x i s t s

that for a l l w

R ( X ) and

E

all x

M\K

E

N

For a canonical Dirac b a s i s we use t h e n o t a t i o n ( G ) x xcM' So throughout t h i s s e c t i o n w e assume t h a t (M,p) i s a Federer measure space.

Then our measure t h e o r e t i c a l Sobolev lemma i s a p p l i c a b l e , c f . Theorem 11.4.5.

111.2.2.

e.

V be an isometry from X i n t o L 2 ( M , p ) with t h e p r o p e r t y t h a t VR is a

Let

k denote an inducing f u n c t i o n of

Carleman o p e r a t o r f o r c e r t a i n R > 0 . L e t

VR and suppose k s a t i s f i e s Condition I I . 4 . l . b i . e . t h e f u n c t i o n

XI+

2 IIk(x)lI

i s i n t e g r a b l e on bounded Borel sets. Then t h e Dirac b a s i s ( C G l , M , p , R , X ) N

a s s o c i a t e d t o V admits a canonical r e p r e s e n t a t i v e (Gx)xcM.

Proof. N

gX

E

Applying Lemma 11.4.4

R(X), x

M,

E

D

=

V there are

that for all w

E

R(X)

and Theorem 11.4.5 with

and a n u l l s e t

a such

and

Since R-'(X)

N

and R ( X ) a r e i n d u a l i t y w e can d e f i n e Gx N

N

(w,gX), = <w,G >

,

w

R(X)

E

E

R-l(X)

by

.

N

I t follows t h a t t h e f u n c t i o n x t + G

i s a r e p r e s e n t a t i v e of [ G I and hence

N

(Gx)xcM i s

0

a canonical Dirac b a s i s .

111.2.3. Theorem. L e t (CGI,M,p,R,X)

x

kt

IIR(c(x))I1

be a Dirac b a s i s . Let G

2 .

E

[ G I . I f t h e Borel f u n c t i o n

is i n t e g r a b l e on bounded Borel sets, then ( C G l , M , p , R , X ) N

c o n t a i n s a canonical Dirac b a s i s (G )

x xcM'

66

DIRAC BASES

Proof.

By Theorem 111.1.8 t h e r e e x i s t s an isometry V

H

< R f , E ( x ) > l = VRf

-+

L 2 ( M , p ) such t h a t

x

we have

: x

I+

R ( G ( x ) ) induces VR. By assump-

.

I t i s c l e a r t h a t t h e Borel f u n c t i o n

tion,

x

E

VR i s a Carleman o p e r a t o r and f o r a l l f Cx

:

k s a t i s f i e s Condition II.4.1.b.

k

1

From Lemma 1 1 1 . 2 . 2

the assertion

0

follows.

Remark.

111.2.4.

W e assume t h e c o n d i t i o n s and n o t a t i o n s of t h e previous theorem. Let (

x

denote an orthonormal b a s i s i n each x

E

M

u

which i s a Schauder b a s i s i n R ( X ) . For

and each r > 0 w e p u t

where V denotes t h e isometry a s s o c i a t e d t o ( C G l , M , p , R , X ) .

i t follows that G = V* x,r such t h a t f o r a l l x set

xB ( x E

r ) . Now Lemma 11.4.4 with

Since

D

=

V yields a null

M\h

The notion of canonical Dirac b a s i s i s a s s o c i a t e d t o t h e g e n e r a l i z e d eigenvalue problem. We d e s c r i b e t h i s connection here. L e t cp be a complex valued Borel f u n c t i o n on

M

which i s bounded on bounded

Borel s e t s . I n L2(M,p) we d e f i n e t h e m u l t i p l i c a t i o n o p e r a t o r M

cp

and

~

by

)

~

CANONICAL DIRAC BASES

x

B (x,r )

f o r each x

k! and each r > 0 t h e a r e i n D ( M 1 . Observe t h a t M i s a normal

Because of t h e c o n d i t i o n s imposed on cp, c h a r a c t e r i s t i c functions

67

E

cp

cp

operator.

Theorem. (IGI,M,p,R,X) denote a Dirac b a s i s .

III. 2.5. Let

1

tion x

I I R ( G ( x ) ) 11

H

2

L

Let G

E

[ G I and l e t t h e Borel func-

be i n t e g r a b l e on bounded Borel s e t s . Let V

(CGI,M,u,R,X).

denote t h e isometry a s s o c i a t e d t o

:

x

+ L2(M,p)

Then t h e r e e x i s t s a canoni-

N

c a l Dirac b a s i s (Gx)xeM null set

L e t cp :

isuch

M

-t

i n [GI with t h e following p r o p e r t i e s . There e x i s t s a

that for all x

E

M\R

C be a measurable f u n c t i o n which is bounded on bounded Borel

s e t s . Then t h e r e e x i s t s a n u l l s e t

Proof.

N

cp

such t h a t f o r a l l x

E

M\N

cp

The f i r s t statement follows immediately from Remark 1 1 1 . 2 . 4

and

Theorem 111.2.3. The second a s s e r t i o n i s v e r i f i e d i f we can prove t h a t t h e r e e x i s t s a n u l l set

N,

such t h a t f o r a l l x

R(X).

M\N2

denote an orthonormal b a s i s i n

Therefore, l e t basis i n

E

Let x E

M and consider t h e e s t i m a t i o n

x

which i s a Schauder

68

DIRAC BASES

Since cp i s bounded on bounded Borel s e t s both

IcpI

and

lcpI2

N,,

such t h a t f o r a l l

on bounded Borel sets. So t h e r e e x i s t s a n u l l s e t x

E

M\N,, l i m v(B(x,r))-'( rCO

Furthermore, f o r a l l y E

I

N,

=

N,,

u N,,

I

Icp(x) -cp(y) 2 d v ( y ) ) = 0

.

B(x,r)

M

So by assumption t h e r e e x i s t s a n u l l s e t

Hence f o r

a r e p-integrable

N,,

such t h a t f o r a l l x

E

M\N,,

t h e wanted r e l a t i o n t follows from

111.2.6. Corollary. We assume t h e n o t a t i o n s and c o n d i t i o n s of t h e previous theorem. Also, we assume, t h a t V i s a u n i t a r y o p e r a t o r and, t h a t V * M V i s c l o s a b l e i n t h e -1 cp H i l b e r t space R i . e . R V * M VR-' i s c l o s a b l e i n X. L e t V* M V denote cp cp t h e c l o s u r e of V* M V w i t h r e s p e c t t o t h e H i l b e r t space R-'(X). Then f o r a l l cp x E M\N w e have

(x),

cp

Proof.

From Theorem 1 1 1 . 2 . 5 we o b t a i n a n u l l set

N

cp

such t h a t f o r a l l

69

DIRAC-RIESZ BASES

x

E

M\Nq

and

Since V * M

rp

R-'(X)

V i s a closed o p e r a t o r i n

we g e t f o r a l l x

E

M\N

The main a p p l i c a t i o n of t h e previous r e s u l t w i l l be given i n t h e next chapter. There w e d i s c u s s t h e generalized eigenvalue problem f o r a f i n i t e number of commuting s e l f - a d j o i n t o p e r a t o r s .

111.3. Dirac-Riesz bases The notion of Dirac b a s i s i s a g e n e r a l i z a t i o n of t h e concept of orthonormal b a s i s f o r s e p a r a b l e H i l b e r t spaces. Besides orthonormal bases t h e r e appear

also Riesz bases i n H i l b e r t space theory. A countable set (gn)nEINi n a

x

s e p a r a b l e H i l b e r t space

e s t a b l i s h e s a Riesz b a s i s , i f t h e r e e x i s t s an

orthonormal b a s i s ( v n ) n E m i n gn =

s

s

vn f o r a l l n

E

x

and a continuous b i j e c t i o n s on

So any Riesz b a s i s i s of t h e form

IN.

x

such t h a t

(s v

~ with )

a continuous b i j e c t i o n . Riesz bases appear a s complete s e t s of eigen-

v e c t o r s of l i n e a r o p e r a t o r s M, which a r e s i m i l a r t o s e l f - a d j o i n t o p e r a t o r s

D

Sps-'.

with pure p o i n t spectrum, i . e . M =

n a t u r a l numbers. Replacing ( I N , T )

A Riesz b a s i s i s l a b e l e d by t h e

by a g e n e r a l measure space (M,u) t h e notion

of Dirac-Riesz b a s i s comes up. This type of continuum b a s i s i s u s e f u l i n t h e treatment of t h e generalized eigenvalue problem o f o p e r a t o r s M which are similar t o (general) s e l f - a d j o i n t operators. As we have seen a Dirac b a s i s f o r a Sobolev t r i p l e

f i x e d by an isometry V from

X

R(X)

5

x

_c

R-'(X)

is

i n t o L (M,p) such t h a t VR i s a Carleman 2

o p e r a t o r . I t t u r n s o u t t h a t a Dirac-Riesz b a s i s i s f i x e d by two i n j e c t i v e o p e r a t o r s J, and J2 from assumptions. Naturally,

X

i n t o L2(M,u) which s a t i s f y some a d d i t i o n a l

t h e d e f i n i t i o n of Dirac-Riesz b a s i s i s more involved

than t h e d e f i n i t i o n of Dirac b a s i s .

~

DIRAC BASES

70

111.3.1. D e f i n i t i o n .

R be a positive operator on x, let (M,p) be a a-finite measure space and let J and J be linear operators from X into L2(M,p) such that R(X) is con1 2 tained in D(Jl) n D(J 1. We call (J, ,J2,R) a Carleman triple if 2

Let

- JIR -

and

J2 R

are Carleman operators ;

for all w,v

(Observe that

E

R(X), (Jlw,J v)L2 - (w,v)x

JIR

and

J2R

*

are bounded on X ) .

111.3.2. Proposition. Let (M,p) be a a-finite measure space and let

J

:

x + L,(M,p)

be a conti-

nuous linear bijection. Then for any positive Hilbert-Schmidt operator

x

R on

the triple (J,(J-l)*,R) establishes a Carleman triple.

Let (Jl,J2,R)be a Carleman triple with respect to the measure space (M,p) and the separable Hilbert space

X.

Then there exist two X-valued functions

p, and p2 which induce J 1R and J2R respectively. It means that for each f €

x

J1

f =

Cx

J2

f =

C x * (f,p2(x))Xl

I+

(f,pl(X))~I

and

Let w,v

E

.

R(X). Then we have Jlw = J,R(R-'w)

and

J,W

=

J2R(R -1 w ) .

Thus we

obtain the Parseval-like equality

(x)1 (R-'v,p, (x)) dp (x)

Since

R(X)

p1,p2

:

and

R-l(X)

.

are in duality, there exist Bore1 functions

M + R-l(X) such that for all x

E

M

and all w

E

R(X)

Let [P 1 and [P 1 denote the equivalence classes corresponding to P1 and P 1 2 2' In our terminology, ([P1l,CP21,M,p,R,X) is a Dirac-Riesz basis.

DIRAC-RIESZ BASES

Definition.

111. 3 . 3 .

L e t R denote a p o s i t i v e bounded o p e r a t o r on

into R-l(X).

x,

(M,IJ) denote a o - f i n i t e

let

t w o equivalence c l a s s e s of Borel f u n c t i o n s from

measure space and [ G l , [ H ]

M

71

The s e x t u p l e ( [ G I , [ H I , M , I J , R , X )

i s c a l l e d a Dirac-Riesz b a s i s

x

i f f o r a c e r t a i n orthonormal b a s i s (u ) in which i s a Schauder b a s i s k kEIN i n R ( X ) t h e following c o n d i t i o n s a r e f u l f i l l e d : Let (pk denote t h e Borel function x

I+

<\,G(x)>

a. For a l l w

E

R(X)

and $k t h e Borel f u n c t i o n x

I+

. W e demand

both s e r i e s

m

m

a r e convergent i n L

2

(M,IJ).

b. For a l l k , e E IN

We mention t h e following consequences of t h i s d e f i n i t i o n .

Corollary.

111.3.4.

Every Dirac b a s i s is a Dirac-Riesz b a s i s .

Proof.

Let

111.3.5.

([GI,M,u,R,X)

-.

Every Riesz b a s i s i n

Proof.

be a Dirac b a s i s . Then we have

x

i s a Dirac-Riesz b a s i s .

Let (gn)nEINbe a Riesz b a s i s i n

I , f o r M t h e s e t of n a t u r a l numbers

IN,

x.

Then f o r R w e t a k e t h e i d e n t i t y

and f o r IJ t h e counting measure T.

W e put g : n*gn,

ncIti.

By d e f i n i t i o n t h e r e e x i s t s a continuous b i j e c t i o n i s an orthonormal b a s i s i n

x.

Also

we put

s

such t h a t

(s-1 gn ) nelN

DIRAC BASES

12

Then the Riesz basis ( g ) is expressed by n nrIN ( 191,

[hl,DJt T I 1 , x)

in our terminology.

0

For a Dirac-Riesz basis ([G],[H],M,LI,R,X),

let (

u

~ denote ) ~ the ~ Schauder ~ I

basis in R(X) as indicated in Definition 111.3.3. We put A

and $, and

J,

:

x t+ <\,H(x)>.

Then on

R(X)

'pk

:

xt+

we introduce the linear operators

J1

by m

and m

Since the family (u ) is a Schauder basis in R(X) the following equalik kclN ties are valid for all w E R(X) and all x E M,

and m

It follows that the functions x tives of

I+

<w,G(x)> and x

t+

<w,H(x)> are representa-

J 1w and J2w, respectively. Straightforward calculation yields m

The operators

J 1 R and J2 R are everywhere defined and for all f

Thus we find that

J 1R

and

J2R

E

X

we have

are bounded Carleman operators. The triple

DIRAC-RIESZ BASES

73

(J J ,R) is a Carleman triple in the sense of Definition 111.3.1. 1’ 2 111.3.6. Theorem. Let

R

denote a positive bounded operator on

X,

let (M,u) denote a o-finite

measure space and [GI ,[HI equivalence classes of

R-l (X)-valued Borel func-

tions. Then (CGI,CHl,N,p,R,X) is a Dirac-Riesz basis iff there exists a Carleman triple (J,,J2,R) such that for each X-valued Borel function x x

.

* R(H(x))

Proof.

I+

R (c(x))

^G

[ GI

E

and each

^H

[ HI the

E

induces the Carleman operator J

induces the Carleman operator

J2R.

1

R

and

The proof is a compilation of results preceding this theorem.

Remark. If

J

is an injective bounded linear operator from

X

into L

2

0

(M,p)

such that j(x) is closed in L , ( M , p ) and if R is chosen such that J R and (J-1 ) * R are Carleman operators, then (J,(J-1 ) * ,R) is a Carleman triple. In particular we can take any Hilbert-Schmidt operator

R.

The concept of Dirac-Riesz basis gives rise to the following type of expansions. 111.3.7. Theorem. Let ([GI ,[HI ,M,U,R,X) denote a Dirac-Riesz basis with corresponding Carleman I

1

triple (J1,J2,R) and let G

6

[GI and let H

E

[HI. Then for all w

E

R(X) we

have the expansion

which is valid in the weak sense that for all v

Let both

JIR

and

J2R

E

R(X)

be Hilbert-Schmidt operators. Then the above expansion

is valid in the strong sense, i.e.

74

DIFAC BASES

Proof.

The first part of the theorem follows from the v-lidity of the

Parseval-like identity as proved after Definition 111.3.1. For the proof of the second part, we observe that x x

I+

*

,.

IIR(H(x)) I1

and

I<w,?i(x)>[ are square integrable. So we obtain the estimation

Remark. The roles of [GI and [HI can be interchanged in the previous theorem.

111.4. Canonical Dirac-Riesz bases Throughout this section we assume that (M,p) is a Federer measure space. Then under certain conditions the Carleman triple

(J, ,J2 ,R)

generates a

so-called canonical Dirac-Riesz basis. The notion of canonical Dirac-Riesz basis is the natural generalization of the concept of canonical Dirac basis. 111.4.1. Definition.

-

-

Let ([Gl,CHI,M,p,R,X) denote a Dirac-Riesz basis. Then the representatives G of [GI and H of [HI establish a

exists a null set

K

canonicaZ Dirac-Riesz basis if there such that for all x E M\K and all w E R ( X ) both

and

u

u

For the canonical Dirac-Riesz basis we use the notation {Gx,HxIxCM. Since we suppose that (M,p) is a Federer measure space, our measure theoretical Sobolev lemma is applicable, cf. Theorem 11.4.5. A s in L e m a 111.2.2 we prove

CANONICAL DIRAC-RIESZ BASES

e.

111.4.2. Let

15

(Jl,J2,R) be

a Carleman triple and let p1

the Carleman operators

J1R

:

M

-+

X

and p2

:

M

+

X

induce

J2 R,

respectively. Suppose p1 and p2 satisfy 2 Condition 11.4.l.b, i.e. both functions x I+ Ilp (x)I I x and x t+ Ilp2(x)II are 1 integrable on bounded Borel sets. Then the associated Dirac-Riesz basis ([GI,[HI,M,p,R,X)

-.

-p,,?,

and

w

u

contains a canonical representative {Gx,HxIxEM.

Following Lemma 11.4.4 and Theorem 11.4.5 there are functions :

M

+

R(X)

and there is a null set

such that for all w

R(X)

E

and

and for all x E

and

- -

Now we define G , H in x x

R-l(X)

by means of the relation

and

Thus we arrive at a sufficient condition such that a Dirac-Riesz basis contains a canonical representative. 111.4.3. Theorem. 1

A

Let (CGI,[HI,M,IJ,R,X)be a Dirac-Riesz basis. Let G both x

L

H

sets of

IIR(G(x)) 11

M,

Proof. The

2

and x

E

[GI and H

E

[HI. If

I1 R(H(x)) II 2 are integrable on bounded Borel 1

I+

then (CGI,[HI,M,p,R,X) contains a canonical Dirac-Riesz basis. assumptions imply that the associated Carleman triple satisfies

the assumptions of the previous lemma.

0

76

DIRAC BASES

J1 R-l(X)

Let ( J , J , R ) denote a Carleman t r i p l e . The l i n e a r o p e r a t o r s 1

2

R ( X ) continuously i n t o L 2 ( M , ~ ) . Let J; and J;

: L,(M,p)

t h e i r dual mappings with r e s p e c t t o t h e d u a l i t y R ( X ) x w

E

R ( X ) and Ccpl

+

R

-1

(x).

and J, map denote

So f o r a l l

L2(M,p) we have by d e f i n i t i o n

E

Then J i and J' a r e continuous l i n e a r mappings. 2

L e t x f M and

r > 0 . The c h a r a c t e r i s t i c f u n c t i o n

B(x,r) i s square i n t e g r a b l e , whence Ji[xB(x,r)

1

E

[x6 (x,r )1 E

and

R-'(X).

Let ( u k l k E m denote an orthonormal b a s i s i n

R(X)

x ~ ( ~ of, ~ t h e) c l o s e d b a l l

L,(M,u)

. Then f o r a l l x

E

M

x

which i s a Schauder b a s i s i n

and a l l r > 0

I f t h e Carleman o p e r a t o r J I R has an inducing f u n c t i o n , which i s square

-

i n t e g r a b l e on bounded Borel s e t s , then Condition I I . 4 . 1 . b Lemma 11.4.4 with x

E

D

=

J, y i e l d s a n u l l s e t N and elements

M, such t h a t f o r a l l x

Replacing

J, by J, we g e t

E

i s s a t i s f i e d . So

R-l

(x),

MI\

u

H

-

G x of

E

R-'(X),

x N

E

M,

with a similar property.

u

Because of t h e c i t e d lemma t h e s e t {G ,HxJxeM

i s a canonical Dirac-Riesz

basis. Like Dirac bases, also Dirac-Riesz bases a r e connected t o c e r t a i n g e n e r a l i z e d eigenvalue problems. Let cp denote a complex valued Borel function on

M, which

i s a bounded Borel

set. A s i n Section 1 1 1 . 2 we d e f i n e t h e normal m u l t i p l i c a t i o n o p e r a t o r M

cp

L,(M,I.I)

by

in

CANONICAL DIRAC-RIESZ BASES

W e observe t h a t

111.4.4.

cx

B(x,r)

1

D(M

E

cp

)

77

f o r all r > 0 and all x

Theorem.

(CGI,[HI,M,U,R,X)

denote a Dirac-Riesz b a s i s . Let 2 Suppose t h e f u n c t i o n s x I+ II R(G(x)) II and x II R(G(x) Let

M.

E

E

[GI and

E

[HI.

2

II are i n t e g r a b l e on (J ,J , R ) denote t h e corresponding Carleman t r i p l e . 1 2 Now t h e r e e x i s t G E R-' (x) and ';ix E R - ' ( X ) , x E M, which e s t a b l i s h a canonical Dirac-Riesz b a s i s . A l s o , t h e r e e x i s t s a n u l l s e t N such t h a t f o r

-

bounded Borel sets. Let

N

all x

E

M\N

Let cp denote a Borel f u n c t i o n which i s bounded on bounded Borel sets. Then there exists a null s e t

Proof. The -

N

cp

such t h a t f o r all x

E

M\N

cp

f i r s t statement follows from t h e remarks preceding t h i s theorem.

We have v e r i f i e d t h e second i f we can prove t h a t t h e r e e x i s t s a n u l l s e t such t h a t f o r a l l x

Let ( \ ) k E N in

R(X).

E

M\N

E

M.

2.1

2,j

denote an orthonormal b a s i s i n

Let x

N

Then w e have

x

which i s a l s o a Schauder b a s i s

DIRAC BASES

Now t h e p r o o f p r o c e e d s a l o n g t h e same l i n e s a s t h e p r o o f o f Theorem 111.2.5.

0

W e o m i t it here.

111.4.5.

CoroZ la*.

Assume t h e n o t a t i o n s a n d c o n d i t i o n s o f Theorem 1 1 1 . 4 . 4 .

J2

:

x

+

L2(M,u)

J, i s a c o n t i n u o u s l i n e a r b i j e c t i o n from X

Jil. Then o n t o L (M,p) w i t h JP Jil. 2

Next assume t h a t t h e unbounded l i n e a r operator l i n e a r operator f r o m Then f o r a l l x -1 J2

R-'(X)

into

R-' ( X )

M\N

E

Y

MVJ2

Y

G~ = v ( x ) G~

.

L e t J;l

=

J-' M J

cP2

2

M J

cp2

Jil.

From D e f i n i t i o n 111.3.1 i t f o l l o w s t h a t J* = 1 Because of t h e p r e c e d i n g theorem w e g e t t h e R - l ( X ) - l i m i t s

and

cp

. Since

and

t h e c l o s e d n e s s of

.

J*M J y i e l d s t h e d e s i r e d r e s u l t . l q 2

i s closable as a

denote i t s closure.

.

Proof.

for a l l x E M\N

A l s o , assume t h a t

is a continuous l i n e a r b i j e c t i o n with i n v e r s e

CANONICAL DIRAC-RIESZ BASES

79

Some b i b l i o g r a p h i c a l notes and comments The concept of Dirac b a s i s occurs f o r t h e f i r s t time i n t h e Ph.D. t h e s i s of t h e f i r s t author [ E 11.

However, it does n o t bear t h i s name t h e r e and i s not

i s o l a t e d from o t h e r r e l a t e d concepts. For t h e development of the concept of Dirac b a s i s see C E G 4 1 , L E G 9 1 and [EG 101. For measurability and i n t e g r a t i o n of H i l b e r t space valued f u n c t i o n s s e e Yosida [ Y ]

o r Dunford-Schwartz IDS], V o l . I . For t h e concepts o f Schauder

b a s i s and Riesz b a s i s s e e Singer [ S i l .

This Page Intentionally Left Blank

81

CHAPTER I V

THE GENERALIZED EIGENVALUE PROBLEM FOR S E L F - A D J O I N T OPERATORS

A n a t u r a l problem i n a theory of generalized functions i s t h e so-called generalized eigenvalue problem. H e u r i s t i c a l l y , t h i s problem can be f o r mulated as follows. Consider t h e t r i p l e 8 c

x

c Y i n which

x

is a Hilbert

space, 8 i s a space of t e s t f u n c t i o n s and P i s a space o f generalized funct i o n s ( d i s t r i b u t i o n s ) . Let P be a s e l f - a d j o i n t o p e r a t o r i n

be i n t h e spectrum of

P

x

and l e t A

E

IR

with m u l t i p l i c i t y mX. The q u e s t i o n i s whether t h e r e

e x i s t mX (generalized) eigenfunctions i n Y . In t h i s c h a p t e r we s o l v e t h i s problem i n t h e b a r e s e t t i n g o f a Sobolev t r i p l e

R(X)

c

x

c

R-l(X).

The main

r e s u l t can be l o o s e l y formulated a s follows. (The terminology i s explained

i n t h i s chapter.) L e t t h e r e be given n commuting s e l f - a d j o i n t o p e r a t o r s

s e p a r a b l e H i l b e r t space such t h a t t h e o p e r a t o r s

R-l(X).

L e t a(P,

,...,P

x. Pe

a d d i t i o n , w e can take R such t h a t R

with t h e property t h a t f o r each

?ie denotes

...,P

in a

(x)

) denote t h e j o i n t spectrum of

mA denote t h e m u l t i p l i c i t y of each X = ( A , , . . . , 1,)

Here

Pl,P2,

Then t h e r e e x i s t s a p o s i t i v e bounded o p e r a t o r R -1 are c l o s a b l e a s l i n e a r o p e r a t o r from R into

the closure of

e

-1

=

(x)

E

(P,

,...,Pn)

a(P1,

...,P

and l e t

) . Then, i n

c o n t a i n s a canonical Dirac b a s i s

l,...,n

Pe i n R

-1

(x).

The proof of t h i s statement involves t h r e e important i n g r e d i e n t s :

-

The commutative m u l t i p l i c i t y theory f o r s e l f - a d j o i n t o p e r a t o r s . The measure t h e o r e t i c a l Sobolev lemma with M = lRnx ID where ID denotes a countable index s e t .

82

THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

-

x

The e x i s t e n c e of an orthonormal b a s i s i n

such t h a t each o p e r a t o r o u t

of a f i n i t e s e t of mutually commuting s e l f - a d j o i n t o p e r a t o r s has a column f i n i t e matrix with r e s p e c t t o t h i s b a s i s . The p r e s e n t chapter h a s been organized such t h a t i n each s e c t i o n only one of t h e s e i n g r e d i e n t s i s discussed. The proof of t h e main r e s u l t i s given i n the f i n a l section.

Commutative m u l t i p l i c i t y t h ,

IV.l.

I n t h i s s e c t i o n w e d e s c r i b e t h e commutative m u l t i p l i c i t y theory f o r a f i n i t e number o f s t r o n g l y commuting s e l f - a d j o i n t o p e r a t o r s ( i . e . t h e i r s p e c t r a l r e s o l u t i o n s mutually commute). Throughout t h i s s e c t i o n w e t a k e a f i x e d n

E

Let

IN.

u

denote a f i n i t e B o r e 1 measure on

lRn such t h a t u ( l R n ) <

-.

nf,

The support of

plement of t h e l a r g e s t open set

0

i.e.

u,

p i s a o - f i n i t e measure on

denoted by supp(u) i s t h e com-

u(0)

f o r which

= 0. Another c h a r a c t e r i z a -

t i o n i s t h e following.

*.

IV.1.1. For x

IRn

E

and r > 0 we p u t B ( x , r ) = {y E lRn

Supp(U1 = {x E

mn

I Vr>o

: p(B(x,r)) >

01. Here

mn .

Proof.

Put S U P P ' ( U )

Then t h e r e e x i s t s r

SUPP'

= Ix

E

mnl Vr>o

:

I

Ix- yIn 2 r ) .

I - In

Then

denotes any norm i n

u ( B ( x , r ) ) > 0 ) . Let x E s u p p ' ( p ) * .

> 0 such t h a t p ( B ( x , r x ) ) = 0. So w e have

(u)* =

U XESUPP' (11)

L e t K > 0. The s e t supp'

(u)* n

*

B(x,rx)

B(O,K)

.

i s bounded i n lRn.

By Besicovitch

covering lemma (Lemma 11.3.4) it follows that t h e r e i s a countable s e t {x. 3

Ij

E

Ri) c supp'

( P I * such t h a t m

supp'(u)* n and hence

B(o,K)

c

U B(xj,rj) j=l

83

COMMUTATIVE MULTIPLICITY THEORY

Since supp' ( p ) h a s p-measure SUPP' (U)*

=

*

m

=

supp' ( p ) * n

zero. C l e a r l y , supp'

B(o,K),

(u) *

i t follows t h a t t h e s e t supp' ( p ) *

i s an open set o f

mn,

whence

SUPP(U)*.

Conversely, l e t x such t h a t

K,l

B(x,E)

s u p p ( p ) * . Since s u p p ( p ) * i s open, t h e r e e x i s t s

E

c supp(p)*. So

~ ( B ( x , E ) = 0 and x

E

E

> 0

supp'(p)*. It yields

SUPP(V)* c supp' ( P I *.

0

Two Borel measures p(l) and p ( 2 ) on IRn

p

N

are called

( N ) = 0 i f and only i f

equivalent i f f o r a l l

= 0. W e use t h e n o t a t i o n 1 * p ( 2 ) and denote t h e equivalence c l a s s of p ( l ) by [ p " ) ] . The equi-

Borel s e t s

we have p

valence c l a s s e s

[ p ( l ) l and

[ p ( l ) ] and

E

IV.1.2.

u(2)

E

[ p ( 2 ) ] are called

r p ( 2 ) 1 we have ;(l)

p,(N)

d i s j o i n t i f f o r each 1 ;(2)

which means

e.

L e t p ( l ) and p ( 2 ) denote two f i n i t e Borel measures on IRn with p (1) Then we have

Proof. Let tions

'pl

c q ] E L 2 ( I R n , p ( l ) + u(~)) and l e t

$

E

[ q l . Define t h e Borel func-

and q2 by

'p1(X)

=

0

,

elsewhere

T(X)

,

if x

0

,

elsewhere

,

and

So t h e mapping U

:

Iq1

H

E

([q11,['p21)

supp(p

.

i s an isometry from L ( I R ~ , u ( ' ) + ~ ( ~ ) ) 2

84

THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

into L~(IR~,~(')) L ~ ( I R ~ , ~ ( ~ ) ) . Conversely, if [q,]

6,

E

and

[cp,]

c2

cp(x)

L2(IRn,p(l)) and [q21

E

ccp

E

L.

=

E

L2(mn,p(2)), then for each

1 we define the Borel function 2

ql(x)

,

if x

cp2(x)

,

if x

,

elsewhere

We observe that for all

E

[PI

cp by

supp(p(l) \supp(p

E

(2))

supp(u(2) \supp(U (1))

,

,

.

we have supp(c) = supp(c'). So it makes

sense to introduce supp([~~l) := supp(G) where p is any representative of CUI.

Let (P l , . . . , P

)

denote an n-set of strongly commuting self-adjoint operators

in the separable Hilbert space

x.

We mean that their spectral projections

mutually commute. Now the notion of unifOrT7I multiplicity for such an n-set is defined as follows.

IV. 1.3. Definition. The n-set

(P1,...,P

Bore1 measures p (

r. ..

is of uniform multiplicity m if there exists m finite

,p (m), which

are mutually equivalent, and a unitary

operator U,

such that each self-adjoint operator

u P u*,

e

e

= 1,

...,n

equals the maximal

operator of m-fold multiplication by the coordinate function id

e

in this direct sum.

Remark. In the previous definition we may take m = IV.1.4.

-.

Let the n-set (P,,

...,P

finite Borel measure p

m

)

:

e

m.

be of uniform multiplicity m. Then there exists a

on IRn and a unitary operator

u,

COMMUTATIVE MULTIPLICITY THEORY

such that

u Peu*, e

=

1,.

. .,n ,

85

equals multiplication by ide in each direct

summand.

Proof. Take Borel measures

-

u(r)

u ( 1 ) ,...,p (m) as

in Definition IV.1.3. Since

and since both measures are finite, following Theorem 11.3.5

p(‘)

there exists a p(S)-integrable function h rs > 0 on ~upp(p(~))(= s~pp(p(~))) such that for all [ q I , [ $ ] E L2(IRn,u(r))

v

Define the operator rs from L2(IRn,y(r)) onto L2(IRn,p(s)) by

Then Let

vrs

is unitary and

2

v*rs[$I

= [hLz$],

[$I

E

Lz(IRn,p(s)).

denote the maximal operator of multiplication by the function idQ in L, ( J R ,P (r)) Then the operator 2 v* satisfies rs 1 rs

.

v

So the self-adjoint operator

qe

in Lz(lRn,p(r)) is unitarily equivalent to

the self-adjoint operator of multiplication by the function ide in

..., p (m)belong to the same equivalence class C P ~ I ,

Since all measures u (1) ,

the above considerations show that we can take one

out of [ p 1 in m m Definition IV.1.3. This is precisely the statement of the lemma. Let Pl,...,P

(ffj(A))AEJR

P

j = 1,

0

denote n strongly commuting self-adjoint operators and let denote the spectral resolution of the identity corresponding to

...,n.

1’ resolution ( H ( A )

For the n-set (Pl,...,Pn) we define the joint spectral )

...,A

in the following standard way: Let Al,

n Borel set.s in IRn. Then we put

denote

86

THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

..

Since H . (A,)H (A ) = /fk (Ak) H , (A,),k,j = 1,. ,n , H is a well-defined 1 3 k k 3 3 projection valued set function on the collection

G

=

{A, x A 2

x

... x A n I A 3, c

IR Borel, j = 1

,...,n} .

The smallest o-algebra which contains the collection G is precisely the Borel o-algebra in lRn generated by its usual locally compact topology. Since any increasing net of orthogonal projections possesses a (strong) limit,

can extend H to the whole Borel a-algebra in IRn. We obtain a

be

projection valued measure on Eln with the property that

Pef

=

i

Ae d H(X)f

,

=

1 ,...,n , f

E

.

D(Pt)

IFP

By means of the joint spectral resolution H, for each Borel function x on lRn we can define the operator

on the domain

These observations lead to the following spectral representation theorem. 1v.1.5. Theorem.

...,7

Let Tl,

denote n strongly commuting self-adjoint operators in

there exists a family t p

Ia

E

a:n,

L,

Then

ID} of finite Borel measures on IRn, where

ID is a countable index set, and a unitary operator p )

x.

such that for each

e

=

1,.

u

from

x

onto

..,n the operator u Te u*

in this

direct sum equals the maximal operator of multiplication by the function idt in each direct summand.

Proof.

Let H denote the joint spectral resolution of the n-set

For any f <{H(A)f pf

:

A

E

x,

f # 0, let

IA

I+

x f denote

(TI,...,T

).

the closure of the linear span

c lRn, Borel)> and let p f denote the finite Borel measure 2 IIH(A)fll , A c IRn, Borel. On <{H(A)f A c IRn, Borel}> we define

the linear operator 11 f by

I

COMMUTATIVE MULTIPLICITY THEORY

L2 ( I R n , p f ) ,

U extends t o a u n i t a r y f onto L2(IRn,pf). F u r t h e r , t h e subspace reduces each

Since t h e simple f u n c t i o n s a r e dense i n

xf

o p e r a t o r from

Te

operator

xf

(T T X f )U*f equals t h e maximal o p e r a t o r of m u l t i p l i c a t i o n f e

and

1

by t h e function idL i n L 2 ( l R n , p f ) ,

= 1,.

..,n.

W i t t h e a i d of a Zornal argument we s e e t h a t t h e r e e x i s t s a maximal count-

a b l e family {x

fa

Ia

xfa

ID} such t h a t

E

a b i l i t y follows from t h e s e p a r a b i l i t y of had g

x

E

such t h a t g 1

xf a

for a l l a

Xfa f o r a l l Bore1 s e t s A

E(A)g 1

for a l l a

Now p u t p

(Count-

ID.

Since i f we

then we would a l s o have

and a l l a

IRn

C

ID,

E

E

ID,

E

and hence

x9 1 xf a

This implies t h a t g = 0.

ID.

E

xfB f o r a # 8 , a,B x.) Now x = a z I D X f a .

I

= pfa,

a

let

ID,

E

u n i t a r y o p e r a t o r and ll

:

a

x,

+ Lz(IRn,p

a

)

denote t h e corresponding

t h e orthogonal p r o j e c t i o n from

x

onto

xf a ‘

Then t h e

l i n e a r o p e r a t o r U,

i s a u n i t a r y o p e r a t o r from

x

onto

e

L

aEID 2

(IRn,p

)

which s a t i s f i e s t h e r e -

0

quirements. Let the n-set space

(Tl,...,T

L (lRn,pa) aEID 2 0

of t h e n-set

) admit a s p e c t r a l r e p r e s e n t a t i o n i n t h e H i l b e r t

as described i n the above theorem. The j o i n t spectrum

(Tl,...,T

) i s defined by

We have t h e following c h a r a c t e r i z a t i o n i n terms of t h e measures p

IV.1.6.

Proof.

a. For convenience we consider t h e maximum norm i n IRn,

i.e.

.

88

THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

Then for all X

Let A 4 o(T1,

E

IRn and r > 0,

...,T ) .

Then there exists

n

n

H ([A - E , X , + E I ) e=1 e e It follows that for all a

B(X,$o)

C

supp(p,)*

= 0

E

> 0 such that

.

ID we have p (B(X,E)) = 0. Therefore

E

for all

a E

ID, whence X

L (a;IDsupp(pa)) .

Conversely, let X L azIDsupp(pa). Then there exists a fixed for all a

ID, p (B(X,o))

E

and X 4 u (Tl,.

=

0. This means H(B(X,E))f

=

E

> 0 such that

0 for all a. Hence

..,Tn).

0

U supp(p ) we can define its multiplicity rnA as the ae ID cardinality of the set { a E ID I A E supp(p ) 1 . By definition the points on

For a point h

E

the "boundary" a!ID supp ( p a )

supp ( p a ) have multiplicity

-.

The above set up is not very satisfactory. The spectral representation theorem yields a countable family of measures

{p

Ia E

ID1 which is not

unique at all. Therefore one has looked for a refinement of this representation theorem. The commutative multiplicity theorem is such a refinement. In a finite dimensional Hilbert space E each commuting n-set of self-adjoint operators

(B1,-..,B

)

has an orthonormal basis of joint eigenvectors. It

makes sense to call an element X of lRn an eigentuple of the n-set

(B,,

...,B,),

if there exists a non-trivial vector eX BeeX = X e

,

L

= 1,

E

E such that

...,n .

The set of all eigentuples is the joint spectrum of ( B l , list all eigentuples of (81,...,B

...,B

1 . In order to

in a well-ordered manner one can list

subsequently all eigentuples of multiplicity one. two, etc. The following theorem is the generalization of this procedure in the case of n strongly

COMMUTATIVE MULTIPLICITY THEORY

89

commuting self-adjoint operators inan infinite dimensional Hilbert space. IV.1.7. Theorem (Commutative multiplicity theorem). Let (T,,

...,T

denote an n-set of strongly commuting self-adjoint operators

)

in the Hilbert space

x.

x = xm x, @

Then

@

x,

x can be

split into a direct sum

...

@

(some of them can be trivial) such that the following assertions are valid:

-

in

-

(Tl,...,T ) restricted to xm' Xm and has uniform multiplicity m.

The n-set

The equivalence classes sponding to each

[!.I

m

m =

m,

1.2, ... acts invariantly

1 of finite Bore1 measures on lRn corre-

xm are mutually disjoint.

We omit the proof of this theorem which is lengthy and which consists of

Im = tedious bookkeeping. The problem is to find a set { f mrj 1 5 j < m+l} which is maximal in the sense that

m,

1,2,. ..

,

m

and also, satisfies

With the aid of Theorem IV.l.7 we can (and will!) always consider the following standard splitting of n-set (T l,...,T measure

Gm.

L2(lRn,cm)

By @

n

x=

x, x, @

e~

... with

respect to the

In each equivalence class Cp 1 we choose one fixed m onto we denote the unitary operator from m (m times) such that um(Pe m ) u i equals the 0 L2(lRn,cm)

):

um

...

Xme

x

rx

maximal operator of m-fold multiplication by the function idC '

We mention the following result. 1v.1.8. CoroZZary.

Proof.

cf. Lemma IV.1.6.

n

90

THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

IV.2.

An a p p l i c a t i o n o f t h e measure t h e o r e t i c a l Sobolev lemma

Chapter I1 d i s c u s s e s a measure t h e o r e t i c a l Sobolev lennna. Here w e u s e t h e r e s u l t s of t h a t c h a p t e r i n t h e following concrete case: L e t ID denote a countable s e t on which w e impose t h e d i s c r e t e topology and l e t IRn c a r r y t h e usual Euclidean topology. By M w e denote t h e t o p l o g i c a l space I R n x ID with product topology. W e observe t h a t a set Q. c lRnx ID i s open i f f f o r

all p

I

ID t h e s e t i? = {x E lRn ( x , p ) E i?) i s open i n lRn. S i m i l a r l y , a P I R n x ID i s a Borel s e t i f f f o r a l l p E ID t h e s e t

E

B of

subset

B

= IXE IRn P d a s follows

I

(x,p)

B1

E

i s a Borel s e t i n

mn.

On M w e introduce a metric

n * denotes t h e Kronecker d e l t a and any norm on IR Pq In checks t h a t t h e metric d induces t h e topology o f M.

I

where 6

For each p

E

let v

ID,

denote a f i n i t e Borel measure on lRn. P

.

One r e a d i l y

Note t h a t v

P

is a r e g u l a r measure following Theorem 11.3.2. We d e f i n e t h e o - f i n i t e regul a r Borel measure v on

M as follows.

Let

B

denote a Borel s u b s e t of M. Then

we d e f i n e

A function cp on

cpp

:

x

*

M

i s a Borel function i f f f o r a l l p

cp(x,p) i s a Borel function on lRn.

integrable i f f for a l l p

E

ID t h e f u n c t i o n cp

E

ID t h e f u n c t i o n

A Borel f u n c t i o n cp on

P

P

is square v - i n t e g r a b l e and P

Thus we a r r i v e a t t h e following r e l a t i o n

L,(IR",v

L2(M,v) = PE ID

P

)

i s v-

P

Therefore, a Borel f u n c t i o n cp on M i s square v-integrable t h e Borel function cp

M

i s v - i n t e g r a b l e and

i f f for a l l p

E

ID

AN APPLICATION OF THE MEASURE THEORETICAL SOBOLEV LEMMA

Now l e t B ( ( x , p ) , r ) denote t h e closed b a l l i n

M

with c e n t r e ( x , p )

and r a d i u s r > 0 (with r e s p e c t t o t h e m e t r i c d ) . Let B ( x , r )

E

91

mnx

ID

denote t h e

c l o s e d b a l l i n lRn with c e n t r e x and r a d i u s r > 0 (with r e s p e c t t o t h e norm

Theorem.

1v.2.1.

(M,v)

The measure space cp :

cpp

M

-+

admits a r e l a t i v e d i f f e r e n t i a t i o n theorem: L e t

B be a Borel f u n c t i o n such t h a t f o r each p E

: x I-+ cp(x,p)

ID t h e Borel f u n c t i o n

is i n t e g r a b l e on bounded Borel s e t s of IRn.

Then t h e r e

N such t h a t f o r a l l r , 0 < r < 1 and a l l ( x , p ) E cp t h e closed b a l l B ( ( x , p ) , r ) has p o s i t i v e v-measure and f o r a l l ( x , p )

exists a null s e t

Proof. By -

Theorem 11.3.8 f o r each p

t h a t f o r a l l r > 0 and a l l x

v -measure and f o r a l l x

E

(0

such u - n u l l s e t N, P P t h e closed b a l l B ( x , r ) has p o s i t i v e

TP

lRn\N,

P

cp

M\N

E

ID t h e r e e x i s t s a

E

IRn\N

E

M\N

P

Put N

=

cp

(x,p)

U

PEID

M\N

E

N

x {p}. Then 'pp

N is a cp

v-null s e t . Observe t h a t f o r a l l

we have

cp

0

Let

x

denote a separable H i l b e r t space and

L 2 (M,v).

u

a u n i t a r y o p e r a t o r from

Further we f i x a p o s i t i v e bounded o p e r a t o r

R on

x

x

onto

such t h a t UR i s

a Carleman o p e r a t o r . For R w e can always t a k e a p o s i t i v e Hilbert-Schmidt operator. Let

k

:

M

+.

f o r each f URf

E

tion x

x

E

t h e function ( x , p ) I+ ( f , h ( ~ , p ) i)s~ a r e p r e s e n t a t i v e of

On h we impose t h e condition t h a t f o r each p E ID t h e func2 IIh(x,p) I I x , x E lRn, i s v - i n t e g r a b l e on bounded Borel s e t s o f lRn. P

L,(M,v). I+

denote an inducing f u n c t i o n of t h e o p e r a t o r UR. we r e c a l l t h a t

x

92

THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

in Throughout t h i s s e c t i o n w e t a k e a f i x e d orthonormal b a s i s ( u ) k kelN which i s a Schauder b a s i s i n R ( X ) . L e t (UR\)-

: ( x , p ) I+ ( \ , k ( ~ , p ) ) ~ ,( x , p ) E lRnx ID.

x,

Then w e have

m

-

t h e function x

For e a c h p E ID,

H

1 I (UR\)-

(x,p)

l2

i s i n t e g r a b l e on

k= 1

bounded Bore1 s e t s .

Following Theorem IV.2.1 t h e r e e x i s t s a v - n u l l s e t N = U N x {p}, where PEID P e a c h N d e n o t e s a v - n u l l s e t , w i t h t h e f o l l o w i n g p r o p e r t i e s : There e x i s t s P P a r e p r e s e n t a t i v e i n e a c h class (URu ) k E IN, p E IN, such t h a t k P' I V . 2.2.

a.

v~ E I vDx ~ s u p p ( v ~ ) \:N ~

b' 'kcIN

vpeID vx ~ s u p p ( v) \ N p P

These c o n d i t i o n s on

) :E

N l e a d t o t h e following r e s u l t .

Theorem.

1v.2.3. Let p

:

E

ID and l e t x

E

-1

(r) = v ( B ( x , r ) ) P

supp(v ) \ N P P m

k E= l

"B

(x,r)

a.) :E ) , : E

( r ) belong t o R - ' ( x )

b. l i m IIE(P)

-

r+O

. We p u t

( r )I( -1 = 0

.

'

x

=

?

k=l

R-'\,

k,p

R-'u

k

and

r > 0. Then w e have

.

rl'i)Of. Cf. Lemma 11.4.4 and Theorem 111.2.5.

0

93

AN APPLICATION OF THE MEASURE THEORETICAL SOBOLEV LEMMA

F o r each

e=

I t means t h a t

1,

...,n ,

2r

Qe

let

denote t h e m u l t i p l i c a t i o n o p e r a t o r d e f i n e d by

i s t h e maximal s e l f - a d j o i n t o p e r a t o r of m u l t i p l i c a t i o n by

t h e coordinate f u n c t i o n i d , i n each summand of t h e d i r e c t sum

c

...,n

0 L ( d , v 1 . For each 1 = 1, P p€lD 2 adjoint i n W e show t h a t t h e E(')

x.

t h e o p e r a t o r P1 =

u* Qeu

is self-

are candidate ( g e n e r a l i z e d ) eigenvec-

tors of t h e o p e r a t o r P

e.

Theorem.

1v.2.4. Let

e = 1, ...,n.

The l i n e a r span

i s contained i n D(PL) = U * ( D ( Q L ) ) Let p

Proof. X

m.

E

Then f o r a l l x

Let p L ID and l e t x

( r ) equals U * [ A @ )

B (x, r )

1,

c

X.

supp(v ) \ N we have P P

E

E

Observe t h a t f o r each r > 0 supp(v ) \ N P P'

where A l p ) denotes t h e f u n c t i o n B (x. r )

AB (P) ( x , r ) (y'q) = v p ( B ( x , r ) ) - ' 'B(x,r)

I t i s c l e a r t h a t [A(')

B (x,r)

D(PL)

=

6pq

'

y E d ,

q c m .

1 belongs t o D ( Q L ) . Hence u*CALYL,rl 1 belongs t o

U*(D(Qe)).

Next we prove t h e second a s s e r t i o n . Because of t h e previous theorem i t remains t o prove that

For each r > 0 we have

94

THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

Thus we get the straightforward estimation

Since the first factor in the above expression tends to zero as r m

since the second factor tends to the limit k ~ ~ l c p ~ , ~ l(2x as ) r derive lim r+C

llRPe E F ) (r) - xeR ) : E

(r)II

=

0

+

+

0 and

0 , we

.

Remark. The above theorem remains valid if we replace

0

Pe

by O(Pl), where

8 denotes any continuous function on IR.

Theorem IV.2.4 expresses that the described procedure always yields candi-

d a t e generalized eigenvectors of the strongly commuting n-set (Pl,...,Pn). To obtain genuine generalized eigenvectors, the n-set has to consist of closable operators in R-'(X) IV.2.5.

R-l (x). A characterization of closable operators in

is the following.

a.

P in R - ~(XI is closable iff R P R - ~is a closable operator in X. For the respective closures 7 and R PR-l we have

A linear operator

95

A SOLUTION

F

D(P)

E

c R-l(X)

-

RF

D( RPR-l) c X

E

and R P F = R P R - ’ RF

Proof.

The l i n e a r o p e r a t o r R

. :

R-l(X)

.+

x

i s u n i t a r y with a d j o i n t

R

-1

.

From

I7

this observation a l l a s s e r t i o n s follow.

IV.2.6. Let

L

=

Corollary. 1,.

.., n .

o p e r a t o r from

R

Suppose t h e o p e r a t o r

-1

(x)

into

R-’(X).

supp(v ) \ N and a l l p P P

for all x

E

Proof. By -

Theorem I V . 2 . 4

Pe

=

u qe u*

is closable as a linear

Then we have

E

ID.

we have t h e following r e s u l t s

and

I t follows t h a t E(’)

E

D(PL)

and

pe) :E

=

(P) xeEx

0

-

I n the next s e c t i o n which c o n t a i n s t h e main r e s u l t of t h i s c h a p t e r , we comb i n e t h e r e s u l t s of t h i s s e c t i o n and t h e m u l t i p l i c i t y theorem. Thus, our s o l u t i o n of t h e generalized eigenvalue problem f o r an n-set of s t r o n g l y commuting s e l f - a d j o i n t o p e r a t o r s i n a s e p a r a b l e i n f i n i t e dimensional H i l b e r t space

x

comes very c l o s e t o t h e s o l u t i o n of t h e eigenvalue problem f o r an

n-set of commuting s e l f - a d j o i n t o p e r a t o r s i n a f i n i t e dimensional H i l b e r t space E.

IV.3. Let

A solution

( Pl , . . . , P

)

denote an n-set of commuting s e l f - a d j o i n t o p e r a t o r s i n

Following Section IV.l t h e r e e x i s t s a s t a n d a r d s p l i t t i n g of

x

x.

into direct

THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

96

such t h a t t h e n-set

m =

.. . xm i s

-, 1 , 2 , .

t h a t each

( Pl,. . .,Pn) a c t s i n v a r i a n t l y i n each summand x m'

A l s o , t h e r e e x i s t f i n i t e Borel measures p m , p , p

( ql , . . . , < )

( P,,..., Pn) i s u n i t a r i l y e q u i v a l e n t t o t h e n - s e t m

restricted to

,... such

m 1 2 u n i t a r i l y e q u i v a l e n t t o t h e d i r e c t sum j$l L 2 ( I R n , u m ) ,

@

L2(JRn,pm)

j=l p l i c a t i o n by t h e f u n c t i o n i d

and

where each

2L

denotes t h e maximal o p e r a t o r of m-fold multi-

eIm

L e t ID denote t h e countable s e t { (m, j)

M = JRnx ID and l e t t h e f i n i t e Borel measures v for m =

-,1,2,

v

v

=

@

1

?;

j < m+l.

m j$l L 2 ( I R n , p m )

and

u Plu*

=

W e t a k e a p o s i t i v e bounded o p e r a t o r

x

be defined by v

qe, R

E

ID t h e f u n c t i o n x

bounded Borel s e t s of IRn.

I+

L (M,v)

onto

2

m,j

let =

'm

such t h a t

where m = - , 1 , 2 , .

UR

such t h a t

Also, we assume, t h a t t h e inducing f u n c t i o n f o r each p

+ l},

As i n t h e previous s e c t i o n set

denote t h e u n i t a r y o p e r a t o r from

u(xm) =

m. j

l 5 j < m

Then a s we have seen

PGID P'

Let

... and

... ,

= -,1,2,

k

:

Ilh(x,p) It2, x

E

M

.. and e

= 1,.

..,n.

i s a Carleman o p e r a t o r . +

x

IRn,

has t h e p r o p e r t y t h a t

i s i n t e g r a b l e on

Observe t h a t any p o s i t i v e Hilbert-Schmidt

operator f u l f i l l s t h e s e requirements: Let (

R(X).

~ denote ) ~ an~ orthonormal ~ basis i n

u

W e choose a r e p r e s e n t a t i v e

and a n u l l s e t a l l (m, j)

E

N

m. j

k, j f o r each (m,j) E ID,

ID and a l l

x

E

supp(p,)

x

which i s a Schauder b a s i s i n

i n t h e equivalence c l a s s

\N

a s indicated i n IV.2.2. ,

mt I

we have

(m) ) k j Then f o r

(URu

A SOLUTION

97

and

belong t o R - l ( X ) .

Moreover, Theorem IV.2.3 and Corollary IV.2.6

yield the

following

Theorem.

IV.3.1.

L e t m = =,1,2,...

and l e t j

E

IN with 1 S j < m + l . Then f o r all

a.

b. For each

e

,...,n and a l l r

> 0, E ( m ) . ( r )

= 1

e

XI

E

I

D(PL)

c

x

and

..

Suppose t h e o p e r a t o r s Pel = 1,. ,m , a r e c l o s a b l e a s l i n e a r o p e r a t o r s -1 from R-l(X) i n t o R with c l o s u r e s denoted by ?[. Then t h e preceding

(x)

assertions yield

n

C.

(m) Remark. The s e t {EX,

Im

=

-, 1,2,.

.. ,

1

5

j < m

+ 1, x

supp(pm)\Nm, j} ex-

E

tends i n a n a t u r a l way t o a canonical Dirac b a s i s i n t h e Sobolev t r i p l e

R(X) c X

c

R-'(X).

Of course w e need n o n - t r i v i a l generalized eigenvectors. The next lemma s a y s t h a t t h e E ( m ) a r e n o n - t r i v i a l f o r almost a l l x. x, j IV.3.2.

m.

L e t t h e E ( m ) be a s i n Theorem IV.3.1. x, j a p - n u l l s e t 0 such t h a t f o r all x

m

m

1 5 j < m+l,

Proof.

Let j

Let m = E

-, 1 , 2 , . ..

) \ om m

supp(p

.

Then t h e r e e x i s t s

and a l l j

E

IN,

t h e elements E ( ~ )a r e n o n - t r i v i a l . x, j E

IN,

1 5 j < m+l.

Put

om,j =

{x

E

supp(pm) I E A Y i = 0 ) . Then

THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

98

for all k

IN

E

We define the function A0 by m,j I , A

Om,j

(x,m',j') = 0

,

ifxEO m'=m, j'=j, mtj ' elsewhere

.

It follows that

Consequently 0m, m 0m = ,j =u 1 0m,]. . AS

is a l~ m-null set. Now the proof is complete if we take

0

we have seen the generalized eigenvalue problem for an n-set

(Pl,..- ,Pn)

of strongly commuting self-adjoint operators can be solved in any triple

R(X)

c

X

c

R-l(X)

if ve impose certain additional conditions on R. If R is R-1 (x) always contains a complete set of

a Hilbert-Schmidt operator then

candidate generalized eigenvectors. These candidates become genuine generalized eigenvectors if for each in R-'(X),

i.e. if

R PLR-'

a.

=

1,. ..,n the operator

Pa.is

closable

is a closable operator in X. Therefore in order

to give a complete solution of the generalized eigenvalue problem for

(P1,...,Pn) it is sufficient to show that there exists a positive HilbertR such that the closability of R Pa.R-', 1 = 1,. . ,n , is

.

Schmidt operator

guaranteed. Consider the following construction. Let L l ,

...,1

be densely defined Linear operators in

x.

Then we define their

m

joint C -domain as follows. IV. 3 . 3 . De f i n ; tion.

The joint

cm-domain cea ( L , ,

...,Ln)

of L,,

...,Ln

is defined as

SOLUTION

A

(The s e t

11,

...,n}IN c o n s i s t s

... L n ( s ) )

D(Ln(,)

of a l l mappings from IN i n t o 11,...,n ) ;

denotes t h e domain of t h e o p e r a t o r between b r a c k e t s . )

( Pl , . . . , P

For t h e s t r o n g l y commuting n-set

x.

in

99

the j o i n t

)

c

a0

-domain i s dense

This i s a s t r a i g h t f o r w a r d consequence of t h e s p e c t r a l r e p r e s e n t a t i o n

theorem.

IV.3.4. Theorem.

Ll,...,L

Let

x

denote l i n e a r o p e r a t o r s i n

with a dense j o i n t

x

Then t h e r e e x i s t s an orthonormal b a s i s i n

e

,

= 1,. . . , n

Proof.

a0

-domain.

such t h a t each o p e r a t o r L

el

has a column f i n i t e matrix r e p r e s e n t a t i o n t o t h i s b a s i s .

Since Cm(L l l . . . , L n )

i s dense i n

e x i s t s an orthonormal b a s i s ( \ ) k E m

cm ( L l,...,L

c

in

x

and s i n c e

x

x

i s separable there

which i s contained i n

a s follows. W e introduce t h e orthonormal b a s i s (v ) k kcIN P u t v1 = ul. Then t h e r e e x i s t s an orthonormal set {v2 , . . . , v n l ) I { v l } with

n1

5 n

).

+ 2 such t h a t

> 1
> c
3

1 ' * * - r V n 3 >'

.. .

I n g e n e r a l , having produced < v l , ,v >, nk < k ( n + 1 ) "k i n the (k + l ) - s t e p we produce t h e orthonormal s e t {vn I

{ v, , . . . , v

"k

5 (k + 1 ) (n + 1 ) + 1 such t h a t with n k+ 1

=



E

((LLV.,Vi) 1 The n-set

Ii,


...,v"k >,

jcH

(P1,...,P

k

E

IN.

i n t h e k-th s t e p , I

k+ 1

T h i s s e t i s complete

A l s o , by c o n s t r u c t i o n t h e matrix

i s column f i n i t e f o r each

)

+ 1,

+l,..., Vnk+l' k

>


Thus we o b t a i n i n d u c t i v e l y an orthonormal s e t (v ) because uk

...,v"1 > with

has a dense j o i n t

0

c

e=

-domain.

I,.. . , n .

0

So by t h e preceding

theorem t h e r e e x i s t s an orthonormal b a s i s (v k E I N such t h a t t h e o p e r a t o r s

P1,

...,P

have a column f i n i t e (and hence row f i n i t e ) matrix r e p r e s e n t a t i o n

THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

100

with respect to this basis. We define a positive Hilbert-Schmidt operator

R

on X by m

Rf

= 1,.

Ik

<{v k

k= 1

Pk(f,Vk)Vk

,

-sequence with positive components. Let -1 equals R (0(Pe)1 . Observe that ,n. The maximal domain of R Pe R

where ( p k ) k E H

1

1

=

Can be any

.. E

IN}> c R(D(Pe))

2

whence R(D(Pe)) is dense in

x.

We prove that

R P~R-'is closable in X.

Let f

E

D(R

PeR-').

(RP

e

Then for all k

R-lf,vk)

=

IN

E

(f,R-1 PeRvk)

because ((Pevk,vJ;)k,j is a column-finite matrix. It follows that the adjoint of

R PeR

is densely defined. So

R PeR-'

X. BY Lemma 1v.2.5 P is a closable operator in R

e

IV.3.5.

=. ...,P )

Let (Pl,

is a closable operator in

-1

(X).

denote an n-set of strongly commuting self-adjoint operators.

Then there exists a positive Hilbert-Schmidt operator operator pe,

x-

=

1

,...,n ,

R

such that each

is closable in R-~(x).

We arrive at the main theorem of this section. IV.3.6. Theorem. Let

(Pl,...,P

)

denote an n-set of strongly commuting self-adjoint operators

x = xm x, e x2 e ... denote the standard splitting of x and let p - , ~ , p ,... denote corresponding multiplicity measures. Then there exists 1 2 in

x.

Let

0

a positive Hilbert-Schmidt operator in

R-l(X)

with closure denoted by

R

Fe,

such that each operator Pe is closable = 1

,...,n .

For m

=

-,1,2

,...

there

is a p -null set N with the following property: for all x E supp(pm)\Nm m m there exist m non-trivial independent generalized eigenvectors E (m) in x,j (x), 1 5 j < m + I, satisfying

R-'

Proof.

The proof is a compilation of the results of this section.

SOME ILLUSTRATIONS

101

Remarks.

- We

r e c a l l that t h e j o i n t spectrum a ( Pl , . . . , P

)

equals

m

m=l U SUPP(Um) u SUPP(I.l,).

-

For t h e eigenvalues of R i n t h e c o n s t r u c t i o n preceding Lemma IV.3.5 any positive

2

-sequence can be taken. One may say t h a t t h e improper eigen-

v e c t o r s of P l , . . . , P

l i e a t t h e "periphery" of t h e H i l b e r t space

x.

IV.4. Some i l l u s t r a t i o n s d . denotes t h e s e l f - a d j o i n t d i f f e r e n t i a l o p e r a t o r i in dx t h e s e l f - a d j o i n t o p e r a t o r of m u l t i p l i c a t i o n by t h e i d e n t i t y

In t h i s s e c t i o n Lz(IR)

2

and

P

L2 (IR).

function i n

Illustration 1. Let

Rm denote t h e p o s i t i v e bounded o p e r a t o r

where w e t a k e m >

4. The

(I

+q2)-m/2i n x

corresponding Sobolev t r i p l e R

m

(x)

=

5

x

L2 (IR) , 5

Ril(X)

can be c h a r a c t e r i z e d a s follows.

Rm(X)

: The H i l b e r t space of equivalence c l a s s e s of Borel

functions cp on IR f o r which t h e f u n c t i o n 2 m/2 I+ ( 1 + x ) q ( x ) i s square i n t e g r a b l e .

x

x

: The H i l b e r t space of equivalence c l a s s e s of square

i n t e g r a b l e f u n c t i o n s on IR.

R,'(X)

: The H i l b e r t space of equivalence c l a s s e s o f Borel

functions cp on IR f o r which t h e f u n c t i o n 2 -m/2 x I+ ( 1 + x ) cp ( x ) i s square i n t e g r a b l e . For each A

E

IR,

let e

the f u n c t i o n x I+ (1 + x

CeAl E R-l(X) f o r a l l A m

denote t h e f u n c t i o n x

A 2 -m/2 )

E

b+

1 -iAx e , J2n

x

E

IR. C l e a r l y

e ( x ) i s square i n t e g r a b l e , whence A

IR. From Chapter I11 i t follows t h a t ( C e A l ) A E I R

i s a canonical Dirac b a s i s .

P i s s e l f - a d j o i n t i n L 2 (IR); P i s of uniform m u l t i p l i c i t y 1 and t h e spectrum of P i s t h e r e a l a x i s . A c y c l i c v e c t o r f o r P i n L (IR) i s given 2 2 e-$x 1. Formally w e have P[eA] = Ace,], A E IR. by [x b The o p e r a t o r

4% In o r d e r t o check t h a t t h i s equation makes sense a s an eigenvalue equation in

Ril(X)

w e have t o show t h a t t h e o p e r a t o r R PR-' m m

is closable i n

X.

We

102

THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

note t h a t t h e o p e r a t o r

X i n t o X.

from Let

Rm P R i l cannot be extended t o a continuous operator

c

m

(IR) denote t h e space of i n f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s on IR

which'vanish

o u t s i d e a compact s u b s e t of IR.

[+I

sponding equivalence class

For a l l J, -1

D(Rm PRm).

belongs t o

E

cz(IR) t h e corre-

I t i s obvious t h a t

D(R-'

PRm ) mi s contained i n D ( ( R m P R i l ) * ) . Since t h e subspace {[$I I JI E cc(IR)} i s dense i n L z ( I R ) = X t h e o p e r a t o r ( Rm PRm-1 ) * i s densely defined. Hence R m P R i l i s a c l o s a b l e o p e r a t o r i n X and P i s a c l o s a b l e -1 o p e r a t o r i n Rm (x) . L e t P denote t h e c l o s u r e of P i n R - l ( X ) . Consequently, m f o r each A

E

IR w e have

CeAl E

D(h

and

FCe,l

= ACeAl

.

L e t IF denote t h e F o u r i e r transform on L 2 ( I R ) ;

put

E

m triple

= 'FRmIF*. Then

Bm

=

(1 + p2)-m'2.

IF i s a u n i t a r y o p e r a t o r . W e

We o b t a i n t h e c l a s s i c a l Sobolev

E m ( L 2 ( ~ ) ) = Hm (IR) c L2(m) c H-,(IR)

=

ii1(L2(~))

which h a s been d e s c r i b e d i n Section 11.5. The F o u r i e r transform IF restricts t o a u n i t a r y o p e r a t o r from R --1 t o r from R-'(x) onto R ( X I . m m Put 6 A = IF[eA], A The element 6 A , A

E E

(X)

I

onto

R

m

(x)

and extends t o a u n i t a r y opera-

IR. Then ( 6 A ) A E I R i s a canonical D i r a c b a s i s i n

IR, of H

H

-m (IR). (IR) is u s u a l l y c a l l e d t h e Dirac d e l t a func-

-m t i o n a t A. With t h e a i d of t h e F o u r i e r transform it simply can be proved

that all A

2 E

extends t o a c l o s e d o p e r a t o r

q in

H-,(IR)

with

= Fh*. A l s o for

I R we have

E

we n o t e t h a t

D(q)

and

86,

.

=

i s n o t a continuous l i n e a r mapping i n

L e t T denote a s e l f - a d j o i n t o p e r a t o r

0

i n a s e p a r a b l e H i l b e r t space X. L e t R

denote a p o s i t i v e bounded o p e r a t o r on

x.

The spectrum of t h e extended o p e r a t o r

7

of t h e o p e r a t o r T.

H -m (IR).

Suppose T i s closable i n

R-' ( X )

.

can be much l a r g e r than t h e spectrum

103

SOME ILLUSTRATIONS

Illustration 2. we take

X

. L e t Mt denote t h e p o s i t i v e bounded o p e r a t o r

L, (IR)

=

where t > 0.

121

(

(q2)'.)

denotes t h e o p e r a t o r

exp(-t

IQI )

The corresponding Sobolev

(x) x Mi1(X)

triple M 5 admits a s i m i l a r c h a r a c t e r i z a t i o n as t h e one t 2 -m/2 given i n I l l u s t r a t i o n 1, i f w e r e p l a c e t h e f u n c t i o n x I+ ( 1 + x ) by The s e t (Ce,l)XEIR i s a canonical Dirac b a s i s i n M - ' ( X ) . The t -1 o p e r a t o r M PM, i s c l o s a b l e , because PM,) c D ( ( M PM-')*) c o n t a i n s t t t t h e dense l i n e a r subspace

x t+ e - t l x l .

D(Mil

P

If

However, t h e spectrum of

A

E

C with

1x1

C

in

M;'(X)

xH-e

with I I m

i s l a r g e r than IR.

Indeed, f o r a l l

-iAx

Jz;;

g i v e s r i s e t o t h e element re,] E C

then w e have

t t h e function

1

e,:

A

M,'(X),

denotes t h e c l o s u r e of P i n

of

M-'(X). t

We show t h a t re,]

E

o(P)

for a l l

A1 < t.

There e x i s t s a sequence ( c p ) in n nclN i n t h e following sense

c

m

(IR) which tends t o t h e f u n c t i o n e 0

and m

Then it follows t h a t f o r each X

E

C with

II m A (

< t

104

THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

and, s i n c e cp e nA

E

C;(IR),

we have Cq e

n A

1

E

D(P) c

m

m

From t h e above i n e q u a l i t i e s we o b t a i n -1

Mt ( X I -

l i m Cq e 1 = Ce,] nX

nand

-1

Mt ( X )

-

.

l i m PCqneAl = Ace,] IF-

This y i e l d s

Cell

for a l l A

E

E

Q: w i t h

DC?)

and

II~AI

?CeAl =

Ale 1 A

< t.

Taking F o u r i e r transforms we d e r i v e

.

M = t

IFM

t

IF* = e x p ( - t l P l )

and

iil= I F M-1~ So f o r a l l A E C with

*

IF

= exp(t1PI)

II m X I

< t

.

x

and

SOME ILLUSTRATIONS

-- 1 ( L 2 ( m ) ) .

6A :=

~ C e , l E Mt

One may say t h a t

.-- 1 Mt (L2(1R))

The o p e r a t o r

Q

105

contains 6-functions with complex argument. 1-1 i n Mt ( L 2 ( I R ) ) . For

extends t o a c l o s e d l i n e a r o p e r a t o r

a l l A c C w i t h I I m A1 < t we have

Remark. The space e x p ( - t I P I ) ( L 2 ( l R ) ) can be considered as a space of f u n c t i o n s on IR,

cm-

which can be extended t o an a n a l y t i c f u n c t i o n on t h e open

s t r i p of width 2 t around t h e r e a l a x i s . Observe t h a t every

u

E

I

exp (-t PI )

(L, (IR)) can be represented i n t h e form co

u(x) =

J2n

iyx-tl yl

f(y)dy

,

x

E

IR

,

-m

where f denotes any square i n t e g r a b l e function.

Illustration 3 . Consider t h e Sobolev t r i p l e H,(IR) operator

22

.

In

L 2 (IR)

L2 (IR) -C H-,(W.

-C

The s e l f - a d j o i n t

i s of uniform m u l t i p l i c i t y 2 with t h e p o s i t i v e r e a l

a x i s a s i t s spectrum. Put cp

:

x

W

lr+

e-5x2 and cpo

: x t+

4 n$

x e-4x2. Then

t h e s e t {[cpeI,[cpoI) i s c y c l i c f o r Q2. The H i l b e r t space L (IR) s p l i t s i n t o 2

L2 (IR) denotes t h e c l o s e d subspace of equivalence c l a s s e s of square ,e i n t e g r a b l e even f u n c t i o n s and L ( I R ) t h e c l o s e d subspace of equivalence 2,o classes of square i n t e g r a b l e odd functions. In accordance with t h e commutawhere

t i v e m u l t i p l i c i t y theory,

where t h e f u n c t i o n s

and

x

and

L, (IR) i s

xo

u n i t a r i l y equivalent t o

a r e defined by

106

THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS

The o p e r a t o r

22

. 1s u n i t a r i l y e q u i v a l e n t t o the maximal o p e r a t o r o f multi-

p l i c a t i o n by t h e i d e n t i t y function i n each d i r e c t summand. S i m i l a r to2 2 . I l l u s t r a t i o n 1 it can b e proved t h a t 2 extends t o a c l o s e d o p e r a t o r 2 i n H,(IR).

For each p

E

' R I

with p = h 2 w e have

-

2 Dom(2 )

E

and

22

2 6h = h 6

Let ll denote t h e p a r i t y o p e r a t o r i n L 2 ( I R )

The o p e r a t o r ll commutes s t r o n g l y with m u l t i p l i c i t y 1. The j o i n t spectrum

A defined by

22 . The

(n,22 )

2-set

(Il,Q2) e q u a l s I R '

f u n c t i o n y on IR by y ( x ) = ( 1 + x ) e - f x 2 . Then [ y l 2 f o r (n,Q 1.

E

. is of uniform

x {-l,l}.

L 2 (IR)

Define t h e

i s a cyclic vector

W e i n t r o d u c e a Borel measure T on IR by

T(B)

= 0

.

f o r a l l Borel s e t s 6 $ { l ) , { - l ) , { - l , l }

Also we i n t r o d u c e t h e f u n c t i o n w on lR X l R by

o(x,y) =

L--

1;

xe(x)

f o r y = 1 and a l l x

x,,(x)

f o r y = -1 and a l l x

IR

E

E

,

IR,

elsewhere

2 Then t h e r e e x i s t s a u n i t a r y o p e r a t o r from L (IR) o n t o L ( W ,w(x,y)dxdT(y)) 2 2 2 * such t h a t u p f.l equals t h e maximal o p e r a t o r of m u l t i p l i c a t i o n by t h e

function i d l

:

(x,y)

I+

x, x

E

IR, and f.lllf.l* e q u a l s t h e bounded l i n e a r

o p e r a t o r of m u l t i p l i c a t i o n by t h e f u n c t i o n i d 2 : ( x , y ) Theset 2 of 2 and

IA ii. W e

E

IR) c H_,(IR) have

H

y.

c o n s i s t s of j o i n t g e n e r a l i z e d eigenvectors

107

SOME ILLUSTRATIONS

and

i i sx

= siqn(X) 6

X E I R .

Some b i b l i o q r a p h i c a l notes and comments There is an abundance of l i t e r a t u r e on t h e s o l u t i o n of t h e "improper" eigenvalue problem f o r s e l f - a d j o i n t o p e r a t o r s i n H i l b e r t space. Here we mention only Berekanskii [Be 11, Maurin [ M a ] , Schwartz [DS] and BereHanskii [ B e

Gelfand-Shilov CGV 4 1 ,

Dunford-

21.

W e emphasize t h a t we d i d n o t f i n d a r e f e r e n c e which g i v e s a s y s t e m a t i c treatment of t h e general eiqenvalue problem f o r an a r b i t r a r y s e l f - a d j o i n t o p e r a t o r i n t h e s e t t i n g of general Sobolev t r i p l e s . Our f i r s t r e s u l t i n t h i s f i e l d appeared as [ E G e l : Hilbert-Schmidt o p e r a t o r then R

x

If i n the t r i p l e R ( X ) c c R-l(X), R i s a -1 always contains a complete s e t o f

(x)

candidate g e n e r a l i z e d eigenvectors f o r any s e l f - a d j o i n t o p e r a t o r i n

x.

Our impression if t h a t people o f t e n f i x a p r i o r i a d i s t r i b u t i o n space i n which they solve t h e g e n e r a l i z e d eiqenvalue problem f o r s e l f - a d j o i n t operat o r s extending t o continuous l i n e a r mappings on t h e chosen d i s t r i b u t i o n space. This i s n o t always a very f r u i t f u l p o i n t of d e p a r t u r e : We look f o r a l a r g e family of Sobolev t r i p l e s

R(X)

c

x

c R-l

(x)

i n which t h e r e e x i s t

candidate generalized eigenvectors f o r a f i x e d s e l f - a d j o i n t o p e r a t o r P (any Hilbert-Schmidt o p e r a t o r R > 0 g i v e s r i s e t o such a t r i p l e ) and next -1 Thus P i s a closable operator i n R

r e s t r i c t t o those t r i p l e s f o r which

(x).

we can keep the d i s t r i b u t i o n space a s small a s p o s s i b l e . F u r t h e r t y p i c a l a s p e c t s of our approach a r e t h e use of r e l a t i v e d i f f e r e n t i a t i o n of f i n i t e Bore1 measures on IRn and t h e extensive use of commutative m u l t i p l i c i t y theory. For a d i s c u s s i o n of commutative m u l t i p l i c i t y theory we r e f e r t o [Brol, [ N e 2 1 and [Col.

This Page Intentionally Left Blank

109

CHAPTER V

D I R E C T RESOLUTIONS I N SOBOLEV T R I P L E S

The concepts w e introduce i n t h i s chapter a r e n o t i n s p i r e d by t h e o r i g i n a l Dirac formalism. However, they a r e c l o s e l y r e l a t e d t o t h e concept of Dirac b a s i s . A s k e t c h of t h e c o n t e n t s of t h e p r e s e n t c h a p t e r i s presented i n t h e f i r s t section.

V.l.

P r e l i m i n a r i e s and summary

F i r s t w e r e c a l l some b a s i c t o p i c s of t h e theory of d i r e c t i n t e g r a l s o f

H i l b e r t spaces. W e introduce t h e d e f i n i t i o n of Borel f i e l d of H i l b e r t spaces.

v. 1.1. D e f i n i t i o n . Let

M

M

be a Bore1 space, f o r each x E

space with i n n e r product ( * , * )

X'

Y

let

The family

denote a s e p a r a b l e H i l b e r t

(Yx)xEM

e s t a b l i s h e s a Borel

f i e l d of H i l b e r t spaces i f t h e r e i s given a l i n e a r subspace

n

space P =

X€M

E

of t h e product

Y x which possesses t h e following p r o p e r t i e s .

a. For every v E b. L e t u

s

s

the function x

I+

Itv(x) I I x is a Borel function.

P be such t h a t for every v E S.

6

s

t h e function x n

( U ( X ) , V ( X ) ) ~i s

a Borel function. Then u

i n s such t h a t f o r every x c. There e x i s t s a sequence ( v ) n nElN family { v n ( x ) I n E IN] e s t a b l i s h e s a t o t a l s e t i n Yx. The v e c t o r f i e l d s belonging t o (

v

s

E

M the

a r e c a l l e d Borel v e c t o r f i e l d s . A sequence

~ i n )s, ~ which ~ s a~t i s f i e s Property ( c ) i s c a l l e d a fundamental sequence

of Borel vector f i e l d s . The following p r o p e r t i e s a r e essential.

110

v.

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

1 . 2 . Proposition.

Let (

v

~ be ) a fundamental ~ ~ ~ sequence of Borel v e c t o r f i e l d s . For a v e c t o r

field v

E

P t o be a Borel v e c t o r f i e l d it i s necessary and s u f f i c i e n t , t h a t

the function x V. 1.3.

( V ( X ) , V ~ ( X ) )i ~s

I+

a Borel f u n c t i o n f o r each n

IN.

Proposition.

Let M be a Borel space and l e t x t+ Y

M.

E

Let (

v

denote a f i e l d of H i l b e r t spaces over

~ be ) a sequence ~ ~ ~ of v e c t o r f i e l d s possessing t h e following

properties.

a. The f u n c t i o n s x n,m

( v n ( x ), v m ( x ))x, x E M a r e Borel f u n c t i o n s , where

I+

lN.

b. For every x

M t h e sequence ( vn ( x ) )nEIN i s t o t a l i n Y X ‘

t

Then t h e r e e x i s t s p r e c i s e l y one Borel f i e l d s t r u c t u r e

s

such t h a t v

E

s

f o r a l l n t IN.

Yx, x

L e t !.I denote a o - f i n i t e Borel measure. Again, l e t X I +

Borel f i e l d of H i l b e r t spaces with Borel f i e l d s t r u c t u r e v

E

s

i s s a i d t o be square u-integrable

i f t h e function x

s. I+

t

M, denote a

A Borel f i e l d

Ilv(x) 11,”

i s p-

i n t e g r a b l e . The square i n t e g r a b l e v e c t o r f i e l d s e s t a b l i s h a complex v e c t o r space K. For v,u

K, t h e function x

E

We put =

(v,U)K

I

*

(V(X),U(X))~,

x

E

M, i s u-integrable.

( v ( X ) , U ( X ) ) ~ dP(x)

M

and t h u s w e d e f i n e a s e s q u i l i n e a r form on

K.

With t h e s e s q u i l i n e a r form

t h e v e c t o r space K i s a semi-inner product space. Passing t o equivalence classes of Borel vector f i e l d s , we o b t a i n a H i l b e r t space Y. We

write

I

@

Y

=

Yxdp(x)

.

M

I f t h e space M i s a s t a n d a r d Borel space, i . e . i t s Borel s t r u c t u r e i s subo r d i n a t e t o a Polish space topology ( t h e topology of a s e p a r a b l e metric complete s p a c e ) , then Y i s a s e p a r a b l e H i l b e r t space. L e t t h e r e be given a countable number of H i l b e r t spaces xk, k c IN. Then

t h e i r d i r e c t sum

x

m

=

e~

k=l

xk

can be considered as t h e H i l b e r t space of se-

PRELIMINARIES AND SUMMARY

quences ( f l , f 2 ,

...)

where each f

k

xk

belongs t o

111

and where

m

k= 1

x,.

S t a r t i n g from t h i s p o i n t of view, each I I - I I denotes t h e norm i n k H i l b e r t space i s c a n o n i c a l l y embedded i n t o x i n t h e following way: t o

Here

xk

each f k

xk

E

i

we connect t h e sequence

k

(f ) = (0,

k

ik ( fk )

x

in

d e f i n e d by

...,O , f k I O , * * * ) rn

CB f can k=l k Thus we can

w i t h f k i n t h e k-th e n t r y of the sequence. Now the d i r e c t sum m

Z i ( f ) i n x. k=l k k from x onto They s a t i s f y t h e

be considered a s a norm converging s e r i e s introduce t h e orthogonal p r o j e c t i o n relations

n;ne

=

Ilkske,

k,e

E

nk

x,.

IN.

D i r e c t sums of H i l b e r t spaces a r e important i n r e p r e s e n t a t i o n theory, e.g. of compact groups and of c e r t a i n C*-algebras. However, i n most c a s e s t h e g e n e r a l concept o f d i r e c t i n t e g r a l of H i l b e r t spaces i s involved. The conc e p t of d i r e c t i n t e g r a l i s much more d e l i c a t e than i t s d i s c r e t e c o u n t e r p a r t , t h e concept of d i r e c t sum. Analogous t o t h e t r a n s i t i o n

e2 -> L,,

a number of

measure t h e o r e t i c a l problems arises. L e t t h e r e be given a Bore1 space M and a a - f i n i t e measure p on M. For each x

x

M, l e t

denote a H i l b e r t space. Under t h e given c o n d i t i o n s on t h e P = xeM n x t h e d i r e c t i n t e g r a l x = e dp ( x ) can be introduced. The d i r e c t i n t e g r a l x c o n s i s t s of equivalence c l a s s e s of E

l i n e a r space of v e c t o r f i e l d s

x

MIxx

square i n t e g r a b l e v e c t o r f i e l d s . Consequently we cannot p r o j e c t x

E

x

onto

x

,

M, i n a c o n s i s t e n t manner. Also, it i s impossible t o regard t h e i n t e g r a l

expression

where x

b+

w(x) denotes a square i n t e g r a b l e v e c t o r f i e l d , a s an o r d i n a r y

i n t e g r a l of an

x

valued function.

(The terminology w i l l be explained i n t h i s

chapter. 1

L e t (CGl,M,p,R,X)

for a l l w

E

R(X)

denote a Dirac b a s i s . L e t G

E

[ G I . Then i n a weak sense

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

112

1

w =

<w,Gx>

.

Gx d p ( x )

M I f we l e t

nx

:

R(X)

+

R-l(X) denote t h e l i n e a r o p e r a t o r

then w e can rewrite the i n t e g r a l expression i n t o

w

=

M Now l e t

I

llxwdu(x)

Yx = l l x R ( X ) .

Then

R-l(X) f o r a l l x

E

M.

Then f o r a l l v,w

E

R(X)

(nXw ,

.

Y

I n eachX

i s a one dimensional H i l b e r t subspace o f

Yx

w e introduce t h e i n n e r product

t h i s yields

n x v ) x = <w,G

>

I t follows t h a t t h e v e c t o r f i e l d x

w

E

R(X)

H

.

llX w i s square i n t e g r a b l e . So f o r a l l

t h e d i r e c t i n t e g r a l expression

r

w =

8

nxwdp(x)

M i s an ordinary i n t e g r a l of a weakly i n t e g r a b l e R-l

( X ) -valued

function.

One of t h e aims of t h i s chapter i s t o extend t h e above i d e a s by i n t r o d u c i n g a family (llx)xcM W i t h TIx : integral

To (llx)xcM we l i n k a d i r e c t

8

Each H i l b e r t space

nX R ( X ) .

R(X) + R-l(X), x EM.

Y X' x

E

M, i s a H i l b e r t subspace o f R - l ( X )

Conversely, f o r each d i r e c t i n t e g r a l

which c o n t a i n s

DIRECT RESOLUTIONS OF THE IDENTITY

113

t h e r e a r e p o s i t i v e bounded o p e r a t o r s R such t h a t f o r each x

space

x

is continuously embedded i n

embeddin:

from

j

x

(x) . The

i n t o R-l

R-'(X),

E

M t h e Hilbert

i . e . t h e r e e x i s t s a continuous

family of embeddings ( j x ) x t induces M

a family of generalized p r o j e c t i o n s (llx)xEM.F u r t h e r , f o r each w

E

R(X) the

d i r e c t i n t e g r a l expression

can a l s o be considered as a (weak) i n t e g r a l of an

R -1

(X)-valued f u n c t i o n ,

lB I

It may be clear t o the reader t h a t t h e embeddings

j X' x

E

M, a r e n o t

uniquely defined. Also i n t h i s case r e s t r i c t i o n s on t h e measure space

(M,v)

l e a d t o a "canonical choice". We o b t a i n r e s u l t s i n connection with t h e c l a s s i c a l Sobolev t r a c e lemmas and with r e p r e s e n t a t i o n theory of commutative von Neumann a l g e b r a s .

V.2.

D i r e c t r e s o l u t i o n s of t h e i d e n t i t y

Let R denote a p o s i t i v e bounded o p e r a t o r on t h e s e p a r a b l e H i l b e r t space In the Sobolev t r i p l e

R(X)

x

5

5

R-l(X)

x.

w e introduce t h e notion of d i r e c t

r e s o l u t i o n of t h e i d e n t i t y .

v.2.1.

Definition.

M l e t t h e r e be given

L e t (M,p) denote a o - f i n i t e measure space. For each x

E

a continuous l i n e a r mapping ll

The o p e r a t o r valued

X

function x

I+

ll

from R(X) i n t o R-l(X).

i s c a l l e d a d i r e c t r e s o l u t i o n of t h e i d e n t i t y i f it satis-

f i e s t h e following conditions. a. The o p e r a t o r s llx, x Kw,n

E

M, a r e nonnegative, i . e . f o r a l l w

E

R(X)

w> 2 0.

b. There e x i s t s an orthonormal b a s i s

(t) kc IN

in

x,

which is Schauder b a s i s

114

DIRECT RESOLUTIONS IN SOBOLEV TRIPLES

in R(X) such that for each k,e

E

IN the function x

I+

is p-

integrable with

A

ExampZe. If (CGI,M,p,R,X) is a Dirac basis, then for any G tion x (E(x)

H

G(x)

;(x)) (w) = <w,E(x)>

Q

nX

Let x i +

E

[GI the func-

G ( x ) is a direct resolution of the identity. Here

Q

E(x)

w

8

E

R(X) -

denote a direct resolution of the identity and let (

u

~

)

denote an orthonormal basis as indicated in Definition V.2.1.b. We take x

M

E

fixed for a moment. In the linear subspace llxR(X) of R-l (X) we intro-

duce the sesquilinear form

where F =

n

w and

G =

n

v.

by

The form ( * , - ) x is well defined because

(F,G)X

does not depend on the particular representation F = l lX w and G = il v. Indeed, let w,w'

n

v =

n

E

R(X)

and v,v'

E

R(X) such that n x w

=

nxw' and

v ' . Then we have <w,n v> = <w,n v'> X

and <w,n v> = cv,n w> =

v.2.2. Let x

%.

E

Lemma. M. Then

Let w

E

for all w

E

=

<Wl,n V> X

.

R(X) the following inequality is valid

R(X). Consider the following estimation

~

~

~

DIRECT RESOLUTIONS O F THE IDENTITY

=

cR2

n

w , T I x R 2 llxw>'

115

IIllxwIIx 2

X

CoroZZary.

v.2.3.

denote a sequence i n R ( X ) such t h a t (llx wn) n E m i s a Cauchy

L e t (wn)nEm

-

R ( X ) with r e s p e c t t o I1 I I x . Then t h e r e e x i s t s F

sequence i n llx

E

R-'

(x)

such t h a t l i m I I F - n x ~ n I I - l = 0 .

P r o o f . Thiz i s an immediate consequence of t h e completeness o f R-'(X)

and

0

of Lemma V.2.2.

Theorem.

v.2.4.

x

There e x i s t s a subspace completion of llx

Proof.

(x)

of

such t h a t

nx R ( X )

c

Let ( T I x ~ n ) n E m

-. Let F and

respectively.

and

Xx

is a

R(X).

(nXvn) n

and

E

denote ~ Cauchy sequences i n TIx

G denote t h e

R

-1

(x) - l i m i t s

R(X).

all w -n v 11 0 as x n x n x (nxwn) ~~m and (llx vn) n e m t

These Cauchy sequences a r e s a i d t o be e q u i v a l e n t i f n -+

xx,

of

-+

(Existency follows from t h e previous c o r o l l a r y . ) Then c l e a r l y

F = G. So t o each equivalence c l a s s of Cauchy sequences i n TI R ( X ) c o r r e -1 Now i n a s t a n d a r d way t h e v e c t o r sponds p r e c i s e l y one element of R

(x).

space of a l l l i m i t s i n p l e t i o n of l l x

R-'(X)

of Cauchy sequences i n llx

R ( X ) becomes a com-

0

RCX).

v. 2.5. Corol law. The canonical i n j e c t i o n j

Proof.

From Lemma V.2.3

x

:

x x c;R-l(X)

i s continuous.

we d e r i v e f o r a l l F

E

x n

m.

V.2.6. Let x

E

M.

a. The mapping TI Let

:l

:

xx

+

:

R-l(X)

R(X)

-+

xx

i s continuous

denote t h e dual of

n

. with r e s p e c t t o t h e d u a l i t i e s

116

X x'

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

N

Xx

R(X)'

and

b. For a l l w

E

R-l(X).

R(X) we

have t h e r e l a t i o n

=

n'xll

IIwII12

,

ll w

x

w.

Proof. a. Follows from t h e i n e q u a l i t y

llnxwII: b. Let w E

R(X).

(nx v ,

= <w, n X w >

For a l l v

nxw)

I t follows t h a t

E

=
IIRllxRII

5

R(X)

,n



=

w

E

R(X)

.

we have t h e r e l a t i o n

w>

.


,n

w>.

Remark. Because of t h e preceding lemma i t m a k e s s e n s e t o c a l l t h e o p e r a t o r s llx,

x

M,

E

generalized projections.

Next, we c o n s t r u c t a d i r e c t i n t e g r a l which i s a s s o c i a t e d t o t h e d i r e c t resolution

V.2.7.

(nx)xEM.

-. IN. The f u n c t i o n x

a. Let k , e

E

b. Let x

M. The sequence

E

H

(nXr,,nxul)x

(nx % ) k e r n

i s p-integrable.

is total i n

X X'

proos. a. The a s s e r t i o n follows from t h e e q u a l i t y

'nx\, b. Let G

E

x

for a l l k

,n

nxue'x

=
u l>

.

be such that

E

IN.

Recalling t h a t G

= (G , n

x

\)

x

= 0

E

R - l ( X ) w e have f o r a l l k

E

IN

.

Since ( u k ) k e m i s a Schauder b a s i s i n

R(X)

it follows t h a t G = 0 .

0

DIRECT RESOLUTIONS O F THE IDENTITY

117

Proposition V.1.3 y i e l d s t h e following r e s u l t .

v.2.8.

Theorem.

s,

There e x i s t s p r e c i s e l y one Borel f i e l d s t r u c t u r e

i n t h e f i e l d of Hilbert

x

spaces n such t h a t t h e v e c t o r f i e l d s x ll u k E I N , belong t o XEM x x k' Then a vector f i e l d v belongs t o s, i f f f o r a l l k E IN t h e function x t+

( v ( x ) ,nx \ ) x

i s a Borel function on

Corollary. L e t w E R ( X ) . Then t h e v e c t o r f i e l d x

sn'

M.

v.2.9.

field structure

prooS. Since in

X, w

$,

(\)kEIN

kEl

llX w , x

E

M, belongs t o t h e Borel

i s a Schauder b a s i s i n R ( X ) and an orthonormal b a s i s

m

=

I+

introduced i n Theorem V.2.8.

where convergence o f t h e series t a k e s p l a c e i n R ( X 1 .

(w,\)\

I t follows t h a t f o r each

i?

E

IN and x

M

E

m

x

Hence x

t,

(nXw, "xue)x

e

0

i s a Borel function.

N o w following Section V.l

the direct integral

i s well-defined. For each k

E

f i e l d . Let

['p,],

I N we p u t 'pk

k

E

IN,

: x I+

nx \ .

Each

So t h e l i n e a r o p e r a t o r V defined by m

1

k= 1

(f,Uk)

i s a square i n t e g r a b l e v e c t o r

denote t h e equivalence class of

Assumption V.2.1.b

Vf =

'pk

CPkl

,

f 6

x

['p

k

1

in

x.

Then by

118

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

i s an isometry from

For each w

X

i n t o X.

R ( X ) and each x

E

M, t h e s e r i e s

E

m

1

(W,Uk)

nx uk

=

nxw

k= 1 converges i n

x X'

I t follows t h a t t h e Borel vector f i e l d

x

H

n X w is a

r e p r e s e n t a t i v e of t h e c l a s s Vw.

The next theorem c o n t a i n s a summary of t h e r e s u l t s o f t h i s s e c t i o n .

v.z.10. Theorem. Let x I+

nX

denote a d i r e c t r e s o l u t i o n of t h e i d e n t i t y with r e s p e c t t o t h e

Sobolev t r i p l e R ( X ) c

x

c

I7-l ( x ) and t h e o - f i n i t e measure space (M,u).

For each x

(Cf. D e f i n i t i o n V . 2 . 1 . )

E

M t h e nondegenerate s e s q u i l i n e a r form

is d e f i n e d by ( l l x w , l v) = Cw , n v>, w,v E R ( X ) . Then x x t h e r e e x i s t s a subspace of R - l ( X ) which i s a comp:etion of ll R ( X ) with in nxR(X)

x

c

nxR(X)

xx.

Put

x

=

n

w belong t o

e Mj xx dpx(XI.

t h a t Vw = [ x

sn

Further, t h e r e e x i s t s a Borel f i e l d s t r u c t u r e

t h e f i e l d s x I+

I+

s,

f o r each w

E

Then t h e r e e x i s t s an isometry

nxwl, w

E

R(X).

such t h a t

R(X).

v

from

x

into

x

such

W e have t h e Plancherel formula

F i n a l l y , we p r e s e n t some expansion r e s u l t s .

v.2.11.

Corollary.

Let w E

R(X).

Then we can w r i t e

where convergence of t h e i n t e g r a l is i n t h e weak R v

E

R(X)

-1

(XI-sense,

i.e.

for all

A GENERAL CONSTRUCTION OF A DIRECT RESOLUTION

R

If

119

i s a Hilbert-Schmidt o p e r a t o r , then t h e following s t r o n g e r r e s u l t i s

valid.

v.2.12. Let

a.

R be a Hilbert-Schmidt operator.

w =

Let w

R ( X ) . Then we can w r i t e

E

nywdu(y)

M a s a strong R

-1

(X)-integral, i . e . the function y

H

Iln w II Y

-1

i s p-

integrable.

P r o o f . Let (vk)kEINdenote a n orthonormal b a s i s of eigenvectors of R with corresponding p o s i t i v e eigenvalues pk, k Since v k

E

R(X) for a l l k

E IN,

d p ( x )

IN.

E

"

2 pk <

Then we have k&

m.

we d e r i v e from t h e Plancherel formula

=

bkt

,

k,e

E

IN

.

M Therefore t h e Borel f u n c t i o n x

m

H

k&

p:


, nx vk>

1.5

U-integrable.

It

follows t h a t

M SO

I

f o r each w

(Rnx R )

trace

du(x) <

-.

R(X)

E

V.3. A general c o n s t r u c t i o n o f a d i r e c t r e s o l u t i o n Let 14 denote a s t a n d a r d Borel space ( c f . Section V.1) and l e t

(Yx)xEM

denote a Borel f i e l d of s e p a r a b l e H i l b e r t spaces. We consider a f i x e d uf i n i t e Borel measure integral Y =

Q

Yx

u

on M. As we have observed i n S e c t i o n V . l ,

the direct

dp(x) i s a separable H i l b e r t space. With t h e next d e f i -

n i t i o n we extend t h e notion o f Carleman o p e r a t o r .

v. 3 . 1. Definition. A l i n e a r operator

c

from Y i n t o Y i s c a l l e d a d-Carleman o p e r a t o r i f i n each

DIRECT RESOLUTIONS IN SOBOLEV TRIPLES

120

class CCqI in Y there exists a representative x n (CCql)

A

(x), x

E

M, such

that each mapping

is a continuous linear operator from Y into Y V.3.2.

.

=.

a. Let L

:

Y

-+

Y be a bounded linear operator, for which there exists an

orthonormal basis ([qk1)5Em a representative (LET,])

with the following property: there exists in each class LCq 1 such that for all x k

E

M

Then L is a d-Carleman operator. b. Let L

:

Y

+

Y be a Hilbert-Schmidt operator. Then L is a d-Carleman

operator.

Pro0 f. a. For each C1b1 E Y and for each x

Then the linear mapping [I)]

H

E

M we define

(x) from

(L[$])”

Y

into Y

is Hilbert-

Schmidt. So L is a d-Carleman operator. b. Let ([$k])kEm

? IILC$,III$

k=1

denote an orthonormal basis in Y. Then the series is convergent. Let (LCI) 1 ) - be a representative of LC$,I. k

Then we have

N

So there exists a null set N such that for all x

N

E

M\N the series

A GENERAL CONSTRUCTION OF A DIRECT RESOLUTION

i s convergent. Now w e d e f i n e f o r each k

Then t h e r e p r e s e n t a t i v e s

(LC$,1)

121

IN

E

u

s a t i s f y t h e requirements of P a r t ( a )

0

of t h i s theorem. Hence L i s a d-Carleman o p e r a t o r .

Theorem.

v.3.3.

Let R be a bounded p o s i t i v e d-Carleman o p e r a t o r on t h e d i r e c t i n t e g r a l @

Yx d p ( x ) . Then t h e r e e x i s t s a family of continuous e v a l u a t i o n s Y = (ix)xEM with ix: R ( Y ) + Yx. The following a s s e r t i o n s are v a l i d .

a. For each [w]

E

R ( Y ) t h e vector f i e l d x

I+

4

w is a r e p r e s e n t a t i v e of Cw].

b. There e x i s t s a n u l l set N such t h a t f o r a l l x dense i n

M\N t h e space i R(Y) i s

E

Y X'

Proof. a. For each

Cql

E

Y t h e r e e x i s t s a r e p r e s e n t a t i v e (RCq1)- of R [ q ] such t h a t

the l i n e a r operator C q l * each x

E

- (x)

(RCq])

M. Now t h e e v a l u a t i o n ix

where [w] = R C q ] , Cq1

E

Y. Then

i s continuous. F u r t h e r , X H

:

i s continuous from

R(Y)

+

Yx,

x

E

M.

Y into Y

for

i s defined by

i s continuous, because

i R

:

'f

-+

Yx

[w] i s a r e p r e s e n t a t i v e of [w].

b. L e t ( v . )

denote a fundamental sequence of square i n t e g r a b l e v e c t o r I jcm f i e l d s i n the Borel f i e l d s t r u c t u r e corresponding t o t h e d i r e c t i n t e g r a l

y. ( W e observe t h a t D e f i n i t i o n V.l.1.c

only provides a fundamental se-

quence of Borel v e c t o r f i e l d s . However, s t a r t i n g with such a fundamental sequence t h e e x i s t e n c e of a fundamental sequence of square i n t e g r a b l e v e c t o r f i e l d s can be proved by s t a n d a r d arguments.) L e t ( [ q k I l k E I N denote an orthonormal b a s i s i n

Y. Since R is i n j e c t i v e

t h e sequence (RCqkl)kcpl i s t o t a l i n Y.

E

Let j

IN. Then t h e r e e x i s t s a

I

( c w ~ l ) n E m i n t h e l i n e a r span < { R L q k 1 k E IN]> which tends t o Lv.1 i n Y. Taking a subsequence if necessary it follows t h a t t h e r e I e x i s t s a n u l l s e t N . such t h a t f o r a l l x E M \ N . t h e sequence ( i x r w ; l ) 3 3 nf IN

sequence

122

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

converges t o v . ( x ) i n 1 t h e sequence ( v1, ( x ) ),

<{ (R[g,])w

(x)

I

k

YX'

NOW

we p u t

N

=

jzmNj.

i s contained i n t h e

Then f o r a l l x

Y X -closure

W]>. So t h e l a t t e r c l o s u r e e q u a l s

E

E

M\N

of

Yx f o r a l l 0

x E M\N.

R > 0 be a bounded d-Carleman o p e r a t o r on Y.

(ix)xEM denote a corresponding family of continuous e v a l u a t i o n s , ix : R ( Y ) + Yx. Since R-l (Y) i s a r e p r e s e n t a t i o n of t h e s t r o n g dual of R ( Y ) , t h e dual mapping it of A.X' x E M, with r e s p e c t t o t h i s d u a l i t y , i s a continuous l i n e a r Let

o i e r a t o r from Y

Let

i n t o R-l(Y). Following Theorem V.3.3.b

there exists a null

N such t h a t f o r a l l x E M\N t h e subspace i X R ( Y ) i s dense i n Yx. So t h e embedding i h : Yx -t R-'(y) i s i n j e c t i v e f o r a l l x E M \ N .

set

Let x

M\N.

E

Put

zx

= i;(Yx).

Then

zX

i s a H i l b e r t space endowed with t h e

i n n e r product

Because o f t h e i n e q u a l i t y

zx

t h e space

i s a genuine H i l b e r t subspace o f R-'(Y),

space topology of

R-' (Y)

.

z

i.e. t h e Hilbert

is s t r o n g e r than t h e topology on

Theorem. R > 0 denote a bounded d-Carleman o p e r a t o r on Y.

z

induced by

v.3.4. Let

an a s s o c i a t e d family o f continuous e v a l u a t i o n s ,

i:

:

Yx

nx Then x

H

l"l

:

Let

R(Y)

(ix)xEM denote +

Yx with dual

For each x E M we d e f i n e

R-'(Y).

+

ix

= i x' ix

.

i s a d i r e c t r e s o l u t i o n o f t h e i d e n t i t y i n t h e sense of Defini-

t i o n V. 2 . 1 . Also,

t h e r e e x i s t s a n u l l set N c

H i l b e r t space Cw1,CvI E

R(Y)

zx

= i:(Yx),

we have

M such t h a t nxR(y) i s dense i n t h e

which i s a H i l b e r t subspace o f R-l(Y). For all

CONSTRUCTION OF A DIRECT RESOLUTION

A GENERAL

<[w]

,n

CVl>

(ixCwl

=

,i x c v l ) y

=

(nx[wl , n x C v l ) z

X

Pro f. -

show only t h a t x

Sobolev t r i p l e

R(Y)

_c

Y

I+

R-'

-C

n

(v")

R(y). Let k , e

Hence x

*
<[w]

E

-

X

i s a d i r e c t r e s o l u t i o n with r e s p e c t t o t h e and t h e a - f i n i t e measure space (M,u).

L e t ( [ q k l ) k E m denote an orthonormal b a s i s i n

in

123

Y which i s a Schauder b a s i s

IN. Then w e have

,nxcqel> i s p-integrable with

,n

[wl>

2 0

I l i x C W l l l Y2

=

-

X

The o t h e r a s s e r t i o n s simply follow from t h e above c o n s i d e r a t i o n s .

0

Remark. I n l i n e with Chapter 11 one could t h i n k of an i n t r o d u c t i o n o f dCarlaman type o p e r a t o r s and s t r o n g d-Carleman o p e r a t o r s i n t h e s e t t i n g of d i r e c t i n t e g r a l s of H i l b e r t spaces. We l e a v e t h i s t o c r e a t i v e r e a d e r s o r s p a r e it f o r f u t u r e t i m e .

v.3.5.

An elementaw i l l u s t r a t i o n . 2

The H i l b e r t space L 2 (IR ) can be considered a s a d i r e c t i n t e g r a l of H i l b e r t spaces i n t h e following way

2 L 2 ( l R ,dx dx 1 2

=

M

r

L2(lR,dx2) dxl

. 2

L e t IF1 denote t h e s i n g l e F o u r i e r transform on L 2 ( l R ) ,

a l l y given by

(IFl0) ( x 1 , x 2 ) =

1 -

Jz;;,

eiXly1 0 ( y 1 , x 2 ) dyl

.

which i s h e u r i s t i c -

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

124

2

L2 (IR 1.

The o p e r a t o r IFl i s u n i t a r y on Let

P denote t h e s e l f - a d j o i n t d i f f e r e n t i a l o p e r a t o r i

put

Rm

1

= 2

(1 +P:)-m/2

on L 2 ( I R ) . and p u t

)

.

We

5. Then Rm i s a p o s i t i v e bounded o p e r a t o r 2 Rm i s a d-Carleman o p e r a t o r . L e t C41 E L 2 ( I R )

RmC@l. Let x1

=

2

L2 (IR

where m >

W e prove t h a t

CnJ

a in 3x1

IR. Then we g e t t h e e s t i m a t i o n

E

It follows, t h a t f o r each x1

E

IR t h e function

i s square i n t e g r a b l e and

with Cm =

1

J2n

dY1.

Thus w e have shown, t h a t f o r each

[@I

E

2

L2(IR )

there exists a representative

(RmC@l)" i n R C41 and corresponding l i n e a r mappings i + L2(IR),

x1

m

E

X1

IR, such t h a t

i x RmC@l

=

Cx,

+

(R,L@I)

1

-

(x1,x2)1

E

L2(IR)

2

: Rm(L2(IR ) ) +

.

A l s o , w e have t h e e s t i m a t i o n

I I i x RmC@l II L2(IR) 1

5

Cm i l l C ~ l l l L 2 ~ m ~[ @ ~ I E L 2 ( W2 )

CANONICAL DIRECT RESOLUTIONS

which y i e l d s c o n t i n u i t y of t h e e v a l u a t i o n s ix 1 Hence t h e o p e r a t o r Rm i s a d-Carleman o p e r a t o r . L e t [$I

E

L2(IR)

and l e t x

2

R i l ( L 2 (IR

) ) +

L, (IR).

So t h e r e e x i s t s an element E ( [ $ ] , x , )

of

with t h e p r o p e r t y

))

Cnl

E

For each x1

E

for all

2

Rm(L2 (IR

IR. The l i n e a r f u n c t i o n a l

E

i s continuous on R ( L 2 ( I R ) ) . m 2

:

125

R m ( L 2 ( I R R L ) ) . Thus we d e r i v e f o r each x

M t h e l i n e a r space { E ( [ + l , x l )

subspace of R

-1

m

2 (L2(IR ) )

I [$I

E

1

E

M

L2(IR)

1

is a H i l b e r t

with i n n e r product

H e u r i s t i c a l l y we can w r i t e

and {E([$l,xl)

I [$I

E

L2(IR)} = L 2 ( I R )

0

<6x >

.

1

Here IS

denotes t h e Dirac d e l t a a t x x1

V.4.

1’

Equivalence c l a s s e s of d i r e c t r e s o l u t i o n s , canonical d i r e c t r e s o l u t i o n s

L e t (M,p) denote a Federer measure space and l e t ( Y ) denote a Bore1 x X€M CB Y dp(x). f i e l d of H i l b e r t spaces. We consider t h e d i r e c t i n t e g r a l Y = M x v ( x ) dp(x) is not an o r d i n a r y For each element c v l E Y t h e expression

I

i n t e g r a l expression a t a l l . For x the f i b e r s Y

and Y

# x2 t h e v e c t o r s v ( x ) and v ( x ) are i n 1 2 So a d i f f e r e n t i a t i o n theorem f o r d i r e c t i n t e g r a l s

X2’ x1 can c e r t a i n l y n o t e x i s t , i . e . t h e expression

126

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

has no meaning. Let

R denote a p o s i t i v e bounded d-Carleman o p e r a t o r on Y. Then t h e r e e x i s t

continuous e v a l u a t i o n s

i

:

R(Y)

+

Yx,

x

M, such t h a t f o r each [rql

E

E

Y

i R C q l is a r e p r e s e n t a t i v e of RCrql. For almost a l l x E M t h e i' i s a l i n e a r isomorphism from Y o n t o z where zx i s a H i l -1 subspace of R ( Y ) . Furthermore, x w llx = i' ix,x E M i s a d i r e c t

the f i e l d x

H

dual mapping bert

r e s o l u t i o n of t h e i d e n t i t y . The d i r e c t i n t e g r a l e x p r e s s i o n

e

1

Cvl =

ixCvl du(x)

,

Cvl

E

R(Y)

M can t h e r e f o r e be considered a s a weak i n t e g r a l of t h e R-l (Y)-valued function x w

if i Cvl x x [vl =

=

n

J

nxCvl du(x)

E

R ( Y ) , ( C W I , C V I=) ~

Cvl,

M ( i . e . f o r a l l Cwl

Therefore one could say t h a t ll

"projects"

d u ( x ) ) . RtY) into

zx.

Further, the

l i m i t expression

makes s e n s e ; it i s t h e d i f f e r e n t i a t i o n r e l a t i v e t o

u

of an R-l(Y)-valued

measure. In t h i s s e c t i o n w e prove a r e l a t i v e d i f f e r e n t i a t i o n theorem as i n d i c a t e d , l e a d i n g t o t h e notion o f canonical d i r e c t r e s o l u t i o n .

v.4.1. Let Let

Definition.

R denote a bounded p o s i t i v e o p e r a t o r on t h e s e p a r a b l e H i l b e r t space (M,u) denote a o - f i n i t e measure space. L e t ll'l) and denote two

x.

d i r e c t r e s o l u t i o n s o f t h e i d e n t i t y with r e s p e c t t o t h e t r i p l e

R(X) c X

5 R-l

(x)

and the measure space ( M , p ) . Then l l ( ' )

and ll")

are c a l l e d

127

CANONICAL DIRECT RESOLUTIONS

e q u i v a l e n t , i f t h e r e i s a n u l l s e t N such t h a t =

n'l'

x

X

x

'

.

M\N

E

we denote t h e equivalence c l a s s of a d i r e c t r e s o l u t i o n ll by ([ll],M,u,R,X). By abuse of terminology also ([lll,M,p,R,X)

w i l l be c a l l e d a d i r e c t resolu-

t i o n of t h e i d e n t i t y . W e show, t h a t t o each equivalence c l a s s ([nl,M,p,R,X)

one d i r e c t i n t e g r a l of H i l b e r t spaces. L e t I l ( l )

(xi1) ) x E M

and

( x i 2 ) ) x E M denote

E

corresponds p r e c i s e l y

[n]

and 11(2)

which i s obtained by zompleting

r e s p e c t t o t h e inner product
1 xi1)

llLJ) w>,

,

v,w

e

x(')

=

,

Since ll'" X

=

and l ) '

denotes t h e

llLJ) R ( X )

with

R ( X ) . Now p u t

E

I

0

dp(x)

,

x(2) =

M

ll")

[IT]. Let

t h e Borel f i e l d s of H i l b e r t spaces a s s o c i a t e d

t o n ( l ) and 11(2), r e s p e c t i v e l y . W e r e c a l l t h a t x ( 1 ) , j = 1,2 H i l b e r t subspace of R-'(X)

E

xi2)

.

dp(x)

M

a r e e q u i v a l e n t t h e r e e x i s t s a n u l l s e t N such t h a t

11(2) f o r a l l x

E

M\N.

I t follows t h a t

So r e c a l l i n g t h e d e f i n i t i o n of d i r e c t i n t e g r a l of H i l b e r t spaces w e can

i d e n t i f y X ( l ) and X ( 2 ) .

Consequently, each c l a s s (Cnl,M,p,R,X)

t o p r e c i s e l y one H i l b e r t space

x.

corresponds

F i n a l l y , we note t h a t

M for all w

E

R(X).

I n t h e remaining p a r t of t h i s s e c t i o n we assume t h a t (M,p) measure space ( c f . D e f i n i t i o n 1 1 . 3 . 1 ) . Let (CnI,M,p,R,X) r e s o l u t i o n of t h e i d e n t i t y and l e t

fi

E

i s a Federer

denote a d i r e c t

[Ill. W e impose t h e following c o n d i t i o n

V.4.2.

The Borel f u n c t i o n x

I+

I I R l l x RIIX+X i s i n t e g r a b l e on bounded Borel s e t s of M.

128

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

Let (vkIkcm denote an orthonormal b a s i s i n

x

which i s a Schauder b a s i s i n

R ( X ) . The Bore1 f u n c t i o n s

a r e i n t e g r a b l e on a n u l l set

N

M.

So t h e r e e x i s t Bore1 f u n c t i o n s

such t h a t f o r a l l x

E

'pU,

K,

k,e

E

IN,

and

M\N

v.4.3.

and

For convenience, we p u t rpke(x) = 0 , K(x) = 0 f o r x We note t h a t t h e d e f i n i t i o n of t h e f u n c t i o n s choice of t h e r e p r e s e n t a t i v e

,.

'pU

E

N.

does not depend on t h e

n.

The following r e l a t i o n s are valid.

For each x <{vk I k

E

E

M we d e f i n e a s e s q u i l i n e a r form

IN]>

Then f o r all u

by

E

<{v k

sx(u,u) = 0 and

Ik

E

IN}> if x

E

N ,

s

on t h e l i n e a r span

129

CANONICAL DIRECT RTSOLUTIONS

f o r a l l x E M\N. all x E

I t follows t h a t s ( u , u ) 2 0 f o r a l l u

M. F u r t h e r , f o r a l l x

<{vk

E

Ik

IN}> and

E

M\N

E

Therefore t h e q u a d r a t i c form U P s ( u , ~ ) u,

x,

E

i s bounded on t h e dense

I

k E IN}> f o r each x E M. Consequently, f o r each l i n e a r subspace <{v k M t h e r e e x i s t s a nonnegative bounded o p e r a t o r 8 such t h a t I18 I1 5 K ( x ) ,

x E

and s

( f . 9 ) = (f

The o p e r a t o r 8

=

, Bxg)

0 for x

E

,

f E

N; f o r a l l k,e

N

we introduce the l i n e a r mapping Ilx

v.4.4.

Rn” w

=

BxR-’w

X I

,

w

E

:

R(X)

R(X)

E

+

g

E

X

.

IN and a l l x

R-l(X). x

X’

t h e mappings TI

M,

M, by

E

.

-

N

Since R n ” R = 8

E

a r e continuous. Also, x t+ Ilx, x

a r e p r e s e n t a t i v e of t h e c l a s s [Ill, because f o r a l l k , e

E

X E M N

( R I l x R v k , ve) = (pkL(x) = (RnxRvk ,v,) Because of Condition V.4.2

for a l l w E R(X).

So by

f o r a l l r > 0 and a l l x

. E

M

E

M,

is

IN and almost a l l

w e have

130

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

we denote a well-defined continuous l i n e a r mapping

-nx(r) from R ( X )

into

R-' ( X ) .

-

Now we a r e well prepared f o r the main theorem of t h i s s e c t i o n , which indicates t h a t x

v.4.6. Let

(M,!.I)

Tl

t+

i s a kind of canonical r e p r e s e n t a t i v e i n t h e c l a s s

Theorem.

R denote a bounded p o s i t i v e o p e r a t o r i n a s e p a r a b l e H i l b e r t space x, l e t denote a Federer measure space and l e t ( L l l l , M , u , R , X ) denote a d i r e c t

r e s o l u t i o n of t h e i d e n t i t y which s a t i s f i e s Condition V . 4 . 2 .

^.

e x i s t s a (canonical) r e p r e s e n t a t i v e

n

N

c

property: There e x i s t s a n u l l s e t v,w

E

Then t h e r e

E [Ill, which p o s s e s s e s t h e following M such t h a t f o r a l l x E M\N and a l l

R(X),

Proof. - W e use t h e n o t a t i o n s and d e f i n i t i o n s a s introduced i n V.4.3-5. f o r each L

E

L

x

E

c

e=1

,

(f,vt)vt

f

E

E

L by

x .

N denote t h e n u l l s e t presented i n V.4.3. M\N. L e t E > 0 .

F i r s t we take L

Also,

P

IN we i n t r o d u c e t h e orthogonal p r o j e c t i o n

PLf =

Let

[n].

Let v,w

R ( X ) and l e t

E

IN so l a r g e and f i x e d , t h a t

II ( 7 - PL) R - l wII <

E

and

II ( I - PL) R-'

vII <

.

E

Then we t a k e ro > 0 so small t h a t f o r a l l r , 0 < r < ro, 1

and a l s o u(B(x,r))-'(

B (x,r) We p u t f = R - l v and g = R-'

I

w.

IIRfi RII Y

1

du(y)

< K(x)

+ 1

5

k,t s L ,

.

Consider t h e following s t r a i g h t f o r w a r d

estimation I(f ,RixRgIX

-

p(B(x,r))-'(f

,RE

X

(r)Rg)l

5

I+II+III+IV+V,

CANONICAL DIRECT RESOLUTIONS

s

E

IIf IIx Ilgllx

,

-

5 E IIg llXltRRx

RttX+X

,

131

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

132

Now adding I - V y i e l d s t h a t f o r a l l r with 0 < r < ro

where

c

= IIvII1 IlwII 1

+ (Ilvll, + I I W I I , ) ( I I R ~ ~ R +~ I K ( x ) + l )

. 0

Thus t h e r e s u l t follows.

-

Remarks.

a. I f t h e dimension of each v e c t o r space ll

R(X)

i s a t most 1 , then Theorem

V.4.6 i s a s p e c i a l form of t h e measure t h e o r e t i c a l Sobolev lemma pres e n t e d i n Chapter 11. b. I f t h e support of t h e a - f i n i t e measure p c o n s i s t s of atoms o n l y , then the d i r e c t integral

-

i s i n f a c t a d i r e c t sum of H i l b e r t spaces. The corresponding d i r e c t

r e s o l u t i o n tClll,M,p,R,X) by

n

=P.

x<Mj, j e m

7 '

X

c o n t a i n s t h e canonical r e p r e s e n t a t i v e ll defined

where t h e M . denote t h e mutually d i s j o i n t p-atoms w i t h 7

where each

~~j xx

P 3.

denotes t h e orthogonal p r o j e c t i o n from

du(x) = X . .

0

jem

V.5.

m

=

." M j

j=l

and

onto

Here we can t a k e R = 1. Observe t h a t

7

x=

x

M

x.. J

Generalized e i g e n p r o j e c t i o n s r e l a t e d t o commutative von Neumann algebras

L e t t h e r e be given a d i r e c t r e s o l u t i o n

-

(Cnl,M,p,R,X).

I f t h e m e a s u r e space

(M,u) i s a Federer measure space, t h e n t h e r e e x i s t s a canonical representative x

H

ll

i n t h e c l a s s Ell].

-I n t h e previous

s e c t i o n we have c o n s t r u c t e d

t h i s canonical r e p r e s e n t a t i v e ll. In t h i s s e c t i o n t h e approach i s d i f f e r e n t . A a - a d d i t i v e p r o j e c t i o n valued measure

P

on a t o p o l o g i c a l Bore1 space M l i e s

133

GENERALIZED EIGENPROJECTIONS

a t t h e b a s i s of our c o n s i d e r a t i o n s . For i n s t a n c e ,

P can be t h e s p e c t r a l

measure of one s e l f - a d j o i n t o p e r a t o r o r of an n-set of mutually commuting s e l f - a d j o i n t o p e r a t o r s . Under c e r t a i n c o n d i t i o n s , i t i s p o s s i b l e t o cons t r u c t a canonical Radon-Nikodym d e r i v a t i v e

t o a s u i t a b l e Borel measure p on

M.

n

:

x

, x

H

E

M,

with r e s p e c t

The family ( l l x ) x E M i s a canonical d i r e c t

r e s o l u t i o n . Each ll i s a generalized e i g e n p r o j e c t i o n of t h e commutative von Neumann algebra generated by t h e p r o j e c t i o n s

P ( A ) where A

c

M

is a Borel set.

I n t h i s s e c t i o n M denotes a o-compact l o c a l l y compact (and hence s e p a r a b l e ) m e t r i c space. A s observed i n Section 1 1 . 3 on

M

any o - f i n i t e Borel measure

M,

i s r e g u l a r . L e t P denote a o-additive p r o j e c t i o n valued s e t f u n c t i o n on i.e.

-

For each Borel s e t A c

M, P ( A )

separable H i l b e r t space

-

i s an orthogonal p r o j e c t i o n on t h e

x. m

For A t h e d i s j o i n t union m

m

Z j=l

P(A.) J

=

I, i f

jLl

,'J

3=1

Aj =

A . we have P ( A ) = j & P ( A j ) . 3

M.)

W e take a p o s i t i v e bounded o p e r a t o r

R

with t h e p r o p e r t y , t h a t f o r each

bounded Borel s e t A t h e p o s i t i v e o p e r a t o r p o s i t i v e Hilbert-Schmidt o p e r a t o r Borel measure p on o-finite.

M by p ( A )

(In particular,

R

R P(A) R

i s t r a c e c l a s s . E.g.

any

s a t i s f i e s t h i s condition. W e define the

= trace(R

M

C l e a r l y , each Borel s e t of

P ( A ) R ) . The measure space ( M , p ) i s has f i n i t e p-measure

and p i s a

r e g u l a r Borel measure. I n a d d i t i o n , we assume t h a t ( M , p ) is a Federer meas u r e space. There e x i s t c o n d i t i o n s on t h e m e t r i c space

M

such t h a t ( M , p ) i s

a Federer measure space f o r o - f i n i t e r e g u l a r Borel measures p with t h e p r o p e r t y t h a t bounded Borel sets of

M

have f i n i t e p-measure.

In Chapter 11,

w e presented t h e following examples:

-

f i n i t e dimensional v e c t o r spaces with metric d induced by some norm

-

Riemannian manifolds (of c l a s s 2 2 ) with t h e i r usual metric.

L e t ( v k ) k c m denote an orthononnal b a s i s i n

x.

Then f o r each k,e

E

IN we

d e f i n e t h e a d d i t i v e set f u n c t i o n

The s e t f u n c t i o n s Qke a r e a b s o l u t e l y continuous with r e s p e c t t o p , i . e .

V.

134

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

P(A) = 0

c*

a k ~ ( A ) = 0. So by t h e d e f i n i t i o n of Federer measure space and

t h e theorem of Radon-Nikodym t h e r e e x i s t s a p-null set N' and Borel functions $

kL

such t h a t f o r a l l x

For convenience we p u t

and a l l k , e

= 0 for a l l x

E

N1

E

lN

and k , L

E

IN.

m.

v.5.1. Let x

Gk.e(x)

M\N'

E

E

Proof.

M\N'

and l e t k , L

E

IN.

Then w e have

1

For a l l r > 0 w e have I Q k L ( B ( x , r ) ) 2 5 a k k ( B ( x , r ) ) Q U ( B ( x , r ) ) . I t

follows t h a t

The f u n c t i o n s

Gkk

are nonnegative. For each Borel s e t A w e have

Since (M,p) i s a Federer measure space, t h e r e i s a n u l l s e t N that for all x

V.5.2. Let x E

E

M\N

Corollary

M\N.

Then t h e s e r i e s

k,i=l IGlCe(x)I2is convergent. m

3

N ' such

135

GENERALIZED EIGENPROJECTIONS

Proof.

The a s s e r t i o n i s a consequence of t h e following e s t i m a t i o n :

v. 5.3. Definition. The o p e r a t o r s

B

:

X

x

-t

x,

x

M, a r e d e f i n e d by

m

Bx f =

k,j=l

'ke(x)(f'v

) vk ,

f e X , X E M \ N .

we observe t h a t 8 i s a Hilbert-Schmidt o p e r a t o r f o r each x The l i n e a r o p e r a t o r s

Here

are related t o

P

M.

a s follows.

e.

v.5.4. Let x

8

E

E

M\N. Then

111 111 denotes t h e Hilbert-Schmidt norm.

Proof.

For a l l r > 0 we have

L e t E > 0 and t a k e a f i x e d A E IN s o l a r g e t h a t m

k=A+ 1 Next, t a k e ro > 0 so small t h a t f o r a l l r , 0 < r < r o , and a l l k , t with k,k? s A

E

IN

136

D I R E C T RESOLUTIONS I N SOBOLEV TRIPLES

and, moreover,

W e g e t t h e following e s t i m a t i o n

By assumption ( * ) we have

: i

k=A+l c=1 and by assumption ( * *

w e have

T E

k=A+1 e=1

Thus i t follows t h a t f o r a l l r with 0 < r < ro

The p r o j e c t i o n valued o-additive s e t function t a t i o n of t h e i n v o l u t i v e Banach a l g e b r a L,(M,p). d e f i n e t h e f i n i t e Bore1 measure

It i s clear t h a t

u

fr

9

on

M

P

g i v e s rise t o a *-represenIndeed, l e t f , g

x

and

by

d p f , g = ( f , g I X . Moreover, t h e measure p

continuous w i t h r e s p e c t t o p .

E

frg

i s absolutely

GENERALIZED EIGENPROJECTIONS

L e t 0 denote a Borel f u n c t i o n on

The o p e r a t o r L ( 0 ) i n

x

M

137

which i s bounded on bounded Borel s e t s .

i s defined by

Ws observe t h a t L ( 0 ) i s a normal o p e r a t o r i n

x.

Since 0 i s bounded on

bounded Borel sets, the following i n e q u a l i t y i s v a l i d f o r each r > 0 , x

M

I

xB(x,r) dpf,f

E

M

'

I n p a r t i c u l a r i t follows t h a t R L ( 0 ) P ( B ( x , r ) ) R i s a t r a c e c l a s s o p e r a t o r on X.

v.5.5.

Theorem.

For each

[el

E

L,(M,p)

we d e f i n e t h e normal o p e r a t o r

JCOI

on X by

1

where any 0 i n [ O ] can be taken. The l i n e a r mapping

i s a*-representation ofthe i n v o l u t i v e Banach algebra L,(M,p) t a t i v e von Neumann algebra of bounded o p e r a t o r s on

Proof.

The proof i s a c o m p i l a t i o n of a number of s t a n d a r d arguments. Hence

0

it i s omitted.

V.5.6.

o n t o a comu-

x.

e.

Let 0 be a Borel function on

M

t h e r e e x i s t s a p-null s e t N,

such t h a t f o r a l l x

which i s bounded on bounded Borel sets. Then E

M\NO,

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

138

Proof.

Because of Lemma V.5.4 we a r e ready, i f w e can prove t h a t t h e r e

exists a null

N,

3

N such t h a t f o r

all x

M\N

E

0

Therefore we e s t i m a t e as follows

2

,,j=,

-

I ? & ( x ) I 5 (kgl ~ , ( x ) ) 2 = 1 f o r all x c M\N. Now indeed t h e r e e x i s t s a n u l l s e t N 3 N such t h a t f o r a l l x E M\N 0 We note t h a t

expression

j

p(B(x,r))-'(

IO(x) - o ( y )

the

1

B(xlr) tends t o zero as r Lemma V.5.1-4,6

+

0.

c o n t a i n s t h e a u x i l i a r y r e s u l t s which we need i n t h e con-

s t r u c t i o n of generalized e i g e n p r o j e c t i o n s a s announced. Their d e f i n i t i o n i s as follows.

V. 5.7. Definition. Let x c M. The o p e r a t o r

Rn" w W e observe t h a t

=

-

nX :

R(X)

,

w

BxR-'w

R n" R = B

X'

-+

R-'(X)

E

R(X)

i s defined by

.

m

whence

i s continuous.

GENERALIZED EIGENPROJECTIONS

First we prove that the family

(fix) XEM

139

is a canonical direct resolution.

Theorem.

v.5.8.

Let v,w

E

R(X). Then the Borel function x

w

I+

is p-integrable and

M Moreover, for all x

Proof.

M\N

E

Let A be a Borel set. Then from Tonelli's theorem and from the esti-

mation

m

5

(k,L=l 1

IvkL(y) 1')

4

IIR-'vllX IIR-'wIIX

,

Y

E

M ,

we derive m w

w> dp(y) =

A

1

k,l=1

QkL(A)

-1

(R

v , vl) (vk,R-'w)

=

In particular it follows that

Let x

E

M\N. Then the second assertion follows from Lemma V.5.4 and the

relations


,n N

w> =

((

R-lv

@

R-'w,

RfiX R

)) =

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

140

The interchange (*) of summation and i n t e g r a t i o n i s permitted, because of

,

t h e r e s u l t ( + ) . By ( (

))

we denote t h e i n n e r product i n t h e H i l b e r t space

of Hilbert-Schmidt o p e r a t o r s ,

(( A , B )) = t r a c e (B*A)

t

111 All12

= trace(A*A)

.

0

-

From t h e previous theorem we d e r i v e t h a t t h e family (TI ) i s a canonical x xeM d i r e c t r e s o l u t i o n . L e t x E M. A s i n Section V.2 w e d e f i n e t h e non-degenerate N

i n t h e v e c t o r space

s e s q u i l i n e a r form ( - , - ) ,

(F,G)x =
where F =

xx

n

-

,nx w>

-

R ( X ) w2th r e s p e c t t o t h e form

c R-l(X)Xof

xx.

by

v and G = ll w.By Theorem V.2.6 t h e r e e x i s t s a completion

subspace of R-'(;). into

nxR(X)

I n t h e f i r s t p a r t of t h i s s e c t i o n w e

-

So

x

i s a Hilbert

nx projectsXR(X) densely show t h a t t h e o p e r a t o r s n

As we have seen t h e o p e r a t o r

-

can be regarded a s g e n e r a l i z e d e i g e n p r o j e c t i o n s and t h e H i l b e r t spaces

x

a s g e n e r a l i z e d eigenspaces.

v.5.9. Let

Theorem.

(3 :

s e t s of

M

+ C denote a Borel f u n c t i o n , which i s bounded on bounded Borel

M.

Let

'((3)

t o Theorem V.5.5. and a l l w

Proof.

E

denote t h e normal o p e r a t o r i n Then t h e r e i s a n u l l set N

(3

x

as d e f i n e d previously

such t h a t f o r a l l x

E

M\N 0

R(X)

By Lemma V.5.6

there exists a null s e t

N (3 such t h a t f o r a l l x

E

M\N,

GENERALIZED EIGENPROJECTIONS

Let x

M\N

E

and l e t w

E

141

R ( X ) . Then t h e following i n e q u a l i t y y i e l d s t h e

wanted r e s u l t

v.5.10.

Theorem.

Let 8 :

M

C denote a Bore1 function which i s bounded on bounded Borel

+.

sets. Suppose t h e o p e r a t o r L (0) i s c l o s a b l e i n

R- 1

(x)

No

such t h a t f o r a l l x

on

M with

Then t h e r e e x i s t s a n u l l s e t

E

-

with c l o s u r e L ( 0 ) .

M\N,

and a l l G

E

xx

w e have

Proof.

Taking t h e function

NE such t h a t f o r a l l x

E

E

N

Thus it follows t h a t f o r a l l x

Let x in

xx.

-

E

M\N,.

=

O(x)fix w

The embedding

I t follows t h a t each

3

u

E

M, we g e t a n u l l set

NE such t h a t f o r a l l x

E

M\NO

M\N 0

E

.

-

xx + R - ' ( X ) i s G E xx belongs

continuous and 11 R ( X ) i s dense xt o t h e domain of L ( 0 ) and

0

L ( 0 ) , = 0 ( x ) G.

Let (

1, x

M\N

Also, there e x i s t s a n u l l s e t

-L ( 0 ) TIx w

=

E(X)

~ denote ) ~ an ~orthonormal ~ basis i n

x

which i s a l s o a Schauder b a s i s

N

in

R ( X ) . Then f o r each

Section V . 2 . )

x

The family

M t h e sequence (llx uk) k E is~ t o t a l i n Xx. (Cf. (Xx)xEM e s t a b l i s h e s a B o r e l f i e l d of H i l b e r t spaces.

E

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

142

We p u t

0

X

xx d p ( x ) .

=

M

x

Then

-

i s a s e p a r a b l e H i l b e r t space because of t h e c o n d i t i o n s on t h e measure

The Bore1 v e c t o r f i e l d s pk

space ( M , p ) .

:

([pkl)kEm in

b l i s h an orthonormal system

x H

x.

nx \ ,

x

E

M, k

m,

E

The isometry U from

x

estainto

x

i s defined by

For each f , g

E

x

we have t h e following r e l a t i o n s

(f,g)x =

M

1

I

duf,g =

( ( U f ) * ( y ) , ( U g ) - ( y ) ) yd p ( y )

I t follows from our previous d e f i n i t i o n s t h a t f o r a l l f , g

lo1

E

d

x

and a l l

L_(M,p)

Hence we can w r i t e f o r a l l f

U J(CO1) w =

M!

x

E

and, i n p a r t i c u l a r , f o r all w

V.5.11.

.

M

E

R(X)

O(y) fi w d p ( y ) Y

.

Lemma.

The l i n e a r o p e r a t o r

:

x

+

x

is unitary.

Proof. We have t o prove t h a t t h e orthonormal x. Therefore, it is s u f f i c i e n t t o prove t h a t

system ( [ p k l ) k E m t h e set

is total in

GENERALIZED EIGENPROJECTIONS

143

is t o t a l i n X.

So suppose t h e r e e x i s t s a square i n t e g r a b l e v e c t o r f i e l d cp with t h e property t h a t for a l l k

-N

SN and a l l 101

E

Then it follows t h a t ( q ( y ) zero. Taking

m

=

-

E

L_(M,p)

, nYXY )

-

vanishes except on a s e t N,

v ~ ~ ~ : v( q (~y ) ,~nY ~uk Y, =N0 *

Since

(n

v.5.12.

i s a t o t a l sequence i n

of L,(M,p) E

xY

. it follows t h a t

Corollary.

The mapping [ O ]

COl

of measure

u N we get k=l k

H

i n t o B(X) i s a *-representation

U J ( C O 1 ) U* from L,(M,p)

as a commutative von Neumann subalgebra of B ( X ) . For each

L,(M,p)

we w r i t e i n correspondence with t h e g e n e r a l l i t e r a t u r e 0

where 7

denotes t h e i d e n t i t y i n

Y

B (xY 1,

Y

6

M.

Remark. The previous c o r o l l a r y rephrases von Neumann's s p e c t r a l theorem. V.5.13. Let

[el

Theorem. E

L,(M,p).

a. For a l l v,w

E

R ( X ) we have

144

DIRECT RESOLUTIONS I N SOBOLEV TRIPLES

b. I n a d d i t i o n , suppose t h a t

R J (COl) R

R i s a Hilbert-Schmidt o p e r a t o r . Then we have

O(y)

=

REy

Rdp(y)

M w i t h convergence i n t h e sense of t h e t r a c e norm. SO f o r a l l

w

R(X)

E

with convergence of t h e i n t e g r a l i n t h e s t r o n g R

-1

(x)

sense.

Proof. a. This a s s e r t i o n i s a consequence of t h e following o b s e r v a t i o n s

Cx

I+

-

nx w l

=

Uw

,

w

E

R(X)

and

(six w , sixv)x =

*

<w

,nX* ,

M ,

W,V

c R(X)

.

b. A l s o t h e second a s s e r t i o n follows from some simple o b s e r v a t i o n s :

and

Here

denotes any orthonormal b a s i s i n

x.

0

Epilogue The o r d i n a r y i n t e g r a t i o n theory f o r s c a l a r valued o r v e c t o r valued f u n c t i o n s can be approached o u t of i n t u i t i o n : going from f i n i t e sums v i a i n f i n i t e sums ( s e r i e s ) t o ‘continuous’ sums ( i n t e g r a l s ) . I n t u i t i o n seems t o f a i l i n d i r e c t i n t e g r a l theory. In f a c t , a d i r e c t i n t e g r a l Y =

0

M

Y d p ( x ) i s n o t an x

o r d i n a r y i n t e g r a l because t h e “ a d d i t i o n ” i s taken over d i f f e r e n t f i b e r s Y

X‘

In t h i s c h a p t e r we have shown t h a t t h e r e e x i s t bounded o p e r a t o r s R > 0 such

145

GENERALIZED EIGENPROJECTIONS

Yx can be continuously embedded i n t o R - ' ( y ) . R ( Y ) , w = Mj@ w(y) du(y) can be considered a s a n R - l (Y)-valued function.

t h a t each f i b e r element w of t e g r a l of

Then each t h e (weak) i n -

Some b i b l i o g r a p h i c a l notes and comments

General r e f e r e n c e s f o r d i r e c t i n t e g r a l theory a r e t h e c e l e b r a t e d book of Dixmier on von Neumann a l g e b r a s [Dix], N i e l s e n ' s monograph [Nie] and t h e t r i l o g y of Lions-Magenes

[LM].

To some e x t e n t t h e concept of d i r e c t

r e s o l u t i o n of t h e i d e n t i t y i s already p r e s e n t i n t h e papers of Foias [Foi 11 and CFoi21.

In t h e s e papers Foias d i s c u s s e s decompositions (of t h e

i d e n t i t y ) i n so-called eigen-operators

(decompositions en o p e r a t e u r s

p r o p r e s ) . These decompositions a r e based on measure spaces

i s a Bore1 measure on CC.

(CC,u),

where p

The connection between t h e notion of d i r e c t re-

s o l u t i o n of t h e i d e n t i t y and d i r e c t i n t e g r a l theory, a s discussed i n t h i s c h a p t e r , does n o t appear i n Foias' papers.

W e have proved t h a t under c e r t a i n c o n d i t i o n s on R (which a r e s a t i s f i e d by all

p o s i t i v e Hilbert-Schmidt o p e r a t o r s ) each f i b e r

tegral space

@

Yx du ( x ) Y = R(y) c y 5 R-' (Y)

Y

of the d i r e c t i n -

R-l

can be embedded i n t o t h e spac:

. The i d e a t o embed t h e f i b e r s

Y

(Y)

of t h e H i l b e r t

i n a Schwartz

type d i s t r i b u t i o n space can be found i n t h e work of Thomas [Tho] and i n t h e Ph-D. t h e s i s of K l a m e r [Kla].

These s t u d i e s a r e i n t h e f i e l d of group re-

p r e s e n t a t i o n theory. In t h i s connection we mention a l s o t h e paper [Schw 2 1

of Schwartz. We p o i n t a t t h e f a c t t h a t t h e s e Schwartz t y p e d i s t r i b u t i o n spaces a r e q u i t e l a r g e . Section A . V . 5

has been i n s p i r e d by Berezanskii [Be 21.

This Page Intentionally Left Blank

147

A THEORY OF GENERALIZED FUNCTIONS

The second part of this book contains the description of a theory of generalized functions. This theory differs from the classical distribution theories (Schwartz, Temple-Lighthill, Gelfand-Shilov) in several aspects. Classical distribution theory is based on real or complex analysis, whereas the present theory of generalized functions is based on functional analysis. We mention, that in our theory, the space of test functions and the space of generalized functions are introduced separately and, next, they are made into a dual system. Opposite to classical distribution theory, our generalized functions are not continuous linear functionals on the space of test functions; instead, they are transparent mathematical objects, which we call trajectories. In fact, the mathematical interpretation of the Dirac formalism, which is to be developed in Part C, depends almost completely on this different approach towards generalized functions. Part B is organized as follows. In Chapter B.1 we introduce the analyticity space SX,A and the trajectory space T X , A , where A denotes any positive self-adjoint (unbounded) operator in a separable Hilbert space

x.

We prove that S

X,A

and T X , A are in duality.

In Chapter B.11 we describe four types of continuous linear mappings betwcen analyticity spaces and trajectory spaces. Correspondingly, we introduce four types of topological tensor products and prove four kernel theorems. Chapter B.111 contains a number of illustrations. We consider specific s e l f adjoint operators A in

L 2 -spaces

over Federer measure spaces ( M , I J ) .

'1'hc.n Ltic:

analyticity space S X , A consists of genuine test functions on M and the trajectory space TX,A consists of generalized functions on M. We show that

SOIINJ

148

Gelfand-Shilov spaces the t r i p l e

s X,A

c

x

c

saB T

a r e of type

X, A

s X,A

and i n d i c a t e r e l a t i o n s between

and Lie group r e p r e s e n t a t i o n theory.

I n Chapter

B.IV we g i v e a renewed d e f i n i t i o n of t h e concept of Dirac b a s i s r e p l a c i n g

R ( X ) by SX,* and R - l ( X )

by TX,A.

W e s o l v e t h e g e n e r a l i z e d eigenvalue pro-

blem i n t h e new s e t t i n g of a n a l y t i c i t y and t r a j e c t o r y spaces.

149

CHAPTER I

A N A L Y T I C I T Y SPACES, TRAJECTORY SPACES AND T H E I R D U A L I T Y

I n t h i s c h a p t e r we p r e s e n t a new theory of generalized f u n c t i o n s i n a comp l e t e l y f u n c t i o n a l a n a l y t i c s e t t i n g . O u r t e s t space and distribution space

are described i n t h e language of f u n c t i o n a l a n a l y s i s . I n f a c t , we introduce t h e prototype of a p a r t i c u l a r c l a s s of d i s t r i b u t i o n t h e o r i e s . I n 1974, De Bruijn proposed a theory of generalized f u n c t i o n s which i s based on complex a n a l y s i s . I n t h i s theory, t h e t e s t space i s t h e space introduced by Gelfand and Shilov. The elements of s 4 a r e c a l l e d smooth 4

s 44

functions. A l s o smoothed f u n c t i o n s a r e introduced. A smoothed f u n c t i o n f i s derived from a smooth function g through a p p l i c a t i o n t o g of an o p e r a t o r taken from a s e t of so-called smoothing o p e r a t o r s . T h i s set o f smoothing o p e r a t o r s i s a one-parameter semigroup (Na)a,O.

De Bruijn has proved t h a t

each smooth f u n c t i o n i s smoothed and t h a t each smoothed function is smooth. Put d i f f e r e n t l y

The o r i g i n a l i d e a i n D e B r u i j n ' s theory is t h e i n t r o d u c t i o n of mappings F from t h e open i n t e r v a l (0,m)

N

F(B) = F ( a +

into

LI s4,

which s a t i s f y t h e r e l a t i o n

8) f o r all a,B > 0. Each mapping

F defines a generalized

function. I n our theory we r e p l a c e t h e semigroup (Na)a,O semigroup (e-tA) t>O

by an a r b i t r a r y one-parameter

where A denotes a nonnegative s e l f - a d j o i n t o p e r a t o r i n

a separable H i l b e r t space

x.

I n t h i s chapter we p r e s e n t t h e a n a l y t i c i t y space SX,A,

T X , A and t h e i r d u a l i t y . we note t h a t S space of generalized functions.

X,A

t h e t r a j e c t o r y space

i s t h e t e s t space and T

X,A the

ANALYTICITY SPACES, TRAJECTORY SPACES AND THEIR DUALITY

150

1.1. The a n a l y t i c i t y space SX A I

A s usual, l e t Let

x

denote a separable H i l b e r t space with i n n e r product

A denote an (unbounded) nonnegative s e l f - a d j o i n t o p e r a t o r i n

x

( a , .

'X'

and l e t

denote i t s s p e c t r a l r e s o l u t i o n of t h e i d e n t i t y . Because of t h e s p e c t r a l theorem f o r each everywhere f i n i t e Bore1 f u n c t i o n cp t h e o p e r a t o r

1

cp(A) =

cp(A) dE,

IR i s densely defined on i t s maximal domain D ( g ( A ) )

We introduce t h e following convention. By

a we mean t h e o p e r a t o r

where x

~

For each t

~ denotes , ~ ) t h e c h a r a c t e r i s t i c function o f [ a , b ) 6

E the operator e

e-tA =

1

-tA

e-tAdEA

is defined by

.

IR

Since A is nonnegative, w e have m

So t h e o p e r a t o r e-tA

is bounded f o r Re t > 0 with an unbounded i n v e r s e e

tA

For Re t = 0 , t h e o p e r a t o r e-tA i s u n i t a r y . In t h e sequel w e only consider r e a l parameters t .

.

s X.A

THE ANALYTICITY SPACE

151

Let t > 0. Then t h e o p e r a t o r e-tA i s a bounded p o s i t i v e o p e r a t o r on t h e dense subspace e

-tA

(x)

of

x

W e denote t h e corresponding norm by I I . I l t . space with i n n e r product ( - , * I t .

-tA

(x))t,o

(e

x.

In

we introduce t h e s e s q u i l i n e a r form

The space e

-tA

(x)

is a Hilbert

Thus w e o b t a i n t h e chain of H i l b e r t spaces

with

e

-tA

The embedding e

-TA

(X)ce

-t A

(x) 4

e

(X),

-TA

o

(X)

<

T

<

~

.

i s continuous f o r

o

< T < t.

Definition.

I. 1.1.

The a n a l y t i c i t y space

'X,A

=

tyo

SX,A is defined by e

-tA

(XI

.

I n t h e above d e f i n i t i o n we have c a l l e d

s X,A

t h e a n a l y t i c i t y space. The use

o f t h i s terminology demands some explanation.

Definition.

I. 1.2.

Let v

x.

E

Then v i s s a i d t o be an a n a l y t i c vector of

A if v

E

n

nE IN

D(An)

and i f t h e r e a r e c o n s t a n t s a > 0 and b > 0 such t h a t

IIAnvIIX 5 a b n n !

a.

1.1.3. k t v

E

X. Then v

E

S i f f v i s an a n a l y t i c v e c t o r of A. X,A

Proof.

5

SX,A. Then t h e r e e x i s t s

-tA

(x).

The o p e r a t o r t > 0 such t h a t v E e , n E IN, i s bounded with IIAn e - t A II = sup ( A n e- X t ) = nn t -n e -n 5 A>O n! t-n. I t follows t h a t v E D ( A n ) and IIAnv [ I x 5 n: t-n IIv 11

Let v An .-tA

E

L e t v be an a n a l y t i c v e c t o r of

that

A. So t h e r e e x i s t c o n s t a n t s a , b > 0 such

152

ANALYTICITY SPACES, TRAJECTORY SPACES AND THEIR DUALITY

IIAnv

Let 0 < t < b-'.

-1

IIx

2 a b n n:

.

Then t h e s e r i e s

7 tn

IIAnvIIX

n=O

& An

m

i s convergent, whence

-tA

follows t h a t v

E

"

Ti

n=O n.

v i s convergent i n

x

with l i m i t etA v.

It

0

(XI.

e

SX,A w e impose t h e i n d u c t i v e l i m i t topology brought about by t h e H i l b e r t -tA ( x ) , t > 0. I t i s- t At h e f i n e s t l o c a l l y convex topology on sX,A f o r

On

spaces e

which a l l i n j e c t i o n s

it

: e

(x)

G S X , A a r e continuous. So a convex and b a l -

anced s u b s e t o f SX,A i s open i n t h e i n d u c t i v e l i m i t topology i f and only i f

0 n e-tA(X)

i s an open s u b s e t of e

-t A ( X I .

The r e a d e r , who i s f a m i l i a r with

t h e concept of i n d u c t i v e l i m i t , may observe t h a t

sX,A

i s a non-strict in-

d u c t i v e l i m i t . I t follows t h a t , f o r a g r e a t e r p a r t , t h e g e n e r a l l i t e r a t u r e on i n d u c t i v e limits of H i l b e r t s p a c e s , where mostly s t r i c t n e s s is assumed,

i s not a p p l i c a b l e i n our s i t u a t i o n . W e i n t r o d u c e an e x p l i c i t s e t of seminorms which g e n e r a t e s a l o c a l l y convex topology e q u i v a l e n t t o t h e i n d u c t i v e l i m i t topology. Thus a l l i n t e r e s t i n g p r o p e r t i e s of t h e l o c a l l y convex t o p o l o g i c a l v e c t o r space

SX,A can be ob-

tained.

I.

I. 4. Definition.

B(IR) c o n s i s t s of a l l everywhere f i n i t e Bore1 f u n c t i o n s J, on IR with t h e

property t h a t f o r a l l t > 0

B+(w) J, E

denotes the s t r i c t p o s i t i v e p a r t of

B ( D ) and $ ( X I

For each $

E

B(IR), i . e .

J, E

B+(IR) i f f

2 E f o r some E > 0 .

E(IR) the o p e r a t o r $ ( A )

i s a normal o p e r a t o r i n

t > 0 t h e o p e r a t o r +(A)e-tA i s bounded. So each w

D ( J , ( A ) ) because of t h e r e l a t i o n

E

S

X,A

X and f o r a l l

i s a member of

THE ANALYTICITY SPACE

sX,A

153

where t > 0 has to be taken sufficiently small. Now it makes sense to introduce the following norms on

SX,A.

I. 1.5. D e f i n i t i o n . For each J,

E

B (IR) (w) =

s J,

we define the norm s

IIJ,(A)w II

,

J, On

'X,A

by

'X,A

The following lemma is one of the fundamental results of our theory of generalized functions. It leads to full comprehension of the topological structure of the locally convex topological vector space S

X,A'

e.

1.1.6.

a. The seminorms s

J,

J,'

E

B + ( I R ) , are continuous on S X,A-

set Q

s

with the property that for all t > 0 the X,A -tA (x). Then e-tA(X) contains an open neighbourhood of zero in e

b. Let Q be a convex set in there exists J,

E

B+(IR) such that

Pro0f. a. Let {w

E

f

> 0 and let J,

SX,A I s6(w)

<

E

f}

B+(IR). We have to prove that the set is open in SX,A. It means that for each t > 0 the

set {w

-tA e (X) 1 s (w) <

E

J,

-tA ( x ) .

is open in e

€1

So let t > 0 and observe that

From this inequality the assertion follows.

1"

P n = n-1 dEA. n E IN. Let r denote the radius of the largest open ball in Pn(x) which fits in the

b. Introduce the orthogonal projections

154

ANALYTICITY SPACES, TRAJECTORY SPACES AND THEIR DUALITY

:= s u p {p

r

I

CU

E

L e t t > 0. Then there i s

{u

I

Pn(X)

A

IIPn uIIX < p l

*u

E

Q}

.

> 0 such t h a t

E

,

I I ~ I I + <~ € 1 c n n e-’tA(X)

-ttA

b e c a u s e il n e-itA(X)

c o n t a i n s an open neighbourhood o f 0 i n e -4nt Thus it follows that f o r all u E P w i t h IIPn U l l x < E e

(x)

II UII 2‘rt = II Pn u II

’t 2

e

=

A t

d ( E A u ,uIx <

E

2

(x).

w e have

.

n- 1

Hence u

Q and w e o b t a i n r

E

>

E

-fnt

e .

.

Next we d e f i n e t h e f u n c t i o n 0 as f o l l o w s

Then f o r a l l X

whence (3 E

E

n

Cn-l,n),

E

IN, w e g e t

B+ (m).

W e now show t h e i m p l i c a t i o n

II@(A) u llx < 1

(*)

Let u

E

e

-tA

(x)

*

u

E

n ,

u

E

f o r some t > 0. W e have

SX,A OJ

E

n= 1

. IIPnuIIt2 <

b e f i x e d . Then

F u r t h e r , because o f o u r a s s u m p t i o n ( * ) -2 llPnuIIx 2 f m i n ( n rn

, 1) ,

n

E

IN

.

-.

Let 0 < T < t

THE ANALYTICITY SPACE

So w e have 2 n

u =

2

Pn u

N

a

E

f o r every n

2

1

1 7( 2 n

pnu)

n=l 2n

+

E

s

155

X.A

lN. Now w e can w r i t e u a s

(n=N+1 2n

%)uN

with m

n=N+1 By ( * * ) w e o b t a i n t h e estimate

I t follows t h a t uN

+

0

i n e-TA(X) i f

N +

-

and hence u

N

E

il n e-TA(X)

f o r s u f f i c i e n t l y l a r g e N. Thus we s e e t h a t u i s a subconvex combination of elements i n u

a,

so t h a t

n.

E

0

Theorem.

I. 1.7.

The i n d u c t i v e l i m i t topology of norms IsJ, 19

Proof. From s e t {u

E

S

B+(IR) I .

E

sX,A

i s brought about by t h e s e t of semi-

t h e f i r s t a s s e r t i o n of t h e previous lemma it follows t h a t each

X.A

I s $ (u) <

€1,

$

B+(m),E

E

> 0 , i s open i n S

On t h e o t h e r hand, l e t Cl be an open s e t i n

X,A' S X , A . The 0 c o n t a i n s a convex

a

open s e t w

n

a.

From t h e second a s s e r t i o n of t h e previous lemma it follows t h a t

contains a s e t {u

E

SX,A

I s X (u) <

11 f o r c e r t a i n

x

E

B+(IR).

0

We have observed already t h a t S X , A i s contained i n t h e domain D ( J , ( A ) ) of n any $ E B + ( I R ) . W e want t o prove t h a t SX,A equals D ( $ ( A ) ) . ConseWB+ (IR) quently, SX,A is t h e maximal common dense domain of t h e o p e r a t o r s $ ( A ) , J, E B+(IR).

1.1.8. Let

x

Herein, t h e following lemma i s e s s e n t i a l .

a. be a bounded Bore1 function on IR+ with t h e p r o p e r t y t h a t

sup I X ( A ) $(A) 120 such t h a t

I

<

m

f o r a l l J,

/ X ( A ) I sCe-ta

,

E

B+(IR). Then t h e r e e x i s t s t > 0 and

A 2

o

.

c >

0

156

ANALYTICITY SPACES, TRAJECTORY SPACES AND THEIR DUALITY

Proof.

Suppose t h e c o n t r a r y , i . e .

Ak > 0 such t h a t

Then f o r each k L IN t h e r e e x i s t s

Since t h e f u n c t i o n h

I+

x(A)

may a s w e l l assume t h a t h

etA, t > 0 , i s bounded on bounded i n t e r v a l s we f o r a l l k E IN and t h a t t h e sequence

< hk+l

k ( h k ) k E I N tends t o i n f i n i t y . Put bo = 0 and d e f i n e t h e f u n c t i o n cp by

Then f o r each t > 0

sup e A20

-tA

cp(A)

and hence cp L B + ( I R).

= sup

However, f o r each k E W

This i s a c o n t r a d i c t i o n .

Theorem.

1.1.8. Let

v

x.

denote a s u b s e t o f

{$(A) v

Iv

Proof.

x

E

v

such t h a t

tion

0

Put

Suppose t h a t

v j i s bounded i n

x

f o r each

i s a bounded s u b s e t of e

Pn =

/"

n- 1

dE

A

-tA

v

i s contained i n D ( $ ( A ) ) and t h a t

JI

E

and rn = sup II Pnv VE

by

v

B+(IR). Then t h e r e e x i s t s t > 0

(X).

11 x,

m

For each

$J

E

B+(IR) t h e function

3

defined by

n

E

W. we d e f i n e t h e func-

THE ANALYTICITY SPACE

B+ (IN. So

i s again i n

for a l l X

I t f o l l o w s t h a t f o r a l l $I

Cn - 1,n) we o b t a i n

E

B+(IR) t h e f u n c t i o n

E

157

'X,A

x

' J , i s bounded. Applying

t h e p r e v i o u s lemma, t h e r e e x i s t t > 0 and C > 0 , s u c h t h a t ~ ( h 5) C e

t 0. Thus w e o b t a i n f o r a l l v N

1

,,etApn

E

v

and a l l N

~II; 1 5

e

2t

e

E

2(n-l)t

<

2

II Pn vIIx

eL N

2 ~2 t

T1 - II Pn v II 2 n

n=l, n rn#O

v

So

5

2t

2 S

Kt

2

L 6

.

-tA

(x)

with

2

\Ivllt

5 Kt

for all v

E

v.

0

CoroZZary.

I. 1.9.

D ( J , ( A ) ) for each

E

= $JEB:(IR)

I. 1.10. Let

e

i s a bounded s u b s e t o f e

Let w

'X,A

2

IN

n= 1

n= 1

w i t h Kt = C

-ta ,

Then w

J, E B + ( I R ) .

E

SX,A. P u t d i f f e r e n t l y ,

D($(A)).

Theorem.

B b e a s u b s e t o f SX,A. Then B i s a bounded s u b s e t o f

s X.A

i f f there

e x i s t s t > 0 , s u c h t h a t B i s a bounded s u b s e t o f t h e H i l b e r t s p a c e e

Proof. c) By a s s u m p t i o n t h e r e e x i s t t > 0 and K > 0 s u c h t h a t f o r a l l w

L e t J,

E

B + ( I R ) . Then f o r a l l w

E

B

E

B

-tA

(x).

158

ANALYTICITY SPACES, TRAJECTORY SPACES AND T H E I R DUALITY

a) L e t

s X, A'

B i s a bounded s u b s e t o f

Hence

B b e a bounded s u b s e t of SX,A.

bounded on

B, i . e . f o r a l l J,

x.

i s a bounded s u b s e t o f

I t means t h a t a l l s e m i n o r m s s

J,

are

B+(m)

E

So from Theorem 1 . 1 . 8 t h e wanted r e s u l t

0

follows.

I n t h e n e x t theorems w e d e s c r i b e c o n v e r g i n g s e q u e n c e s , Cauchy s e q u e n c e s and

SX,A'

compact sets i n

I. 1.11.

Theorem.

s X,A'

i s a n u l l sequence i n Then ( U n ) n e m S X , Ai f f t h e r e e x i s t s t > 0 s u c h t h a t ( un ) nEIN i s c o n t a i n e d i n e-tA(X) and IIunII + 0 a s n + m.

Let (

~ d e)n o t e~ a s~e q u~e n c e i n

u

Proof. c')

L e t J,

E

B+ (IR).

Hence u

+

=+I Suppose ( J,

E

B+(IR).

Then w e have

0 as n

-

+

~ i s) a n~ u l l~ s e q~u e n c e i n

u

sX , A '

i n the inductive l i m i t topology o f

So t h e s e t { u

In

E

IN}

sX,A'

Then

is bounded i n

II$(A) un I I x + 0 f o r a l l SX,A and h e n c e t h e r e

e x i s t T > 0 and K > 0 s u c h t h a t

Taking J,

E

1 i n p a r t i c u l a r it f o l l o w s t h a t

I

derive

IIu 11

+

0 as

as

n

+

n X

n

+

-.

Thus w e

a

Va,o

Vt,O

:

e2Xt d ( E x u n , un)

*

0

m

.

0

Now w e show t h a t

IIu 11 n t

follows

+

0 f o r a n y t < T. T h e r e f o r e , w e s p l i t

a IlunII

2

m

e

=

0

2Xt

d(Exun

,un) +

e2Xt d ( E X u n , un)

a

.

IIu )I2 as n

t

THE ANALYTICITY SPACE

sX.A

159

The second i n t e g r a l i s estimated a s follows m

m

J

e2ht d(E,un

, un)

<

e

J

2a(t-~)

a

e21Td ( E X u n , u n ) 5

a 2a(t-T)

< e

K

2

.

So f o r each E > 0 we can t a k e a > 0 s o l a r g e and f i x e d that -2a(T-t) 2 K < $c2. Next, w e can take N E IN s o l a r g e t h a t f o r a l l n > N

a e2Xt d ( E X u n, u n ) < is

2

.

0

I t then follows t h a t

IIu II n

t

<

E

for a l l n

E

0

IN.

Theorem.

I. 1.12.

a. Let (unlnEINbe a Cauchy sequence i n SX,A. Then t h e r e e x i s t s t > 0 such that b.

(UJnEIN

i s a Cauchy sequence i n e-tA(X).

SX,A i s s e q u e n t i a l l y complete, i . e . every Cauchy sequence i n

sX , A

is a

convergent sequence.

Proof. a. The proof i s based on s i m i l a r arguments as t h e proof of Theorem 1.1.11 and it i s omitted. b. Since e-tA(X) i s a H i l b e r t space t h e second a s s e r t i o n follows from t h e

0

f i r s t one.

1.1.13.

Theorem. K of SX,* i s compact i f f t h e r e e x i s t s t > 0 such t h a t K i s a com-

A subset

p a c t s u b s e t of e - t A ( X ) .

Proof. c) Suppose t h e r e e x i s t s t > 0 such t h a t

K i s a compact s u b s e t of e

-tA

(X).

Following t h e d e f i n i t i o n o f t h e i n d u c t i v e l i m i t topology t h e i n j e c t i o n

*)

-t A

(x)

c + S ~ , i~s continuous. So K being t h e image of a compact t A s u b s e t of e (x) under it, i s compact i n sX,A' L e t K be a compact s u b s e t of SX,A. Then K i s a l s o a bounded s u b s e t of -TA SX,A. So there e x i s t s T > 0 so t h a t K i s a bounded s u b s e t of e ( X ) . we

it

: e

160

ANALYTICITY SPACES, TRAJECTORY SPACES AND THEIR DUALITY

-tA show that K is compact in e (x) for all t, 0 < t < T. Since the injection

SX,A c * X

is continuous, K is also compact in

X.

Let (unInEINbe a sequence in K. This sequence contains a subsequence

- uIIx = 0 . We put j-w j is a bounded sequence in e-'A(X) and

such that for some u

(s,)jcm

E

K, lim II un

v. = un, - u . Then (v.) 7 1 I jeIN IIv.11 + 0 as j + m. Now we are in the same position as in the proof of

I X

the only-if part of Theorem 1.1.11. We conclude that IIv.II + 0 for I t 0 < t < T, i.e. the subsequence (un ) , converges to u in the Hilbert -tA space e (x). Hence K is a compactjsubset of e-tA ( X ) for 0 < t < T.

Remark. A sequence which is bounded in null sequence in

SX,A'

sX.A

x

and tends to zero in

is a

In order to make a link with the literature on locally convex topological vector spaces we describe the other properties of

SX,A using

the standard

terminology of topological vector spaces. The terminology is explained in the proof. I. 1.14. Theorem. a.

SX,A is

complete.

b.

SX,A is

bornological.

c. SX,A is barreled. d.

e.

SX,A is on X.

a Monte1 space iff for every t > 0 the operator e-tA is compact

SX,A is

a nuclear space iff for every t > 0 the operator e-tA is Hilbert-

Schmidt on

x.

Pro0 f. a. Let (w

)

be a Cauchy net in

SX,A. Then

($(A)wa), and in particular (w

is complete there exists w(J,), w in and (wa) tends to w. The operators w

E

D($(A)) and w($) = J,(A)w, $I

to w in S

for each J,

E

B+(IR) the net

itself, is a Cauchy net in

E

x

X.

Since X

such that ($(A)w ) tends to w($)

$(A)

are closed. It follows that

B+(IR). Hence w

E

SX,A and

(w

)

tends

X,A'

b. We have to prove that every convex circled subset

every bounded subset B of

s

X, A

Q of

s

that absorbs X,A is a neighbourhood of zero. So let

THE ANALYTICITY SPACE

51 c

s X, A

Bt i s

be such a set and l e t

sX,A

bunded i n

Bt

there exists

sX,A

161

denote t h e u n i t b a l l i n e E

> 0 such t h a t

Bt

E

-t A ( x ) .

c Q.

Since

I t follows

t h a t Q n e-tA(X) contains t h e open neighbourhood of t h e o r i g i n

E

Bt '

So

by Lemma 1.1.4 Q contains t h e open neighbourhood of t h e o r i g i n

C.

f o r some $I

E

B+(m).

v

c

sX , A

A barrel

i s a s u b s e t which i s r a d i a l convex c i r c l e d and closed.

We have t o prove t h a t each b a r r e l c o n t a i n s an open neighbourhood of t h e

v

o r i g i n . Let

sX,A

C

be a b a r r e l . Then f o r each t > 0 ,

b a r r e l i n t h e H i l b e r t space e-tA(X).

So

v n

But then

v

n e

- t A (x)

is a

e-tA(X) c o n t a i n s an open

neighbourhood of t h e o r i g i n i n t h e H i l b e r t space e 1.1.15.)

v

-tA

(x).

(Cf. Theorem

c o n t a i n s an open neighbourhood o f t h e o r i g i n i n

sX , A

by Lemma 1.1.4. d. W e have t o prove t h a t every closed and bounded s u b s e t of

s X.A

is

i f f f o r every t > 0 t h e o p e r a t o r e-tA i s compact. c) Consider t h e diagram

From t h i s diagram we observe t h e following: Let

B be a closed and bounded s e t i n SX,A. Then B i s c l o s e d and - t A (x) f o r some t > 0. So t h e s e t etA(B) i s c l o s e d and

bounded i n e

x.

bounded i n

Since t h e o p e r a t o r e-4tA i s compact, t h e s e t eStA(B)

is a compact s u b s e t of -)

Let (

x

I

B

i s a compact s u b s e t of e-'tA(X).

~ be ) a bounded ~ ~ sequence ~ in

bounded sequence i n {e-tA x

X.so

n

E

IN}

Theorem 1.1.13.)

sX,A'

x.

-tA

Then ( e

So t h e c l o s u r e i n

sX,A

xnInEm of t h e s e t

i s compact and hence s e q u e n t i a l l y compact.

-tA

I t follows t h a t t h e sequence (e

x

~

is a (Cf. contains

)

an SX,A- and hence X-convergent subsequence. Since t h e sequence ( x n I n E m has been taken a r b i t r a r i l y , t h e o p e r a t o r e-tA i s compact.

~

~

ANALYTICITY SPACES, TRAJECTORY SPACES AND THEIR DUALITY

162

e. SX,A is a nuclear space iff for every norm B + ( I R ) , with

x

the canonical injection J

:

the Banach completions of

sX,A

norm s

X'

x

E

respectively.

s

J,'

J, e B + ( I R ) ,

there is a

X t

such that

t J, (i.e. x(X) 2 $ ( A ) , I

.

.

s X G s J , is nuclear.

Here

I

O),

sX

and

with respect to the norm s

sJ,

and

J,

denote s

X'

Let the operator e-tA be Hilbert-Schmidt for all t > 0. By the spectral theorem, the operator <

m

A

has a discrete spectrum 0

5

X1

5

X2

... with

5

for all t > 0. We denote the corresponding normalized

n=1 eigenvectors by (v ) So ( n new' First we construct a function w

v E

~ is)an ~orthonormal ~ ~ basis in

x.

B+(IR) such that v(A1-l is a nuclear

operator. Therefore we choose a sequence (N

n) neIN of natural numbers

as follows

Then the function v

B (IR) is defined by

E

w(A) = 1

,

elsewhere

.

So we get

Now let J, I

J

:

s

E

-

c+s

X J ,

B+(IR). Put

x

= J, . v . Then

x

t $. The injection

is nuclear, because J can be written as

Here (u,w) = (x(A)u

X

, x(A)wIX,

is an orthonormal basis in orthonormal basis in

I

s

^s

X

u,w

I

E

S

X'

We note that

(x(Xk) -1 vk)k e l N

and, similarly, (J,(Xk)-lv ) k kelN is an

J,'

Let SX,* be a nuclear space. Take J, such that the injection

=

1. Then there exists

x

E

B

(IR)

THE TRAJECTORY SPACE T

is nuclear. So

x(A)-'

163

X,A

is a trace class operator on X. But then for -tA -tA J x(A)-' is trace class and = IX(A>e

all t > 0, the operator e

0

hence Hilbert-Schmidt.

Remark. Let R be Then

A

I X0I .

Ihl 2

a

A subset

set of R. A subset

R. XA whenever

topological vector space and let A, 8 be subsets of

is said to absorb B if there exists X o

c

u

E

of R is called radial if

C such that

u

B

c

absorbs every finite sub-

of R is circled if XC c C whenever

IX I

5

I.

I. I . 15. Theorem. a. Every locally convex topological vector space R which is of the second category, is barreled.

b. Every locally convex topological vector space R which is metrizable is bornological.

Pro0f. _ I _

a) Let D be a barrel in R. Then D is radial and circled. S o we can write R as the union

u nD. Since R is a second category space there exists nE IN IN such that the closed set n D has an interior point. Hence D has 0 an interior point y. Since D is circled, -y E D and, since D is convex

no

E

0 =

'ry +?r(-y) is an interior point of

D.

b) If R is metrizable, there exists a countable basis of null neighbourhoods

In E

{Vn

IN} which decreases, i.e.

Vn

3

Vn+l.

Let

A

be a convex and

circled subset of R that absorbs every bounded set in R . Then we must have xn

E

vn c nA for some n IN. If this were false there would exist vn with x $ nA for each n IN. Since the sequence (xn1 n€IN has E

E

to tend to zero, it is bounded and hence absorbed by

A.

This yields a

contradiction.

1.2. The trajectory space T

0

X,A

we now introduce our space of trajectories TX,A. The elements in this space are candidates for becoming "generalized functions". In the definition of

TX,A the role.

semigroup property of the family {e-tA

It

>

01 plays an important

164

ANALYTICITY SPACES, TRAJECTORY SPACES AND T H E I R D U A L I T Y

1.2. I. D e f i n i t i o n . Let F be an X-valued f u n c t i o n on t h e open i n t e r v a l ( 0 , ~ )with t h e a d d i t i o n a l property t h a t

vt,o vTz0

: F ( t + T ) = e-TAF(t)

.

Then F is c a l l e d a t r a j e c t o r y . The space of t r a j e c t o r i e s i s denoted by and i s c a l l e d t h e t r a j e c t o r y space.

TX,A

T X.A

i s a v e c t o r space under t h e usual

pointwise a d d i t i o n and s c a l a r m u l t i p l i c a t i o n .

Remark. Let F

t

TX,A. Then f o r a l l t > 0 and

0 <

< t

T

The space TX,A c o n s i s t s of a l l s o l u t i o n s of t h e e v o l u t i o n equation

with t h e a d d i t i o n a l c o n s t r a i n t t h a t F ( t )

E

x

f o r a l l t > 0. The i n i t i a l

condition " F ( 0 ) " corresponding t o t h e above e q u a t i o n , need n o t be an element

x.

of x

E

For i n s t a n c e , c o n s i d e r t h e t r a j e c t o r y F

X\D(A) and k

E

:

t

W

Ak e-tAx where we t a k e

IN. Loosely speaking, each t r a j e c t o r y r e p r e s e n t s an

i n i t i a l condition which may be beyond H i l b e r t space.

I. 2 . 2 .

Definition.

For t h e topology on

T

X,A

w e t a k e t h e l o c a l l y convex topology induced by t h e

seminorms q ( F ) = t t F ( t ) I I X , t > 0. So a n e t (F ) i n TX,A converges t o F i f f t f o r a l l t > 0 t h e n e t (F ( t ) )converges to F ( t ) i n x. With t h i s topology T X , A is a l o c a l l y convex t o p o l o g i c a l v e c t o r space.

The seminorms q t a r e i n f a c t norms on

TX,A. To

f o r some t > 0. Then F ( t ) = 0 , and hence operators

- (T-t) A e

F(T)

see t h i s , l e t I I F ( t ) IIx = 0

= 0 for all T >

a r e i n j e c t i v e . The H i l b e r t space

x

0 because t h e

can be embedded i n t o

TX,A, i . e . t h e r e e x i s t s a continuous l i n e a r i n j e c t i o n from X i n t o T t h i s end, w e d e f i n e t h e l i n e a r mapping emb by

X,A' To

TEE TRAJECTORY SPACE T

165

X, A

Continuity and i n j e c t i v i t y o f emb can be shown s t r a i g h t f o r w a r d l y .

I. 2.3.

Proposition.

TX,A-

emb(X) i s a dense subspace of

Proof.

Let F

E

TX,A. Then F ( t )

E

x

f o r each t > 0 - Let

For each t > 0 and each n E IN with 0 <

Therefore, t h e topology of T of seminonns ql,n,

n c IN.

X,A

5 t

> 0 - Then

w e have

is already generated by t h e countable number

the t o p o l o g i c a l v e c t o r space

SO

T X , A is metrizable.

The usual standard arguments show t h a t by

a metric i s defined which g i v e s r i s e t o a l o c a l l y convex topology e q u i v a l e n t t o the topology of T

I. 2.4.

X, A

introduced i n D e f i n i t i o n 1 . 2 . 2 .

Theorem.

The t o p o l o g i c a l v e c t o r space

TX,A i s a Frechet space, i . e . a complete

metrizable l o c a l l y convex t o p o l o g i c a l v e c t o r space.

Proof.

W e only prove t h a t

T

X,A

is complete.

L e t ( F n ) n E m be a Cauchy sequence i n T

is a Cauchy sequence i n

We define F

: t H

x.

X,A'

Then f o r a l l t > 0 ( F n l t ) ) n E m

So t h e r e e x i s t s F ( t )

E

F ( t ) , t > 0. Then t h e r e l a t i o n s

x,

t > 0 , with

166

ANALYTICITY SPACES, TRAJECTORY SPACES AND T H E I R DUALITY

show t h a t F

TX,A.

E

So t h e sequence ( F n ) n E m converges t o F i n

T X,A'

0

A t t h e beginning of t h i s s e c t i o n we have noted t h a t t h e r e i s a connection

between t h e t r a j e c t o r y space

T

and a y e t unspecified s e t of i n i t i a l con-

X,A

d i t i o n s . Now we s h a l l e x p l a i n t h i s statement. Let J,

B + (IR)

E

and l e t F

$(A) F ( t ) , t

E

TX,A. Then J , ( A ) F denotes t h e t r a j e c t o r y

> 0. Observe t h a t

t

kb

x

because w e can w r i t e

$(A) F ( t )

$(A)

=

f o r a l l t > 0 and 0 <

T

.-'IA

TX,A-

( F ( t ) ) i s a well-defined element of

F ( t - T)

< t. Simplifying t h e n o t a t i o n w e w r i t e $ ( A ) f in-

s t e a d of $ ( A ) ( e m b ( f ) ) f o r each f of

$(A)

E

x.

F i r s t we c h a r a c t e r i z e bounded s u b s e t s

A s a consequence of t h i s c h a r a c t e r i z a t i o n it t u r n s out t h a t a l l

TX,A can be w r i t t e n a s $ ( A ) f where f

elements of

E

X and

J, E B + ( I R ) .

Theorem.

1.2.5.

B

a . A set

TX,A i s bounded i f f f o r each t

c

B

b. A s e t

Bo

set

> 0 t h e set { F ( t ) It > O } i s

x.

bounded i n

TX,A i s bounded i f f t h e r e e x i s t s X such t h a t B = {J,(A)f I f E Bo}.

c

of

(I E B+(IR) and a bounded sub-

Proof. a. By d e f i n i t i o n a s e t B c T bounded on b. *)

B.

W e take Pn =

X

In dEA, n

n- 1 f o r a l l n E IN,

r

= sup

X,A

i s bounded i f f f o r a l l t > 0 t h e norm q

E

Since e

IN.

the numbers rn, n

E

A P

A

IIe P n F ( l ) I I

Now w e d e f i n e t h e f u n c t i o n J, by

m J,

=

c

n=1

( 1 + n r n ) X[n-l,n)

.

is

i s a bounded o p e r a t o r on

IN,

FEB are well-defined.

t

THE TRAJECTORY SPACE

Then f o r each t > 0 and a l l A

5

1 + 2e

t

E

For each F

E

IIAe-'tAII

sup IIF($t) I I x

B

.

B +(IR ). m

B, t h e s e r i e s

E

167

X,A

[n - 1 , n )

FE

Thus it follows t h a t Q

T

.El

( 1 + n r n I w 2 11 eA P n F ( l ) 11;

converges.

Hence

x

belongs t o Now t h e set

in

X,

and F = $ ( A ) gF.

Bo

c

x

1

i s defined by Bo = {gF F

IIgFII

2

m

=

1

2 -2 A (1 + n r n ) lie P ~ F ( I II)

m

1

5

n= 1 c) suppose

i s bounded

B

= {$(A)f

x.

Then f o r

f.

If

E

E

B0 } with B,

a bounded s u b s e t of

Bo

0

B i s a bounded subset of T X,A'

Corollaq.

I. 2.6.

Let F

B0

n=l n

each t > 0 and a l l f

Hence

B). Then

E

hecause

E

TX,A. Then t h e r e e x i s t Q

E

B+(IR)

and f

E

x

such t h a t F = $ ( A ) f .

W e a l s o c h a r a c t e r i z e converging sequences, Cauchy sequences and compact s e t s

in TX,A*

168

ANALYTICITY SPACES, TRAJECTORY SPACES AND THEIR D U A L I T Y

Theorem.

1.2.7.

A sequence (

~ converges ) ~ ~ t o ~F i n

F

a sequence ( f )

n n€lN

in

x

TX,A,

i f f t h e r e e x i s t s J,

which i s convergent i n

x

with l i m i t f

E

B (IR)

E

x

and

where

Fn = $(A) f n and F = + ( A ) f .

Proof. *)

Let ( f n ) n c m be an X-convergent sequence with l i m i t f and l e t JI

E

B

(IR).

Then f o r a l l t > 0

II$(A) e

-tA

(fn- f )

IIx

(n

0

-t

.

-+ m )

I t follows t h a t t h e sequence ($(A) f n I n E m tends t o J,(A)f i n *)

L e t ( F n ) n E m denote a converging sequence i n

{Fn I n

E

mj u

gn,g

x

such t h a t Fn =

J,

E

f =

E

B+(IR) by 2 -1

(1 + A

)

IF) i s bounded i n

$(A)

TX,A.

T

w i t h l i m i t F . The s e t

X,A

So t h e r e e x i s t s cp

cp(A) gn, F = cp(A)g. P u t K c p ( h ) , h E m. Put f

= (1 + A 2 )

g. We prove t h a t

[ I f n - f l l X -t 0 a s

n

TX,A.

E

B+(IR) and

IIg 1 I and d e f i n e ncm 2 n X

= sup =

+

(I +A

gn,

-. For each L > 0 we

have the following i n e q u a l i t i e s

and

5

sup ACCO

e2')

2 l l F n ( l ) - F(1) IIx

.

rL1

L e t E > 0. Having f i x e d L > 0 such t h a t (1 + L 2 ) > ( 2 c 2 ) - l ( K 2

w e can take no

E

IN such t h a t f o r n > n

0

+

IIyII 2 )

THE TRAJECTORY SPACE

Then ( * )

T

169

X,A

and ( * * ) y i e l d

IIfn- f l l X

<

for n > n

E

0

0'

Theorem.

1.2.8.

A sequence (Fn)nEINi s a Cauchy sequence i n sequence ( f n ) n e m i n

Proof.

x

and J,

E

T

X,A

i f f t h e r e e x i s t s a Cauchy

B+(IR) such t h a t Fn = $(A) f n , n

IN.

E

The proof i s s i m i l a r t o t h e proof of t h e previous theorem. W e leave

0

it t o t h e reader.

1.2.9. Theorem.

TX,A. Then t h e following statements a r e e q u i v a l e n t .

L e t K be a s u b s e t of

a. K i s compact. b. There e x i s t s $

K

= I$(A)f

1f

E E

B+ (IR) and

x

a compact s u b s e t KO of

such t h a t

KO).

1F

c. For a l l t > 0 , t h e set [ F ( t )

x.

K) i s compact i n

E

Proof. a. * b .

Let K be a compact s u b s e t of Hence t h e r e e x i s t s cp

TX, A'

such t h a t K = cp(A) and p u t KO =

(ao).

E

TX,A. Then K i s a bounded s u b s e t o f Bo of x

B + ( J R ) and a bounded s u b s e t

Define J,

E

( I +A2)-I(B0). Then K

$(A)

B+(IR) by

= $(A) (KO).

=

2

(1 + A ) cp(X)

Let ( f n ) n E w b e a

sequence i n KO. Then t h e sequence ($(A) f n I n c I N c o n t a i n s a subsequence ( $ ( A ) f n j )jEIN

with l i m i t $ ( A ) f i n K, f

E

KO. With s i m i l a r

arguments a s used i n t h e proof of Theorem 1.2.7 it follows t h a t t h e sequence ( f , , ) 3 b.

* c.

The s e t {F(t)

converges t o f i n

,

IF

E

K} c

x

X.

Hence KO i s compact.

i s t h e continuous image of t h e compact

I

s e t KO under t h e bounded operator + ( A ) e-tA. compact i n c.

*

Hence { F ( t ) F

E

K] is

X.

a . Let ( F n ) n E I N be a sequence i n

K.

We have t o prove t h a t ( F n ) n E m con-

t a i n s a converging subsequence with l i m i t F

E

K. Consider t h e se1

it has a converging subsequence ( F n ( 1) nE IN. 2 The sequence (FA($) )ncIN has a converging subsequence !Fn($)) n E m ,

quence (Fn ( 1) ) nE

170

ANALYTICITY SPACES, TRAJECTORY SPACES AND THEIR DUALITY m m in K such that (FnInEm etc. Thus we arrive at sequences (F ) nncm and Fn(m)ncm converges to is a subsequence of (FnInEm for m >

e

an element of the set {G( 1) I G m ( F : ) ~ ~ converges ~ in K.

E

K}. Now the diagonal sequence

0

Finally, we mention some properties of T

X,A

as a topological vector space.

1.2.10. Theorem. a. TX,A is complete. is barreled.

b. T

X,A c. TX,A is bornological.

d. TX,A is Montel iff for all t > 0 the operator e-tA is compact. e. TX,A is nuclear iff for all t > 0 the operator e-tA is a Hilbert-Schmidt operator.

Proof. a.-c. Being a Frechet space, TX,A is complete, barreled and bornological. Cf. Theorem 1.1.15. d.

We prove that T operator

$(A)-'

is Montel iff there exists J, E B (IR) such that the X,A is compact on X. Compactness of $(A;-' implies that

for all t > 0 the operator e-tA is compact, because e-tA = ($(A)e-tA)$(~)-l. c) Let

cp

E

B denote a closed and bounded subset of TX,A. Then there exists

B

(IR) and a bounded subset Bo of

B = {cp(A)x Ix n

E

IN, Fn

=

E

x

such that

Bo}. Let (FnlnEm be a sequence in B. Then for each

cp(A)xn with xn

E

Bo. Since

$(A) -1

is a compact opera-

(sj)

tor, there exists a subsequence j E and ~ y E x such that -1 Since cp $ E B+(IR) the sequence II $ ( A ) xn - yll -+ 0 as j +. 1 in B tends to cp(A) $(A) y in TX,A. It follows that (cp(A)xn ) .

-.

?(A)

J,(b?) y

E

-

B, because B is closed. So we obtain a converging sub-

sequence (FnjljEm of the sequence (Fn)nEm with limit F *)

=

cp(A) $ ( A )

y

in B. so B is compact in T

X,A'

Let (f ) be a bounded subset in x. Then the closure of the set n ncIN {emb(fn) In E m} is a compact set in TX,A. So for all t > 0 the -tA sequence (e contains a convergent subsequence in x. fn)n c m Hence e-tA is a compact operator for all t > 0. So A has a discrete spectrum without accumulation points. Put +(A) = 1

+ i2, A E

IR.

s X,A

PAIRING AND DUALITY O F

Then JI

e.

E

+(A)-'

B+(IR) and

AND

T

171

X,A

i s compact.

follows t h a t A has a d i s c r e t e spectrum with eigenvalues ( h k ) k c m

+=) I t

and with a corresponding orthonormal b a s i s of eigenvectors ( vk ) kcl" m e 'kt i s convergent. where f o r a l l t > 0 t h e series k= 1 W e have t o prove, t h a t f o r each t > 0 t h e r e e x i s t s 0 < T < t such L

1

T, i n t o Tt

T,, with r e s p e c t t o t h e norms q t and T denote t h e completion of T t _XtA q,. Let t > 0. Consider J : T t 4 T Z t . Then J can be w r i t t e n as t h a t t h e canonical i n j e c t i o n J of

i s nuclear. Here

1

-

L

where ( * , . ) - ,

denotes t h e inner product of -Xkt mapping, because e k= 1

T t . Hence J i s a nuclear

-.

=+)

For each T > 0 t h e i n j e c t i o n i n j ,

:

x

GT,

i s continuous. L e t t > 0.

Then t h e r e e x i s t s T > 0 such t h a t t h e i n j e c t i o n

1

i s n u c l e a r . So t h e i n j e c t i o n i n j inj

t

=

J oinj,.

Remark. The space

Tt

:

t-tA

I t follows t h a t e

X 4 T t i s n u c l e a r , because is a t r a c e c l a s s o p e r a t o r .

0

can be w r i t t e n a s etA(X) i n t h e sense of Chapter A . 1 1 .

1.3. P a i r i n g and d u a l i t y of

SX,Aa n d T

X,A

In t h i s s e c t i o n , we p r e s e n t a Riesz-like r e p r e s e n t a t i o n theorem f o r t h e couple ( S X , A , T X , A ) . This theorem s t a t e s t h a t a continuous l i n e a r f u n c t i o n a l on SX,A i s represented by an element of l i n e a r f u n c t i o n a l on

sX , A

and, conversely, a continuous

TX,A by an element of TX,A. The spaces S

X,A

and T

X,A

e s t a b l i s h a dual p a i r . I n o r d e r t o o b t a i n t h e mentioned r e p r e s e n t a t i o n

X,A and T X,A' i . e . a s e s q u i l i n e a r mapping SX,A and l e t F E TX,A. Then t h e complex

r e s u l t we introduce a p a i r i n g o f S i n t o C. Let w

from 'X,A X T X , A number <w,F> i s defined by

E

<w,F> = (etA w , F ( t ) ) x

.

ANALYTICITY SPACES, TRAJECTORY SPACES AND THEIR D U A L I T Y

172

In t h e d e f i n i t i o n of <w,F> w e have t o t a k e t > 0 so s m a l l t h a t w

e

E

-tA

(x).

The d e f i n i t i o n of <w,D does not depend on t h e chqice of t > 0. Indeed, l e t -tA -TA w E e Then f o r a l l T with 0 < T S t w e have w E e and

(x).

(x)

- ( r - t ) A TA e w , F ( t ) I X = (etAw ( e T A w ,F ( T ) ) X= (e The t h u s defined p a i r i n g < * , * > of <w,F> = 0 f o r a l l F

TX,A, then w

E

sX , A =

and T

X,A

0 because

,F(t))X

i s non-degenerate:

x

c+T

XiA

. If

and

Conversely, i f <w,F> = 0 f o r a l l w L SX,A, then ( h , F ( t ) ) X= 0 f o r a l l h

E

x

and a l l t > 0 and so F = 0 .

The following theorem i s a genuine extension of t h e Riesz r e p r e s e n t a t i o n theorem t o our kind of spaces.

I. 3. I . Theorem. a. A l i n e a r f u n c t i o n a l

1

s X, A

i s continuous i f f t h e r e e x i s t s G

E

T

m on T X , A i s continuous i f f t h e r e e x i s t s w

E

s

on

X,A

such t h a t

b. A l i n e a r f u n c t i o n a l

XiA

such t h a t

-

F 'E

T

: m(F) = < w , D

.

X,A

Proof. a . c) L e t G

E

TX,A.

Then G = $ ( A ) g f o r some q

follows t h a t f o r a l l u

E

E

X and some J,

By Theorem 1.1.6 c o n t i n u i t y of t h e l i n e a r f u n c t i o n a l u *)

Let

E

B+(IR).

It

s X,A

I+ < u , Q

follows.

1 be a continuous l i n e a r f u n c t i o n a l on SX,A. Then t h e l i n e a r func-

t i o n a l 1oe-tA i s continuous on t h e r e e x i s t qt

E

x,

x

f o r a l l t > 0. By Riesz' theorem

t > 0 , such that

sX , A

PAIRING AND DUALITY OF

For a l l t > 0 and

T

I t follows t h a t gt+T = e - T A g t .

b. *)

Let w = e

€

SX,A, u

- t A g with g

E

= e

x

So t h e X-valued f u n c t i o n G on ( O , - )

-tA

TX,A. F i n a l l y , we observe

f,

b+

< w , D i s continuous.

T X,A'

Let m be a continuous l i n e a r f u n c t i o n a l on l e t ( f n ) n c m be a n u l l sequence i n

x.

and so ($(A) f n ) n E m is a n u l l sequence i n 3

1 w e o b t a i n m(emb(fn)) + 0 , n +

a continuous l i n e a r f u n c t i o n a l on X. w

E

x

Also,

such t h a t f o r a l l f

f o r each $

E

E

Let $

B+(IFO and

E

Then f o r a l l t > 0

II$(A)e-tAfnllX 9 II$(A)e-tAII IIfn1Ix + 0

$

173

X,A

and t > 0. Then we have

So t h e l i n e a r f u n c t i o n a l F *)

T

> 0 we have

defined by G ( t ) = gt, t > 0 , belongs t o that for all u

AND

m.

So

as n

+

-

TX,A. In particular, for t h e r e s t r i c t i o n o f m t o x is

I t follows t h a t t h e r e e x i s t s

x,

B+(IR) there e x i s t s w

Since $(A) i s s e l f - a d j o i n t i n

X,

$

E

x

such t h a t f o r a l l f

it follows t h a t f o r a l l f

E

E

D(Q(A))

x

ANALYTICITY SPACES, TRAJECTORY SPACES AND THEIR DUALITY

174

E

a l l J,

B+(IR).

E

Thus we s e e t h a t w

D ( $ ( A ) ) and $(A)w = w

Hence w

J,.

Finally, since

I t follows from Corollary 1.1.8 t h a t w

E

0

I. 3.2.

-

-

X,A

=

=

TX,A, because t h e space

I+

S i m i l a r l y , w e do

SX,A.

De f i n i t i o n .

The weak topology on

u

s;,A

provides only a r e p r e s e n t a t i o n o f t h e dual of SX,A.

T'

SX,A.

E

TX,*.

W e note t h a t it would be i n c o r r e c t t o w r i t e

not w r i t e

D($(A)) for

X i s dense i n T X , A , the r e p r e s e n t a t i o n

is valid for a l l F

TX,A

E

II, u

E

E

i s t h e topology induced by t h e seminorms

S X , A , where

The weak topology on F H ( < u , F > ( ,F

sX , A T

X,A

F E

i s t h e topology induced by t h e seminorms

TX,A, where u

Remark. The w a k topology on

TX , A '

sX , A

E

s X,A'

i s t h e weakest l o c a l l y convex topology

f o r vhich a l l l i n e a r f u n c t i o n a l s u H , u

T

S i m i l a r l y , t h e weak topology on

X,A

E

S X , A , a r e continuous.

i s t h e weakest l o c a l l y convex topology

___

f o r which a l l l i n e a r f u n c t i o n a l s F t+ on

TX,A a r e continuous.

I n t h e n e x t theorem we show t h a t i n both spaces

sX , A

and

T

X,A

weakly

bounded s e t s a r e bounded.

1.3.2.

Theorem (Banach-Steinhaus).

a. Let E be a s u b s e t of exists M

W

T

X,A

with t h e property t h a t f o r a l l w

> 0 such t h a t f o r every F

E

E one has I<w,F>I

each t > 0 t h e r e e x i s t s Ct > 0 such t h a t b. Let 0 be a s u b s e t of

S

X,A

IIF(t) IIx S Ct

E

SX,A t h e r e Then f o r

for all F

with t h e property t h a t f o r a l l G

e x i s t s MG > 0 such t h a t f o r every u

E

5 Mw.

E

0 one has II 5 MG.

T

X,A

B.

E

there

Then t h e r e

e x i s t s t > 0 such t h a t 9 i s a bounded s u b s e t of t h e H i l b e r t space e

- t A (x).

Proof. a. L e t t > 0. Then f o r a l l f all F

E

B

E

x

t h e r e e x i s t s a constant M

t ,f

such t h a t f o r

P A I R I N G AND DUALITY OF

sX , A

AND

T

175

X,A

From t h e o r d i n a r y Banach-Steinhaus theorem f o r H i l b e r t spaces it follows t h a t t h e r e e x i s t s a c o n s t a n t Ct > 0 such t h a t

b. Let

IJJ

B+(IR). Then f o r a l l g

E

that for all u

E

x

E

there e x i s t s a constant M

Jl,g

e

Hence it follows t h a t t h e r e e x i s t s a c o n s t a n t C

So 0 i s a bounded s u b s e t i n

e

s

+&A

i s a bounded s u b s e t of e

J,

such

> 0 such t h a t

and hence t h e r e e x i s t s t > 0 such t h a t

(XI.

0

In t h e next theorems w e c h a r a c t e r i z e weak convergence of sequences b o t h i n

S X , A and i n T X , A . I. 3.4.

Theorem.

The sequence ( such t h a t ( (W,Un)t

-f

~ tends ) ~t o ~ zero~ weakly i n

u

-tA

~ i s) contained ~ ~ ~i n e

u

(x)

sX,A

iff there e x i s t s t > 0

and f o r a l l w

E

e

-tA

(x),

0.

Proof.

i n t h e H i l b e r t space e-tA(X) Weak convergence of a sequence ( u ) n €IN means weak convergence of t h e sequence (etX un) n E i n ~ X.

-)

Let F

TX,A. Since etA u

E

t h a t = (etA u n , F(:)) *)

-+

0 weakly i n -+

Following Theorem 1.3.3 t h e s e t {u exists n

E

IN.

T

x

and s i n c e F ( t )

In

E

For each L > 0 we d e f i n e t h e p r o j e c t i o n llL by

J

=

0

L

dEA

.

x

it follows

IN} i s bounded i n SX,A. So t h e r e

> 0 and a c o n s t a n t CT > Onsuch t h a t 11 e T A u n11

nL

E

0.

5 CT

for a l l

176

ANALYTICITY SPACES, TRAJECTORY SPACES AND THEIR DUALITY

Let g

E

x

and l e t 0 < t <

T.

For each L > 0 t h e o p e r a t o r etA nL i s

bounded. Hence we g e t

, e t A n L g>


= (etA

u

F u r t h e r , we have lle-(r-t)AnL

II = e - ( T - t

I t follows t h a t

So f o r each

> 0 , w e f i x L > 0 so l a r g e t h a t

E

-(T-t) L

CT

II gllx < 4 E

and n e x t no E IN so l a r g e t h a t f o r a l l n > n

0

Then i t follows t h a t f o r a l l n > no

1.3.5.

CoroZlaw.

a. A convergent sequence i n S X , A i s weakly convergent. b. A bounded sequence i n S X , A contains a weakly convergent subsequence.

Proof. a. Cf. Theorem 1.1.10 -1.3.4. b. Let

( ~ ~ ) ~ ~ s~X , A ' ~ y a bounded sequence i n

such t h a t ( e

tA

sequence (e

u

So t h e r e e x i s t s a sub-

% j ) j E m which i s weakly convergent i n

t h e previous theorem t h a t t h e sequence

i n SX,A.

Then t h e r e e x i s t s t > 0

~ is a bounded ) ~ sequence ~ i n ~x.

x.

I t follows from

i s weakly convergent

0

P A I R I N G AN0 DUALITY O F

I. 3.6.

s

X,A

T

AND

X,A

177

Theorem.

The sequence ( F n ) n E I N tends t o zero i n T ( F n ( t ) ) n E mtends t o zero weakly i n

x.

X, A

i f f f o r a l l t > 0 t h e sequence

Pro0f. *)

Let w

E

SX,A. Then f o r a l l t > 0 s u f f i c i e n t l y small <w,F > = (etAw, F n ( t ) l X -+ 0 n

9)

Let t > 0 and l e t g

I. 3 . 7 . Corollary

E

x.

,

n

-f

-.

Then we have

.

a . A convergent sequence i n TX,A i s weakly convergent. b. A bounded sequence i n TX,A contains a weakly convergent subsequence.

Pro0f. a. T r i v i a l . denote a bounded sequence i n TX,A. b. L e t (F ) n neIN $ E B+(IR) and a bounded sequence ( g n ) n E m i n n

E

m.

(Cf. Theorem 1.2.5.)

x

such t h a t Fn = $ ( A ) gn,

The sequence (gn)nEINc o n t a i n s a weakly

convergent subsequence ( g n, ,! j E m . cFnj) j t m

Then t h e r e e x i s t s

converges weakly i n

Then t h e corresponding sequence

TX ,A'

0

The next theorem d e a l s with t h e question whether weak convergence of a sequence implies i t s convergence.

1.3.8.

Theorem.

The following t h r e e statements a r e equivalent.

a. For each t > 0, t h e o p e r a t o r e-tA is compact on

x.

b. Each weakly convergent sequence i n SX,A i s convergent. c. Each weakly convergent sequence i n TX,A i s convergent.

Proof a.

* b.

Let (

u

sX , A ' Then f o r some weakly i n x. Since by

~ be ) a weakly ~ ~ convergent ~ sequence i n

tA

t > 0 t h e sequence (e

unInElN

converges

ANALYTICITY SPACES, TRAJECTORY SPACES AND T H E I R DUALITY

178

assumption t h e o p e r a t o r e - ( t - T ) A i s compact f o r 0 < T < t , t h e

x.

un) n r I N converges i n t h e norm topology of sequence ( e T A -TA

means t h a t (u i s a convergent sequence i n e n neIN i n SX,A. b. * a .

L e t ( g n ) n c m denote a sequence i n

t > 0. The sequence e

- t A gn

-P

x

with gn

-P

(x)

0 weakly i n

0 weakly i n SX,A.

This

and hence

x.

Let

Hence t h e sequence

-tA

converges t o z e r o i n S X , A . SO i n p a r t i c u l a r , ( e tAgn) nc IN gn IIx -+ 0. I t follows t h a t e-tA i s a compact o p e r a t o r . IIea.

* c.

L e t (Fn)ncIN be a weakly convergent sequence i n TX,A. Then f o r a l l T > 0 t h e sequence (F ( T ) ) i s weakly convergent i n x. Using t h e n nrIN compactness of each o p e r a t o r e-tA, t > 0 , it follows t h a t each

sequence (F ( t + T I )

nE IN

i s norm convergent i n

T > 0 t h e sequence (F (T))

n

c.

* a.

have l i m n-

1.4.

nem

L e t (gn)nrIN be a sequence i n

,IntA

Then a l s o g

gn IIx = 0.

x

Hence f o r a l l

x.

which converges weakly t o zero.

TX,A. So

+ 0 weakly i n

x.

is norm convergent i n

by assumption, f o r a l l t > 0 w e

Hence f o r a l l t > 0, e-tA i s compact.

0

Sequence space r e p r e s e n t a t i o n

we consider t h e c a s e t h a t t h e nonnegative s e l f - a d j o i n t o p e r a t o r A generates a semigroup which c o n s i s t s of Hilbert-Schmidt o p e r a t o r s . Then t h e spaces

S X , A and T X , A a r e nuclear. Greater knowledge of t h i s s o - c a l l e d nuclear c a s e w i l l t u r n o u t u s e f u l f o r a b e t t e r understanding o f our mathematical i n t e r -

p r e t a t i o n of D i r a c ’ s formalism. Therefore, we reformulate some d e f i n i t i o n s of Chapter I i n t h i s s p e c i a l case.

A s a l r e a d y has been observed, i n t h e nuclear c a s e

X

5 h2 S

... (some of

A has a d i s c r e t e spectrum C < m for a l l

m

the h

may be equal) with

1 n= 1 t > 0. L e t ( v n ) n E m denote t h e corresponding orthonormal b a s i s of eigen-

v e c t o r s with A v

= Xnvn, n

e x i s t s t > 0 such t h a t w

E

E

e

IN.

-tA

A vector w E

(x);

x

belongs t o

it follows t h a t

We have t h e following c h a r a c t e r i z a t i o n .

s

X,A

i f there

SEQUENCE SPACE REPRESENTATION

179

1.4.1. Proposition. Let w

t

x.

Then w

E

sX , A

i f f t h e r e e x i s t s t > 0 such t h a t

Pro0 f. *)

C)

m

2

I

e2Ant (w,vn) There e x i s t s t > 0 such t h a t n= 1 -A t (W,V,)~ =o(e ). Let 0 <

<

m,

whence

< t . There e x i s t s C > 0 such t h a t

T

I

so eAnT ( w , v n ) x ~2 Ce-An(t-r)

n= ? 1 e2An7I (

W

,

V

~

)i s~

.

< m, a l s o t h e s e r i e s Since .-2An(t-T) n= 1 convergent. ~ ~ I t follows t h a t w E e-TA(X). 0

By t h e previous lemma SX,A i s isomorphic t o a dense subspace of t h e H i l b e r t space

e2 of

SI

square summable sequences. Indeed, p u t I ( w n ) n c m E C IN 3t,0 : wn = O(e-A t )

I

=

cp : S X , A

-f

S{(Xn)ncIN1 i s cp(w)

:

n

,

I+

n

( W , V ~ ) ~

S i m i l a r l y , t h e space T

X,A

1.

Then t h e isomorphism

e s t a b l i s h e d by

E

IN, w

E

SX , A

admits t h e following c h a r a c t e r i z a t i o n .

Proposition. T X , A i f f t h e r e i s a sequence (a

I. 4.2. F E

)

n nEIN

with a

A T = O(e

f o r a l l T > 0,

such t h a t f o r a l l t > 0

Proof. *)

For a l l t > 0 w e have F ( t )

Since ( F ( t l )

, vn)

A

t

E

= e n ( 2-t1)

x.

Take a f i x e d t > 0 and p u t

( F ( t 2 ) , v n ) , t h e d e f i n i t i o n of t h e a

does

180

ANALYTICITY SPACES, TRAJECTORY SPACES AND THEIR DUALITY

n o t depend on t h e choice of t. So f o r a l l t > 0 m

1

F(t) =

e

-Ant

a v

n n

n= 1

Moreover f o r a l l T > 0 , a

where t h e s e r i e s a r e X-convergent. s i n c e (F(T) +)

,v

a

n

o(e-An(t-T)

=

So F ( t ) =

m

A T

e

-Ant

)

= O(eAn') we have

and hence

a v is a member of n n

x.

The f u n c t i o n t I+ F ( t ) i s a

0

trajectory.

Remark. Any t r a j e c t o r y F

E

m

z n=l Y

n= 1

an vn'

T

n n

Also t h e space

can be represented by t h e formal s e r i e s

X,A

which converges i n

e-Ant a v

T X , A - ~ e n s e ,i . e .

for a l l t > 0, the series

a r e X-convergent.

T X , A i s isomorphic t o a sequence space.

To see t h i s we d e f i n e

t h e space T{(An)nEINI= { ( a n ) n c m c

Now t h e isomorphism @ :

OF : n

b+

c IN 1 Vt>,,

:

a

= ~

A t

( ne )

I .

T X , A + T I (An)nEm) i s e s t a b l i s h e d by

eAnt ( F ( t )

, vn)x

.

where any t > 0 may b e taken. I t i s n o t hard t o see t h a t @ extends cp. ( 0 r e s t r i c t e d t o emb(X)), then

e2

Moreover, if we p u t

j = O remb(X)

j can be seen a s a u n i t a r y o p e r a t o r from

depending on t h e choice of t h e orthonormal b a s i s ( v ) as exn ncIN pressed by Riesz-Fischer's theorem. The mapping 0 sends t h e t r i p l e

onto

),

~= O ) ( 1 )~.

Let t > 0 and l e t 0 < T < t . Since a

.-Ant

= O(e

SX,A c X c T

X,A

isomorphically onto t h e t r i p l e

x

SEQUENCE SPACE REPRESENTATION

'e,,jAj*

181

2'

Te2,jAj*

TX,A.

Then t h e expression <w,F> denotes t h e complex

Here we have

Let w

E

S X , A and tA

number ( e

F

E

w , F ( t ) ) X where t > 0 has t o be taken s u f f i c i e n t l y small. We

observe t h a t

m

Some comments and b i b l i o g r a p h i c a l notes The d e s c r i p t i o n of t h e Gelfand-Shilov space due t o De Bruijn [ B r ] .

s,,4

a s an a n a l y t i c i t y space is

We remark, however, t h a t t h i s r e s u l t i s not s t a t e d

a s such i n [ B r l . The general theory of t h e dual p a i r of l o c a l l y convex v e c t o r spaces

TX,A

s X,A'

s t a r t e d with t h e University r e p o r t [ G 1 1 of t h e second author of t h i s

monograph. An extended and improved version of t h i s r e p o r t appeared a s t h e t r i p l e of papers [G 2-41.

The name a n a l y t i c i t y space has been introduced by

t h e f i r s t author i n h i s Ph.D.

t h e s i s [E 11. The concept of a n a l y t i c v e c t o r

i s due t o Nelson [ N e 11. A combination of [G2-41

and [ E 11 r e s u l t e d i n CEG91. I n [EGKl t h e theory

of a n a l y t i c i t y and t r a j e c t o r y spaces i s embedded i n a t h e o r y on a much wider c l a s s of l o c a l l y convex v e c t o r spaces t h a t a r e p r o j e c t i v e and i n d u c t i v e

l i m i t s of H i l b e r t spaces. Additional r e f e r e n c e s on d i s t r i b u t i o n theory a r e [ G S 1-23, and [Schw 11.

[Li],

[Te], [ J o l

This Page Intentionally Left Blank

183

CHAPTER I I

L I N E A R MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

important result of the previous chapter is the characterization of the

An

non-strict inductive limit topology for the space SX,A.

It has been proved

that S X , A is the maximal common dense domain of the operators $ ( A ) , J,

E

B ( I R ) , and that the inductive limit topology for S

topology for which all operators $ ( A )

:

SX,A

+

X, $I

E

XtA

is the coarsest

B ( I R ) , are continuous.

These observations have led to complete characterizations of null sequences,

of bounded subsets and of compact subsets of

s

X,A' is very much like a strict inductive limit space.

As

It turns out, that

sX.A

importantly, also null sequences, bounded subsets and compact subsets of

the trajectory space TX,A have got an elegant characterization. The full strength of Hilbert space theory becomes available. In the present chapter, we employ these characterizations to obtain a detailed exposition of continuous linear mappings, of topological tensor products and of so-called kernel theorems. We emphasize that considerations of this type are not current in the classical distribution theories.

11.1. Continuous linear mappings between ananlyticity spaces and trajectory spaces As

in the previous chapter A denotes a nonnegative self-adjoint operator in

the separable Hilbert space

x.

We let

B denote a nonnegative self-adjoint

operator in a separable Hilbert space Y. In this section we present conditions, which imply the continuity of linear mappings of the following types

'X,A

-f

'Y,B

' 'X,A

-f

TY,B

' TX,A

-t

'Y,B

TX,A

TY,B

'

We also study a fifth type: continuous linear mappings from S X , A into which have an extension to a continuous linear mapping from T

X,A

into

'Y,B

TY,B'

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

184

We call them extendible. The next theorem is an immediate consequence of properties of the inductive limit topology of

sX,A

and of the fact that

sX,A

is bornological.

11. I. I. Theorem. Let

R denote a locally convex topological vector space.

L

SX,A *

:

R

linear mapping

A

is continuous

a. iff for each t > 0 the mapping L e-tA

:

x*R

is continuous.

b. iff for each null sequence (un)n6m in SX,A, the sequence ( L un) nclN is a null sequence in R.

Proof. a.

9)

(x)

x

-tA The operator e-tA is a continuous isomorphism from onto e for -tA G S X , A is continuous. So, each t > 0 . Moreover, the embedding e

(x)

if L

:

R

SX,A -+

is continuous, then 1 e-tA

:

x

+

R

is continuous.

Lt

denote the restriction of 1 to e-tA(X). Then by assumption 1t -tA is continuous on e (x). Let a be a convex open null neighbourhood - tA (x) = Lt-1 ( a ) is an e in R. Then for each t > 0 , t h e set L - ' ( Q )

e) Let

open neighbourhood in e-tA(X). Therefore, L - l ( a ) is open in SX,A. b. The proof of b. is a consequence of a. because null sequences in some Hilbert space e-tA(X) are null sequence in SX,A and vice versa. In the next theorem we characterize continuous linear mappings from into Sy,B.

sX,A

11.1.2. Theorem. Suppose P

:

SX,A *

sy,Bis a

linear mapping. The following seven conditions

are equivalent. a.

P

is continuous.

b. Let (

u

is ~ be)a null ~ ~sequence ~ in SX,A. Then the sequence ( P u ) n nElN

a null sequence in S

Y,B'

c. For each t > 0 , the operator P e-tA d. For each J,

$ ( B ) pe-tA

E

:

:

x

-+

Y is bounded.

B + ( I R ) and each t > 0 we have Pe-tA(X) c

x

-+

D ( J , ( B ) ) and

Y is bounded.

e. For each t > 0 there exists

s > 0

-tA such that P e

(x)

C

- SB e ( Y ) and

CONTINUOUS LINEAR MAPPINGS

esg pe-tA

:

x

Y i s bounded.

+

f . There e x i s t s a dense l i n e a r subspace

D

D the l i n e a r functional L

H

g

E

185

p, 9

:

w

Y such t h a t f o r each f i x e d ( P w , ~ ) ,w E SX,A, i s continuous

of

'X,A'

On

*

:

Y +

x

For each t > 0 , t h e o p e r a t o r e-tA

:

x

s X,A

g. For each t > 0 , t h e o p e r a t o r ( P e-tA)

i s bounded.

Proof. a.ob. a.

* c.

Cf. Theorem 11.1.1.

~

e

:

x- s~ ~ i~s , continuous. ~

* d.

P e-d(X) $ ( B ) pe-tA -tA

L e t t > 0. Since

the operator

c

i s continuous. Hence

In particular, ~

-+

continuous, because t h e i n j e c t i o n

c.

-+

Sy,B

s Y,B c

e :

x- +

Y ~i s ~

GY i s continuous.

D ( $ ( B ) ) f o r each

$

E

B+(W,

x.

i s d e f i n e d on t h e whole of Moreover, t h e -1 i s closed, because $ ( B ) i s a bounded o p e r a t o r .

o p e r a t o r $ ( 8 ) Pe

$ ( B ) P e - t A is

By t h e closed graph theorem, t h e l i n e a r o p e r a t o r bounded. d.

* e.

Let

u

denote t h e u n i t b a l l i n

J , ( B ) ?'eetA(U)

sy,B.I t

i s bounded i n

x.

x.

Then f o r a l l J,

So Pe-tA(U)

E

B+(IR) t h e set

i s a bounded set i n

follows t h a t t h e r e e x i s t s s > 0 such t h a t P e

bounded s u b s e t of e

-sB ( Y ) . Hence Pe-tA(X)

-tA

(u)

is a

c e-SB(y) and esB pe-tA

i s a bounded o p e r a t o r .

c.

f . We can t a k e

D

=

Y. Take g

E

Y f i x e d . For each f

E

X and each t > 0 ,

we have

Following Theorem 11.1.1, ?k f.

* g.

The l i n e a r f u n c t i o n a l

f o r some F

E

Prg

cp,, on

i s continuous on

sX , A '

S X , A can be r e p r e s e n t e d by

TX,A. Observe t h a t F ( t ) E SX,A f o r a l l t > 0. Now g

f i x t > 0. Then from ( * ) w e d e r i v e f o r each g

E

D

186

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

and hence g

E

-tA * -tA * D ( ( P e ) ) with ( P e ) g -tA *

dense i n Y , t h e a d j o i n t ( P e

= F ( t ) .Since 0 i s 9 -tA is densely defined. So P e

)

a c l o s a b l e everywhere defined and hence bounded o p e r a t o r on

(Pe-tA) *

Consequently, g.

* c.

is

x.

i s bounded.

P e-tA

The everywhere defined o p e r a t o r

i s bounded because

(Pe

-tA) *

i s bounded.

e.

* b.

T r i v i a l , because of t h e c h a r a c t e r i z a t i o n o f n u l l sequences i n

'X,A*

0

The n e x t two c o r o l l a r i e s a r e important f o r a p p l i c a t i o n s .

Corollq.

11.1.3.

Q

Let

x*

:

Y be a densely defined l i n e a r o p e r a t o r with D(Q)

f o r each t > 0 t h e o p e r a t o r a l l JI Vt>o

B + ( I R ) . Then

E

SB

3s,0

Proof.

:

e

Let w

-tA

2

maps :

Qe

x*

e-tA

sX,A

continuously i n

(x)

3

c

SX,A. Suppose

D ( J I (8) ) f o r

sy,B. ( I t follows

Y i s bounded.) -tA

SX,A. Then w = e

E

-tA

e

is bounded and

f f o r some t > 0 and f

E

x.

that

I t follows

that

-tA

Qe

So

Q

n

W+(m)

sX , A

maps

f E

D($(B))

i n t o S y l B , whence

2

:

=

Sy,B

SX,A

-+

. sy,B is

continuous by Theorem

0

II.1.2.C. 11.1.4.

Corollary

suppose

Q

:

and Q(Sx,,)

x* c

Proof. 2

i s c l o s a b l e i f f D(Q*)

where defined on operator that

'f i s a densely defined c l o s a b l e o p e r a t o r . L e t 0 ~ 2 )3

S y l B . Then 2 maps S X , A continuously i n t o

2 e-tA

* ) g

= e-tA Q*g f o r a l l g

i s densely defined and 2 e

i s a bounded o p e r a t o r . So

Q

E

s X,A

i s every-

D(2*) t h e

- t A i s c l o s a b l e . I t follows

obeys Condition c. of Theorem

0

11.1.2. 11.1.5.

i s dense i n Y . The o p e r a t o r Qe-tA

-tA Since ( Q e

x.

(Qe - t A ) *

s Y,B'

Theorem.

Let K : S X , A +

Ty,8

equivalent. a. K i s continuous.

be a l i n e a r mapping. The followiqg f o u r c o n d i t i o n s are

CONTINUOUS LINEAR MAPPINGS

b. For each t > 0 , s > 0 , t h e o p e r a t o r e-SBKe-tA

c. For each t > 0 , t h e r e e x i s t s Q

$ ( 8 ) - ' Ke-tA

:

x

+

E

x

+

Y

i s bounded.

B + ( I R ) such t h a t t h e o p e r a t o r

Y i s bounded.

d. For each s > 0 , t h e mapping e - s g K

Proof. a. * b.

:

187

S X , A + S y , g i s continuous.

:

T y , B i s continuous as w e l l a s t h e l i n e a r mapping e-" -sB Ke-tA i s a continuous l i n e a r o p e r a t o r Y. Hence e b.

* C.

u

Let t > 0 and l e t

x.

denote t h e u n i t b a l l i n

i s bounded i n Ty,g.

K e-tA(U)

* b.

b.

* d.

c

Y

and Q

T

from from

x

into

Y,B

into

i n t o Y.

Then t h e s e t

By Theorem 1 . 2 t h e r e e x i s t s a bounded

- t A (U) B + ( W such t h a t $(B)-l Ke t h a t t h e o p e r a t o r $ ( B ) - l Ke-tA : X -+ Y i s bounded.

s e t 6,

c.

sX , A

Let t > 0 and s > 0. Then t h e l i n e a r mapping K etA from

=

E

6,.

I t follows

T r i v i a l , because $ ( B ) e-sB i s bounded f o r a l l s > 0 . Let s > 0. Since (e-SBK ) e-tA

:

x

+

s Y,B

c

Y

i s bounded, from

Theorem 11.1.2 it follows t h a t edSBK i s a continuous l i n e a r mapping from S X , A i n t o Sy,g. d.

* a.

Let s > 0 and l e t w t h e r e e x i s t s rp

E

E

SX,A'

Then (Kw) ( s ) = e-"

Kw and by assumption

B+(JR) such t h a t

11.1.6. Theorem.

M M to Let

TX,A

:

x.

+

sy,Bbe

a l i n e a r mapping. L e t M

denote t h e r e s t r i c t i o n of

The following f i v e c o n d i t i o n s are equivalent.

a. M i s continuous. b. For a l l rp,$

E

B+(IR) t h e o p e r a t o r $ ( B ) Mrp(A) is a bounded o p e r a t o r from

X i n t o Y. B+(IR) t h e r e e x i s t s t > 0 such t h a t $ ( 8 ) MetA with domain (TX,A) = o < ? < t - T A ( X ) i s bounded as an o p e r a t o r from x i n t o Y .

c. For a l l J,

-tA

d. For a l l J,

E

E

B+(IR) t h e r e e x i s t s X i n t o Y.

s > 0 such t h a t esBM$(A) i s bounded a s

an operator from

e. There e x i s t s s > 0 and t > 0 such t h a t esB MetA i s bounded on

n

O
e-TA(X)as an o p e r a t o r from

x

into

Y. Hence (M e

tA

)

**

i s a con-

188

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

tinuous l i n e a r mapping from

s

XtA

into S

Y,B

and M = (MetA )

e-tA i s considered as a l i n e a r mapping from T X , A i n t o

Proof. a. * b.

L e t cp,$

E

B+(IR). Then

$ ( B ) Mcp(A)

:

sX,A'

sy,Bi s

TX,A *

Here

continuous. So

i t s r e s t r i c t i o n t o X i s bounded a s a l i n e a r o p e r a t o r from b.

*

a. L e t (Fn)neIN denote a n u l l sequence i n

x

in and cp n u l l sequence ( f ) n ncIN n E IN. SO f o r a l l J, E B+(IR)

I t follows t h a t

E

TX,*.

x

i n t o Y.

Then t h e r e e x i s t s a such t h a t Fn = cp(A) f n ,

B+(IR)

(M Fn) n ~ I Nis a n u l l sequence i n SY,B'

SO

M i s con-

tinuous. a.

* c.

L e t J,

x

:

B+(IR).

It is clear t h a t $ ( B )

Y. Hence ( $ ( B ) Mr)*

into

N

M

E

Ty,B + SX,A

Mr i s

a bounded o p e r a t o r from

i s bounded. Consider t h e l i n e a r mapping

d e f i n e d by

( $ ( B ) M ) * = n $ ( B ) on . I t follows t h a t ( $ ( B ) M*) ( Y ) c r* SX,A. So by Corollary 11.1.4 t h e l i n e a r mapping ( $ ( B ) M ) : : Y * SX,A i s continuous. Therefore, ( J , ( B ) Mr)* ( Y ) c e-tA;X) and tA e ( $ ( 8 )Mr)* : Y + x i s bounded f o r some t > 0. Now t h e proof i s * * extends $ ( B ) Me t A . complete, because t h e o p e r a t o r (etA($(B) Mr) )

W e obtain c

c.

* b.

T r i v i a l , because f o r a l l t > 0 and a l l cp

cp(A) e-tA a.

* d.

L e t J,

E

E

B+(IR) t h e operator

x

continuously i n t o S

i s bounded.

B+(IR).

The o p e r a t o r M $ ( A ) maps

A s a s p e c i a l consequence of Theorem II.1.2.d

exists s

7

Y,B'

it follows t h a t t h e r e

0 such t h a t esB M J, ( A ) is bounded regarded as an operator

from X i n t o Y. d.

* e.

Take a sequence ( t ) in I R ' with t C 0. I n d u c t i v e l y , w e d e f i n e n neIN +" i n IR with s > s n + l , sn C 0 and sequences (s,) nE rm and ( N ) n neIN N < Nn+l, N t

-.

Take any s

1

> 0 . I f f o r a l l N > 0 and a l l f E

x,

Ilfllx = 1

189

CONTINUOUS LINEAR MAPPINGS

then t h e o p e r a t o r e S I BMetlA i s bounded by 1 on t h e dense subspace

G

of

x

defined by G = { x A ( A ) fI f

E

extended t o a bounded o p e r a t o r on and f l

E

x,

x, A x. I f

c

IR Borel). So it can be

not, there exists N

1

> 0

IIf 11 = 1 , with 1 x

Since t h e f u n c t i o n

x

etZX

'[OtNi)

(A)

belongs t o B+(IR) t h e r e

e x i s t s s 2 , 0 < s 2 < %slsuch t h a t t h e o p e r a t o r

t A

sB

Me

'[O,N,)

(A)

i s bounded. I f t h e r e does not e x i s t N

2

> N

1

+ 1 and f 2

x,

E

IIf 11 = 1 with 2 x

t A

then similar arguments a s above y i e l d t h a t t h e o p e r a t o r eS2' M e 2 extends t o a bounded o p e r a t o r on and such f 2

E

x

x.

I f not then such a number N

> s2 > s 3 >

Suppose i n t h i s way we have obtained s N1 < N2 < N j <

2

exist.

... <

Nk and f l

,..., f k x, E

IIflIIx =

...

>

...

=

Sk

> 0,

IIfkIIx = 1

with t h e property t h a t t h e o p e r a t o r

i s bounded and

f o r a l l j = l , . . . , k . Then t h e r e e x i s t s 0 < s

k+ 1

< %sksuch t h a t t h e

operator

i s bounded. F u r t h e r , i f t h e r e does not e x i s t N k + l > Nk + 1 and fk+l

E

x,

llfk+lllx = 1 such t h a t

LINEAR MAPPINGS. TENSOR PRODUCTS AND KERNEL THEOREMS

190

then t h e o p e r a t o r eSk+lBMetk+lA i s bounded on G and hence w e a r e ready i f w e t a k e t = tk+land s = s k+l. I f n o t , w e o b t a i n such a number N k + l

and such fk+l

E

x

and proceed as i n d i c a t e d .

Thus w e o b t a i n t h e wanted sequences (

~ and ) ( ~N n ) n ~ Em

s

~and t h e

sequence ( f ) i n X. n nEIN we d e f i n e J, E B + ( I R ) by

By assumption, t h e r e e x i s t s s > 0 such t h a t t h e o p e r a t o r esB

i s bounded. However, f o r each s > 0 t h e r e e x i s t s ko for all k > k

0

w e have s

IIeSBM$(A) f k

IIy

k

E

M $(A)

IN such t h a t

< s and

S B Met k A

t Ile k

fkII > k

.

So a c o n t r a d i c t i o n i s e s t a b l i s h e d and t h e r e e x i s t s k E IN such t h a t (*)

e.

* b.

is violated.

L e t $,cp

B+(IR). Then w e have

E

0

From t h i s r e l a t i o n t h e r e s u l t follows.

Theorem.

11.1.7.

Let W : TX,A

-+

r e s t r i c t i o n of

a.

W

Ty,B W to

be a l i n e a r mapping. Let

x.

Wr

:

X

-+

Ty,B denote t h e

The following f i v e c o n d i t i o n s a r e e q u i v a l e n t .

i s continuous.

b. For each cp

E

B+(IR) and each

s > 0 t h e operator e

-sB

Wq(A) :

X

-+

Y is

bounded. c . For each s > 0 ,

(e-SBWr)*

( Y ) c SX,A.

d. For each s > 0 t h e r e e x i s t s t > 0 such t h a t t h e o p e r a t o r e-sB WetA : :

x

3

e-tA(X)

e . For each v t

-f

Y i s bounded.

sy,B, t h e

functional F

I+


,WD,

F E

TX,A i s continuous

CONTINUOUS LINEAR MAPPINGS

on

T

Proof. a. * b.

X,A’ Since

w

i s continuous, f o r each s > 0 and each cp c B + ( I R ) t h e

Wcp(A)

mapping e-sB

X

:

+

Y i s continuous. Hence e

bounded as a l i n e a r o p e r a t o r from b. * a .

191

x

x

in

and a n u l l sequence ( f )

n nElN

Wcp(A) is

Y.

into

TX,A.

Let ( F n ) n e l N be a n u l l sequence i n

-sB

Then t h e r e e x i s t s cp

cp(A) f n , n

such t h a t Fn =

B+(IR)

E E

lN.

Let s > 0. Then we have

II ( W F ~ ) ( s ) I I y

=

IIe

-sB

q ( A ) f n IIy

+

.

0

Thus t h e c o n t i n u i t y o f W follows. b.

*

c . Let s > 0. For all cp

Y.

on

Let g

E

Y.

* d.

w,)*

-sB

-sB

Wr)

* (Y)

(etA(e-”Wr)*)* E

wr)

)*g

rSX,*. *

E

D(cp(A))

wr) *


,W

D

f o r a l l cp

E

B+(IRI.

It

-sB

wr) *

i s a continuous

So t h e r e e x i s t s t > 0 such t h a t

i s bounded on Y. Now t h e o p e r a t o r

is bounded on

sy,B.Then

Wcp(A))* i s bounded

i s defined on a l l of Y and

Y i n t o SX,A.

t h e o p e r a t o r etA(e-SB

e. L e t v

c

c SX,A. It follows t h a t ( e

l i n e a r mapping from

4

(Y)

The c l o s e d o p e r a t o r ( e (e

d.

-sB W (e

Then

follows t h a t (e-” C.

-sB

B+(IR) t h e o p e r a t o r (e

E

x

and extends e

-sB

= (eSBv , (WF) ( s ) ) y .

We

tA

.

So w e g e t

sB tA * sB (e g , ( W F ) ( S ) ) = ~ ((e-SBWe ) e g , F ( t ) ) X f o r t > 0 s u f f i c i e n t l y small. So t h e l i n e a r f u n c t i o n a l F

i s continuous on e.

3

T

i s continuous on X.

E

B + ( I R ) . For each g c Y t h e l i n e a r f u n c t i o n a l

Theorem.

-S

I t follows t h a t t h e o p e r a t o r ( e

everywhere defined. Hence e

11.1.8.


X,A’

b. Let s > 0 and l e t cp

L e t t h e l i n e a r mappings

++

-sB

Wcp(A) i s bounded on

x.

8

Wcp(A))* i s

17

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEORENS

192

b e continuous. Then the d u a l l i n e a r mappings

are a l s o continuous.

P a . Compare Theorem I I . 1 . 2 . e w i t h Theorem I I . 1 . 7 . c . II.1.5.b

Look a t Theorem

0

and Theorem I I . 1 . 6 . c .

I t i s a n i n t e r e s t i n g q u e s t i o n whether a d e n s e l y d e f i n e d

l i n e a r o p e r a t o r from

T

from

XtA

( p o s s i b l y ) unbounded

i n t o Y c a n b e extended t o a c o n t i n u o u s l i n e a r mapping

Y,B'

Theorem.

11.1.9. Let

into T

x

x

E be a d e n s e l y d e f i n e d l i n e a r o p e r a t o r from

E

tended t o a continuous l i n e a r mapping and E*(Sy,B)

from T

X,A

i n t o Y . Then into

T

Y,B

iff

E

can be ex-

sy,Bc

D(E*)

SX,A.

c

=. e) I f

5

e x i s t s as a c o n t i n u o u s l i n e a r mapping, i t s d u a l mapping

E'

:

Sy,B

S X , A i s c o n t i n u o u s . For each f

-+

D ( E ) and e a c h v

L

E

sy,Bw e

have ( v , E f ) y = i v , Ef> = ( E ' v ,f I x Hence E* *)

3

E'

.

and E*(Sy,*) c SX,*.

From C o r o l l a r y 11.1.4 it follows t h a t E* maps *

'X,A' TY,B' 11.1.10. A

By t h e p r e v i o u s theorem, i t s d u a l (E Finally w e observe t h a t

*

(E

I

)

I

)

sY,B

continuously i n t o

maps T X , A c o n t i n u o u s l y i n t o

i s a n e x t e n s i o n of E .

0

CoroZZary.

2 : S X , A -+ sy,Bcan be extended t o a c o n t i n u o u s TX,A * T y , B i f f p h a s an a d j o i n t 2* : Y -+ x w i t h t h e

c o n t i n u o u s l i n e a r mapping

l i n e a r mapping

:

p r o p e r t y t h a t D(Q*)

3

sy,Band

2*(sy,g) c SX,A.

TOPOLOGICAL TENSOR PRODUCTS

193

11.2. Topological tensor products o f analyticity spaces and trajectory spaces

x

For two separable Hilbert spaces

and Y we consider the complex vector

space of Hilbert-Schmidt operators. We denote this vector space by For any

z

E

x

B

7

m

2

1

I I Z I I ~ , ~=

I I Z ~1 .1 <~

j=l

3

B

Y.

we have

.

m

y

x

in

Y and any orthonormal basis (e.),

x

The number IIzIIxQy does not depend on the choice of the orthonormal basis cej)jEm8 and inner product

z

IIzIIXBy defines a norm on

!+

x

B

Y. This norm comes from the

m

x

Endowed with this inner product elements in f e

x,

x

0

Y

are 5 a II,5

Q

x,

E

Y is a Hilbert space. Examples of Y, defined by ( 5

rl E

x

The linear subspace of

Q

x

Ba

Y.

The subspace

is called the sesquilinear algebraic tensor product of

x

@

(f,C)X

q,

Y which consists of all Hilbert-Schmidt operators

of the type just mentioned, will be denoted by space

Q q ) f =

and finite linear combinations o f these

x

0

x

Y is a completion o f

Y

B

x

and

Y.

x

@a Y

The Hilbert

with respect to II-IIxBy. Therefore,

Y is called a complete topological tensor product

of

x

and Y. We mention

the following elementary properties.

Proposition.

11.2.1.

a. V f,5Ex V g,nEy b. VXEC VfeX Vgey

(5

: :

@

X(f

n

I f

Q

g) =

So the canonical mapping

x

9)xey = ( f # S ) X (17rg)y.

Q

Y

x

g

(if)0 -f

x

0

Y

=

f

@

(Xg)

.

defined by [ f ; g ]

H

f

0 g

is anti-

linear in f and linear in g. c. VZEXQYV f E X Vgcy Let M

:

x

+.

x

and

L

:

(Z

:

Y +

I f

Y

Q

g)xey =

(Zf ,g)y

denote bounded linear operators. BY M

denote the linear operator from

x

B

Y into

x

Q

Y

defined by (M a

Q

L)Z

L

we

=

LZM*.

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

194

d. VfeX Vgfy

(M

:

0

L ) (f

@

e. The operator norms of M, L and M

IIM

@

LII

0

L are related by

@

L is injective.

.

IlMll IIL II

=

.

(Mf) @ ( L g )

g) =

f. If K and L are injective, then K

Only rather recently significant results have been attained in the theory of closable tensor product of unbounded closable operators. We mention the following definition and results. 11.2.2. D e f i n i t i o n . Let

A

with domain

Y

D(A)

c

X

@

Y

(A

D(A)

denote a densely defined linear operator in

we introduce the linear operator 1)f

B g =

(Af)

A

D(B)

On

g

and linear extension. Similarly, for a densely defined operator main

x.

Ba 7 by

we introduce the linear operator 7

B

@

on

X

@

B with do-

D ( B ) by

and linear extension. 11.2.3. Let a.

A

A

-.

and

@a

I

B be

self-adjoint operators in

and 'I Ba

A

@

and Y , respectively.

B are essentially self-adjoint in

unique self-adjoint extension by b. The operator

X

I+ 10 B

A cza

1 and 1

with domain

D(A)

@

@

x

@

Y. We denote the

€3.

D(B)

is essentially

self-adjoint. We denote its unique self-adjoint extension by C.

If A and

B

are nonnegative, then

A

@

I, I

@

B

and

A

a

A

a

B.

6 are nonnegative. 0

In the sequel

A

and B denote two nonnegative self-adjoint operators in the

separable Hilbert spaces

x

and

Y.

TOPOLOGICAL TENSOR PRODUCTS

195

e.

11.2.4.

For a l l t 2 0 . w e have t h e r e l a t i o n s

Applying t h e r e s u l t s of Chapter I we can introduce t h e spaces

sXmY ,A&'

TXeY,AaB' SX@Y,A@I' etc'

Definition.

11.2.5.

The canonical s e s q u i l i n e a r mapping by [ u , v l + u

€3 V .

Here t h e symbol

SX,A

:

@

€3

This d e f i n i t i o n i s c o n s i s t e n t , because f o r u exist f

SO

x,

E

that U

i s defined

E

S X , A and v

E

sy,g. t h e r e

g E Y and t > 0 such t h a t u = e-tA f and v = e-tB g. T h i s

V E

@

sy,B + SX€3y,AaB

X

i s t h e same a s i n P r o p o s i t i o n 11.2.1.

s~~y,A~g*

Theorem

11.2.6.

'XgY, AaB

i s a complete t o p o l o g i c a l t e n s o r product of SX,A and

sy,B. This

means a. SXey,AaB

i s complete.

b. The s e s q u i l i n e a r mapping 0

:

SX,A

sy,B+ SX€3y,AleB

x

c . The l i n e a r span of t h e image of a, i . e .

SX,A ea

sy,Bi s

dense i n

i s continuous.

t h e a l g e b r a i c t e n s o r product

sXoY, AmB'

Proof. a. By Theorem 1.1.13 a l l a n a l y t i c i t y spaces a r e complete. b. Let (

(

v

~ be) a ~convergent ~ ~ sequence i n S

w

~ be )a convergent ~ ~ ~ sequence i n

e x i s t s t > 0 such t h a t 11 etA (wn - w ) n +

-.

II

s YtB -f

I t follows t h a t

II e

t AmB

(Wn

€3

vn

-

w

€3

v) IIx€3y s

X,A

E s and l e t X,A s Y,B' Then there

with l i m i t w

with l i m i t v

E

0 and 11 etB (vn - v ) 11

+

0 as

196

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

Since t h e l a t t e r expression tends t o zero a s n tends t o i n f i n i t y , t h e proof i s complete. c . For each t > 0, t h e l i n e a r space e of t h e H i l b e r t space e -t(AmB)(X

-tA

(x)

%ee-tB(y)

i s a dense subspace

Y ) . Thus t h e wanted r e s u l t follows.

@

0

Next we introduce a complete t o p o l o g i c a l t e n s o r product of t h e t r a j e c t o r y spaces

11.2.7.

TX A

,

and

Ty,B.

Definition.

The s e s q u i l i n e a r mapping

mT

:

TX,A

Ty,B

x

TX@y,AmB,

+

[F;GI

-t

F

@T

G is

defined by F

mT

G : t t+ F ( t )

@

.

G(t)

This d e f i n i t i o n i s c o n s i s t e n t , because

(F

mT

-TA

G) ( tfT) = (e

= e- T ( A m B )

(F(t)

@

F(t))

@

-TB

(e

G ( t ) ) = e-T

G(t))

=

(F aT G ) ( t )

.

11.2.8. Theorem.

TXey,AmB i s a complete t o p o l o g i c a l t e n s o r product of T X,A and T y , B . This means a.

TXoy,AmB i s complete.

b. The s e s q u i l i n e a r mapping

@

:

TX,A

x

Ty,B

+

TXey,hB

i s continuous.

c. The l i n e a r span of t h e image of aT, i . e . t h e a l g e b r a i c t e n s o r product of

TX,A and Ty,B, i s dense i n T

X@Y, h B '

Proof. a. By Theorem 1.2.10 a l l t r a j e c t o r y spaces a r e complete. b. The c o n t i n u i t y follows from t h e r e l a t i o n

which holds f o r a l l F

E

TX , A , G

E

T y , B and t

> 0.

TOPOLOGICAL TENSOR PRODUCTS

c)

X

@

Y i s dense i n TXey,AmB and

x

B

Y i s dense i n

197

x

0

Y. Thus t h e r e s u l t 0

follows.

A l s o we introduce

mkced t o p o l o g i c a l t e n s o r products of an a n a l y t i c i t y space

and a t r a j e c t o r y space. Therefore we have t o introduce two new types of s p a c e s . H e r e t h i s i s done i n an ad hoc manner. I n t h e comments t o t h i s chapt e r we r e f e r t o a more systematic approach.

11.2.9.

Definition.

W e introduce t h e following l i n e a r subspace of

T XeY, ZeB

I n t h i s space w e t a k e t h e l o c a l l y convex topology generated by t h e seminorms

We observe t h a t $(A)

11.2.10.

1 = $ ( A a I ) f o r a l l Bore1 f u n c t i o n s Jl.

8

Definition. 63

ST

w

B

The s e s q u i l i n e a r mapping w aST G : t

11.2.11.

H

'

'X,A

G(t)

TY,8 + TXaY,le8 i s defined by

.

Theorem.

The space T(SXey,AaI

SX,A and Ty,8. a. T(SXey,AaI

,I

B

8) i s a complete t o p o l o g i c a l t e n s o r product of

This means

,7

@

B)

i s complete.

b. The s e s q u i l i n e a r mapping

is continuous.

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

198

C.

The algebraic tensor product of

'X,A

s X,A

and TylB, i.e. the linear span of

Ty,B under eST, is dense in T ( S X e Y , A e l '

x

Proof. a. Let (0

,1

be a Cauchy net in T(SXgy,Ael

)

8). Then the net (0

@

)

is

also a Cauchy net in T X e y l l e p B . Since the latter space is complete there with Ilaa(t) - O(t) /Ixey XeY, 1eB B+(IR) and each t > 0, the net ( ( $ ( A )

exists @ $ E

E

T

From the closedness of the operator $ ( A ) 6(t)

E

D($(A)

@(t)

6

SXey,Ael

b. Let $

E

@

I).

+

7 ) Oa(t)) converges in

@

@

0 for all t > 0. For each

x

0

Y

1 it follows that

Since this is true for all $

E

B+(IR) we have

for each t > 0. Cf. Corollary 1.1.9.

B+(IR) and let t > 0. Then for all w

E

sX , A

and G

E

T

YiB

From this equality, the continuity of eST follows. c. Let

@ E

T(SXey,Ael

,I

0

8). Then for all t

> 0, O(t)

1

E

S XeY, A e l and the

sequence (@(;)I tends to 6 in T ( S , 'I @ 8). Hence sXeY, A e l is nE IN XeY, A e l dense in T(SXey,Ael , 'I 8). Since SX,A @a Y is dense in SX e Y ,A e l and Y c Ty,B, the assertion follows. 0 11.2.12. Definition. We introduce the following linear subspace of T

XeY, A e l

In this vector space we take the locally convex topology generated by the seminorms u

( E ) = II (1 0

t#$

.

$(B)) E(t) IIxey

11.2.13. Definition. The sesquilinear mapping F eTS v : t

I+

Q

TS

F(t)

:

TX,A

0

v

x

sy,B + TXey,Ael

is defined by

.

(It is clear that for all t > 0 , F(t)

@

w

E

sXeY ,leB')

TOPOLOGICAL TENSOR PRODUCTS

199

The proof of the following theorem runs the same as the proof of Theorem 11.2.11 and is omitted. 11.2.14. Theorem.

,A

The space T ( S XeY,

TX,A and

Sy,8.

I)

B

is a complete topological tensor product of

So

a. T(SXey,Ie8 ,A

Q

I)

is complete.

b. The sesquilinear mapping

TS

Q

:

TX,A

x

Sy,B

-+

T(SXey,lQB ,A

Q

I ) is con-

tinuous. c. The algebraic tensor product of T linear span of

TX,A

sy,Bunder

x

and

X,A

sY,8'

i.e. the image of the

is dense in T ( S XaY,I&3 'A

QTS,

'I).

'

- ~ 8 The linear operator 1 8 e , s > 0 , extends to a continuous linear mapping -sB from TXeY,AaI into XQy,AQI. The image is denoted by ( 1 Q e ) T X ~ Y ,Ael' In this subspace of TXQy,AQI the following norms are well-defined.

with n

E

IN.

It is not hard to prove that ( 1

0

-sB e

)

T X ~ Y ,AQI

becomes a Frechet space in

this way. In fact it equals the trajectory space T

X@e-S8(Y) ,A@I.

Now we are able to introduce a second topological tensor product of

and Sy,8.

T

X,A

11.2.15. De f i n i t i o n . We introduce the following linear subspace of S(TXey,AQI t 1 The space

(1

Q

e-")

S(TXey,AsI , I

Q

8

B)

Q

-sE e

TXeY,AeI

*

8) is an inductive limit of the spaces

x S(TXay,AQI , I

( @ , o * ~ =
8) = s>o U (I

TXey,AQI. For its topology we take the inductive limit topology.

On T(SXQy,AQI ,I

where

Q

TXsY ,AQ~'

, (1

Q

Q

8) we introduce the pairing

Q

, "8

€8 e ) n>xQy

> 0 has to be taken sufficiently small. The number

e@,n% does

not

200

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

depend on t h e choice of

due t o t h e t r a j e c t o r y p r o p e r t y of 4 . F i n a l l y , we

E

introduce t h e embedding of S ( T XeY,AaI

I) 0

emb(r,) : ti-+(e-tA

We observe t h a t f o r F

T

E

and w

X,A

because f o r s > 0 so small t h a t w

F aTS w

(I

E

@

e

-S

E

E

e

E

"

t >

sY,B

we have

( Y ) w e have

TX@Y ,A@l

)

Theorem.

11.2.16.

a. The a l g e b r a i c t e n s o r product of T X , A and

(TX@Y, A@I , I b. L e t Q

E

@

sy,Eunder

QTS is dense i n

E).

,I

T(SXay,Ael

e,

@

8 ) . Then t h e l i n e a r f u n c t i o n a l

: 0 t+ Q

i s continuous on S ( T C.

.

o

,

-S8

I):

T(SXey,l@B , A

into

')

@

XeY,A@l '

@

The embedding d e s c r i b e d i n D e f i n i t i o n 11.2.15 i s continuous.

Proof. a . For each s > 0 , U

s>o

X @a e-"(Y)

x

@a e-"(y)

i s dense i n S(T

b. To prove t h e c o n t i n u i t y of that 0 c

Lo

(r

restricted t o e-sB )

i s dense i n

(I

@

e,

TX@e-S~(y),Aal.

X@Y,A@I

Hence

@

it i s necessary and s u f f i c i e n t t o prove

-sB ) TXay,A@7

e

i s continuous. For a l l

TX@Y ,Am1 we have 1

-A ( g ~ , n > g l2 II ( I a e S E n(;) 1 llXgyII(en s I )

4(s)

llXay

.

From t h i s t h e c o n t i n u i t y follows. C.

Let t > 0 and l e t J, inequality is v a l i d

E

B + ( I R ) . Then f o r a l l W

E

T XaY,Ael t h e following

TOPOLOGICAL TENSOR PRODUCTS

201

So t h e l i n e a r mapping

i s continuous.

From t h i s t h e c o n t i n u i t y of emb follows

Def i ni t i on.

11.2.17.

S i m i l a r t o D e f i n i t i o n 11.2.15 we introduce t h e space

with t h e corresponding i n d u c t i v e l i m i t topology.

On

T(SXay,Ie8 , A

@

I)

x

S(TXeY,Ie8 , A

@

I ) we introduce t h e p a i r i n g

e ~ , c =s <~z ( E ) , ( e E A I ) pXay , where Z

E

T(SX@y,I@8, A @ I ) and 5 E S(TX@y,I@8 , A @ I ) . The embedding of a I ) i n t o T(SX4y,A@r , I 8 8) i s defined by

S(TX@y,IaB, A

emb(<) : t w ( I

@

e-t8)g

,

5

E

s

(

T

~, A ~ 7) ~. ,

~

S i m i l a r t o Theorem 11.2.16 w e have

11.2.18.

Theorem.

s X,A

a. The a l g e b r a i c t e n s o r product of

S(TXey,Ie8 , A

@

b. For each f i x e d E

5

I+
and

Ty,8

under

BST i s

dense i n

I). E

T(S

X@Y, re8

,A

ep

I) the linear functional

%A

i s continuous on S (TXaY,

,A

@

I).

c . The embedding described i n D e f i n i t i o n 11.2.17 i s continuous.

~

~

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

202

11.3. Kernel theorems I n t h i s s e c t i o n we show t h a t t h e elements of each of t h e complete topolog i c a l t e n s o r product o f t h e preceding s e c t i o n can be i n t e r p r e t e d as continuous l i n e a r mappings of one of t h e types w e d i s c u s s e d i n t h e f i r s t s e c t i o n of t h i s c h a p t e r . W e give necessary and s u f f i c i e n t c o n d i t i o n s on t h e semigroups e-tA and e-tB,

which ensure t h a t each t o p o l o g i c a l t e n s o r product

comprises a l l continuous l i n e a r mappings o f t h e corresponding type. In t h i s case w e s a y t h a t a k e r n e l theorem holds.

Case

K ~ :Continuous l i n e a r mappings from

T

L

SXey,AmB. Then w e d e f i n e t h e TX,A i n t o Sy,B i n t h e following way.

Let 8

For each F

T

E

For

T

into S

Y,B'

w e d e f i n e 8F by

X,A

8F = ( ( e T A@

K1

XtA

a c t i o n of 8 as a l i n e a r o p e r a t o r from

E

7)

6) F(T) .

> 0 t h i s d e f i n i t i o n makes sense and _ J e s n o t depend on t h e choice of

(x

1

since 8

T. Moreover,

whence 8F

11.3.1.

E

e

Y) f o r

s u f f i c i e n t l y small, w e have

T

sy,B.

Theorem. 1

a. For each 8 by K

E

1

sXey,AmB,

E

t h e l i n e a r mapping 8 from T

X,A

into S

YtB

defined

i s continuous. 1

b. For each 8

sXeY, AmB'

E

<8F

F E T

X,A

,DY= <6 ,F

@

G>

XBY

andGET

YtB

*

c. I f f o r each t > 0 a t l e a s t one of t h e o p e r a t o r s e-tA,

Schmidt, then

s XeY,AmB

e-tB is H i l b e r t -

comprises a l l continuous l i n e a r mappings from

TX,A i n t o Sy,B. d. SX@X,AmA comprises a l l continuous l i n e a r mappings from T t h e o p e r a t o r e-tA is Hilbert-Schmidt f o r a l l t > 0.

XiA

into S

XtA

iff

203

KERNEL THEOREMS

Proof. a. The o p e r a t o r 8 defined i n K 1 s a t i s f i e s Condition b. of Theorem 11.1.6.

6

Indeed, s i n c e

6

D(cp(A)

4

operator on b. L e t

^e

6

E

, G>y

^e

n Y, w e have

$ ( B ) ) for 0 (cp ( A ) 1.

SXey,AsB.
x

E

cp,$

. Since 6 sXeY, A s B '

e

=

E

B+(IR), wkence $ ( B ) e c p ( A )

i s a bounded

Then f o r T > 0 s u f f i c i e n t l y small w e g e t

TA

eTB

= ( ( (e

) G)F(T) , G ( T ) l Y

=

1

=

c . Let

r

<e , F

0

G

>

~

~

~

be a continuous l i n e a r mapping from

sy,B.

TX , A i n t o

Then by Theorem

t h e r e e x i s t s T > 0 such t h a t t h e o p e r a t o r e T Br e T Ais bounded

II.1.6.e,

on e-TA(X)c X. Now p u t -$TB

r = e

-$T6

{e

.TA)

TB (e

.-$TA]

-~TA

r

The o p e r a t o r between { * I i s Hilbert-Schmidt, -$T8 i s Hilbert-Schmidt. Hence r E e 1

t > 0. I f e

-tA

E

- ~ T Aor

sx8y,AreB.

d. The i f - p a r t is a s p e c i a l c a s e of c. For t h e o n l y - i f - p a r t

because e i t h e r e

consider t h e o p e r a t o r e

-t A

:

TX,A

+ SX,A f o r each

0

SXaX,AreA, it has t o be Hilbert-Schmidt.

Case K ~ :Continuous l i n e a r mappings from SX,A i n t o

Ty,B. TXey,Ale8. Then we d e f i n e t h e a c t i o n of K as a l i n e a r mapping from i n t h e following way. into T

1

1

Let K E

'X,A

Y,B

For each w

E

s X,A

we d e f i n e Kw by

Here f o r each s > 0 w e t a k e 0 < T < s so small t h a t w t i o n of K w ( s ) , s > 0, does n o t depend on 1

p e r t y of K .

T

E

e

-TA

(x).

The d e f i n i -

because of t h e t r a j e c t o r y pro-

204

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

Theorem.

11.3.2.

1

a . For each

K

T XaY,AeBt h e l i n e a r o p e r a t o r K

E

:

SX,A

Ty,B

d e f i n e d by K2

, e - tB

is Hilbert-

-+

i s continuous.

k

b. For each

<w

c T X@,Y,Aag'

,Kv>

=
@

' 'y,g

'X,A' w

we have

,

c. I f f o r each t > 0 a t l e a s t one of t h e o p e r a t o r s e

T XaY,AmB comprises a l l continuous l i n e a r mappings from

Schmidt, then

SX,A i n t o T d.

-tA

Y,B'

TXeX,AeA comprises a l l continuous l i n e a r mappings from S i f f f o r each t > 0 t h e o p e r a t o r e-tA

into

X,A

T

X ,A

i s Hilbert-Schmidt.

proos. a. We u s e Condition b. of Theorem 11.1.5. Let t > 0. Then e-tBKe-tA b. Let

^K

E

c. L e t

Q

:

TX@y,AsB,

II.1.5.b function

Q(t)

t(t) x

@

-T

B

E

w E e-TAand v

Y,B

E

e

Y. ( Y ) f o r some

E

x

T > 0. Then we have

be a continuous l i n e a r mapping. According t o Theorem

t h e o p e r a t o r e-Tg Le-TA is bounded f o r a l l

3 on

I

Then

T

-+

'X,A

=

T

> 0 . Define t h e

( 0 , m ) by

@

Y f o r a l l t > 0 , because w e can w r i t e

By assumption a t l e a s t one of t h e o p e r a t o r s e-4tA, e-'rtB i s H i l b e r t -

.

Schmidt. I t i s c l e a r t h a t and

represents

a(t

+ T ) = e-T(AeB)

as described i n K2.

a(t).

Hence

5

E

T

X@Y, AmB

KERNEL THEOREMS

205

d. The if-part is a special case of c. For the only-if-part consider the linear mapping emb follows that t t+ e-2tA belongs to

TX@X, AaA'

s

:

X,A CtTX,A. It Hence e-tA is Hilbert-Schmidt

0

for all t > 0 . Theorems 11.3.1-2 present the kernel theorems which are current in the classical distribution theories. Here also kernel theorems will be proved for the continuous linear mappings from

ous linear mappings from TX,A into

s

s and for the continuY,B as we know, a novelty in

into

X,A Ty,B- AS far

distribution theory. Case K ~ continuous : linear mappings from

Let

^L

T(SXeY,,oB , A

E

mapping from For each w

SX,A

^L

as a linear

into S y l B in the following way.

SX,A we define Lw by

E

1

-

~w

K3

S

X,A into 'Y,B' Then we define the action of

I).

@

(eTAw)

= L(T)

.

This definition makes sense, if T > 0 is taken so small that w The definition of Lw does not depend on all $ Lw

E

E

B+(IR), it follows, that Lw

T

> 0 . Since

L(7) E

D ( $ ( B ) ) for all $

E

E

D(7

-T A

e

E @

(x).

$ ( 8 ) ) for

B + ( I R ) , whence

sy,B.

11.3.3. Theorem. 1

a. For each

L

E

T(SXay,

sy,B defined by 1

b. Let L

E

,A

T(SMy,IeB , A
, G>Y

@

I)

the linear operator

K3 is continuous. @

I), w

E

SX,A,

1

=

G E

L from SX,A into

Ty,B. Then

.

4 L , w aST G>A

c. If for each t > 0 at least one of the operators e-tA, e-tB is HilbertSchmidt, the space T(SX@y,leg mappings from

S

into

,A

@

1 ) comprises all continuous linear

S

Y,B' X,A d. T(SXeX,leA , A 0 7 ) comprises all continuous linear mappings from

S

X,A -tA is Hilbert-Schmidt. into SX,A iff for each t > 0 the operator e

Proof. a. We use Condition c. of Theorem 11.1.2.

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

206

1

L e t t > 0. Then f o r a l l Q

Q ( B ) Le

^L

b. Let w

-t A

belongs t o X

Q

E

@ST G

E

D(1

@

$(8)). So t h e o p e r a t o r

Y and t h e r e f o r e i s bounded.

,A @ l), w T(S X@Y, l a 8 6 S (TXay, laB , A 8 1 ) .

E

L(t)

B+(IR),

SX,A and G

E

Ty,B. We note t h a t

E

W e get

-

aB < ~ w,G>y = (e ( L ( T ) e T A w ) , G ( a ) ) y =

1

=

Here

T

> 0 has t o be taken so s m a l l t h a t w

small t h a t C.

.

4 1 , w aST G S A

i(T)

Let P : SX,A

-P

(I

E

e-OB)

Sy,B be

e

E

-TA

(x)

and n e x t a > 0 so

(X 0 Y ) .

a continuous l i n e a r mapping. By Theorem I I . 1 . 2 . c

f o r a l l t > 0 t h e r e e x i s t s st > 0 such t h a t t h e l i n e a r o p e r a t o r :

x

-P

.P ( t )

=

Pe

eStB Pe-tA

Y i s bounded.

-tA

,

t >

Now d e f i n e

o

^P

on (0,m) by

.

Then w e can write

A

Because of our assumptions, = (e-TAo

I)

P(t),

P(t)

' _SXoY,IaB' Since

i t follows that

P

^P(t+ T I = p e - ( t + T ) A =

T ( s ~, A e~ I )~. , ~ ~ ~

d. The i f - p a r t is a s p e c i a l case of c. For t h e o n l y - i f - p a r t

t > 0 the o p e r a t o r

consider t h e i d e n t i t y 1. I t follows t h a t f o r a l l

1 e-t A is Hilbert-Schmidt.

Case K ~ :Continuous l i n e a r mappings from

Let

E

mapping from

T

For each F

T

E

X,A X,A

T

X,A into TY,B* 8). Then w e defined t h e a c t i o n o f into T i n the following way.

T(SXey,AoI , I

@

Y,8 we d e f i n e OF by

0

as a linear

207

KERNEL THEOREMS

8F

K4

: s

-

(O(s) e

-+

t A) F ( t s ) ,

s > 0

Here ts > 0 has t o be taken so small t h a t $ ( s )

( e-t,A

E

1)(x

B

0

y). It is

obvious t h a t t h e d e f i n i t i o n o f OF does n o t depend on t h e choice of t

11.3.4.

Theorem.

i E T ( SXey,Ael , 1 0 8 ) t h e l i n e a r o p e r a t o r T y , B a s defined i n K4 i s continuous.

a. For each

b. L e t 8

E

T(SXPy,Aml , 1


,

=

@

.

<8

B), F

E

TX,A and v

, F aTSvPB

0 from

T(SXeY,AeI , I

Schmidt, then

T

X,A

d. T(SXBX,ABI -into

T

X,A

X,A

into

. e-tB i s H i l b e r t -

B ) comprises a l l continuous l i n e a r mappings

T Y,B'

into

,I

B

T

Sy,B. Then

E

c. I f f o r each t > 0 a t l e a s t one of t h e o p e r a t o r s e-tA, from

.

@

A ) comprises a l l continuous l i n e a r mappings from T

X,A

i f f f o r each t > 0 t h e o p e r a t o r e-tA i s Hilbert-Schmidt.

Proof. a. We use Condition c. of Theorem 11.1.7. Let s > 0. Then t h e r e e x i s t s t > 0 such t h a t

t(s)

E

-tA

(e

0

1)(x

B

Y).

I t follows t h a t t h e o p e r a t o r e-sB 0 etA i s bounded on i t s domain D(etA).

b. F i r s t , observe t h a t F B TS v E


Y --

=

c. Let 'f :

S(TXey,ABl

,7

B

8 ) . we have

( e S B v , (;(s)etsA) F ( t s ) ) y =

.8 , F aTS vBB .

T X , A + Ty,B be a continuous l i n e a r mapping. According t o Theorem

11.1.7 f o r each s > 0 t h e r e e x i s t s ts > 0 such t h a t t h e o p e r a t o r

e-sB Y etsA

.Y ( t )

X

:

=

-+

Y

e- t g Y

i s bounded on D(etSA). We d e f i n e

,

?

by

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

208

Then we can write

11.4. Matrix representations In this section we assume that

sX,A

and

that there exists an orthonormal basis eigenvalues of A with eigenvalues A,

sy, 8 are nuclear spaces. It means (u~)~,~ in x which consists of

t 0 satisfying Vt,o

Similarly, there exists an orthonormal basis (

v

:

:

x

-+

e2

and

v

:

<

m.

~ in)Y which ~ ~ consists ~

of eigenvectors of 8 with eigenvalues pk t 0 satisfying V We define the unitary operator U

-xtt

Y, e=1

x

-+

e2

t>O by

. .

k=l

e-'kt

<-.

sX,A is isomorphic to the sequence space se2,UAU* and sy,g is isomorphic to the sequence space se2,,gv*, i.e. for all w E SX,A we have

Then uw

E

se2,uAu*

and for all v

E

sy,g,

Vv

E

se2,vsv*.

The operator

u

extends

TX,A onto Te2,UAU* and, similarly, V extends to an onto Te2,VBu*. In the appendix to the previous chapter

to an isomorphism from isomorphism from

T

Y,B we have given characterizations of these type of sequence spaces.

The four types that we discussed in Section 11.1 admit a matrix representation with respect to the orthonormal basis (u

4.) &IN

and (Vk)ktIN"

Here we

give characterizations of the corresponding four types of infinite matrices. Case MI: Continuous linear mappings from

T

X,A

into S

Y,8'

11.4.1. D e f i n i t i o n . The set TMS{ ( l e ) t E m ("1 ,

(Gke)k,ecm

kEml c

CmXm

consists of all infinite matrices

for which there exists t > 0 such that

MATRIX REPRESENTATIONS

Let

(‘kl)k,&lN

0 <

T

TMSI(A,),Em;

E

209

( p k ) k E I N l and l e t F

TX,A. Then f o r

E

< t both s u f f i c i e n t l y small w e have t h e e s t i m a t i o n

I t implies t h a t t h e s e r i e s

i s convergent i n “ke)k,&lN

On

Y.

TX,A m

So it makes sense t o d e f i n e t h e a c t i o n of t h e matrix

by m

M1

Then O i s a l i n e a r mapping from

11.4.2.

X,A

into

sY,B

by ( * )

Theorem.

a. For each into

TX,A b. Let

T

r

(6ke ) k,&IN E TMSC(hl)eEm; (pk)kEmIN) s y , a defined by M 1 i s continuous.

( rke)k,,EIN

be a continuous l i n e a r mapping from

f i n i t e matrix

belongs t o TMS{

t h e l i n e a r mapping 8 from

TX,A into

sY,a. Then

t h e in-

defined by

; (pk) kcm}

and r e p r e s e n t s

r.

Proof. a. W e prove t h a t t h e r e e x i s t s

T

> 0 such t h a t

Then t h e o p e r a t o r e r g B e T A extends t o a Hilbert-Schmidt o p e r a t o r from

x into

210

LINEAR MAPPINGSl TENSOR PRODUCTS AND KERNEL THEOREMS

Y.

So l e t t > 0 be so small t h a t

Then f o r 0 < T < t w e have t h e e s t i m a t i o n

Now t h e r e s u l t follows Theorem 11. .6.e. b. Since t h e r e e x i s t s t > 0 such t h a t t h e o p e r a t o r (etg

r etA)* i s

bounded,

it follows t h a t

Remark. The space TMSt (Xe)eEIN;

(vk) kcINj can

be i d e n t i f i e d with

’ L 2 d 2 ,UAU*leVB V* Case M2:

Continuous l i n e a r mappings from

11.4.3.

Definition.

The s e t

SMT{ ( X l ) e E m ;

(R,)k,,tIN

c Cmxm

X,A

into

T

Y,B‘

c o n s i s t s of a l l i n f i n i t e matrices

f o r which f o r a l l t > 0

E

exists t >

(pk)kem}

S

o

SMT{ ( k e ) e E IN;

such t h a t w

(p,)

e-d(X).

kEml and

l e t w r SX,A. Then t h e r e

so f o r all s > 0,

o

< r < t

MATRIX REPRESENTATIONS

211

The e s t i m a t i o n implies t h a t t h e mapping

i s a member of T Y, 8’ So it makes sense t o d e f i n e t h e a c t i o n of ( Kkk? k,LclN on SX,A by

Then K i s a l i n e a r mapping from

sX , A

i n t o TylB.

11.4.4. Theorem. a. For each (?kelk,eew

SX,A into T b. Let

2

E

SMT{ ( h e ) t E m ;

(pk)kclN1

as defined by M2 i s continuous.

Y.8

(2(

be a continuous l i n e a r mapping from

f i n i t e matrix

belongs t o

t h e l i n e a r mapping K from

SMTI

k! ) k,&IN

i

defined by

(pk)

sX , A

i n t o TylB. Then t h e in-

and r e p r e s e n t s

2.

Proof. a. W e prove t h a t f o r each t > 0

Then f o r each t > 0 t h e o p e r a t o r e-tB Ke-tA i s a Hilbert-Schmidt o p e r a t o r and hence Condition c . of Theorem 11.1.5 is s a t i s f i e d . So l e t t > 0. Then t h e following e s t i m a t i o n y i e l d s t h e wanted r e s u l t

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

212

b. Since f o r each t > 0 t h e o p e r a t o r e-tB

Case M3: Continuous l i n e a r mappings from 11.4.5.

Definition.

The s e t

SMSI(Ae)ec,;

(Lke)k,ecm

( p k ) k E m }c C m x m

Qe-tA i s bounded, w e g e t

sX,A

into

’Y,B*

c o n s i s t s of a l l i n f i n i t e matrices

with t h e property

1

E

SMSI(AeIeE,;

Let ( L k e )k ,kIN -tA e x i s t s t > 0 such t h a t w E e

that

whence

This implies t h a t t h e series m

m

( p k ) k E m } and l e t w

(x).

E

SX,A.

Then t h e r e

L e t 0 < T < t. Then t h e r e i s s > 0 such

213

MATRIX REPRESENTATIONS

converges i n t h e H i l b e r t space

on SX,A by

(Lke)k,ecm

t h e a c t i o n of

e e B B ( y ) . Therefore, i t makes sense t o d e f i n e

“3

Then

L

11.4.6.

i s a l i n e a r mapping from

into

s Y,B’

Theorem.

a. For each (1 )

k l k,tEIN

S X , Ai n t o b. L e t

s X,A

P

matrix

:

s Y,B

:x,A

-+

E

SMS{ (Xe)e61N;

( p k ) k E m l t h e l i n e a r mapping

L from

as defined by M3 i s continuous.

sy,Bbe

(Pke)k,eEm

a continuous l i n e a r mapping. Then t h e i n f i n i t e

defined by

Proof. a. Let t > 0. Then t h e r e e x i s t s s > 0 such t h a t

( sup k,&M

So we g e t the e s t i m a t i o n

I t follows t h a t t h e o p e r a t o r d. of Theorem 11.1.2

.IrsB

it follows

L e-tA i s Hilbert-Schmidt. t h a t L i s continuous.

b. For a i l t > 0 t h e r e e x i s t s s > 0 such t h a t t h e o p e r a t o r e

i s bounded. T h i s y i e l d s

SB

By Condition

Pe

-tA

:

x

+

Y

214

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

Remark. The space SMS{

*

( p k ) ksm} c a n be i d e n t i f i e d w i t h

,UAU* e 7 ) .

Case M ~ :C o n t i n u o u s l i n e a r mappings from T 11.4.7.

Definition.

The s e t

TMT{(ALILcIN;

(ik,)k,L61N with t h e

property

i s e s t i m a t e d by

Hence f o r a l l s > 0 the series m

is c o n v e r g e n t i n Y . I t follows t h a t m

m

into T

Y,B’

( v k ) k c I N } c C m X m c o n s i s t s o f a l l i n f i n i t e matrices

Then t h e series

m

X,A

MATRIX REPRESENTATIONS

215

T Y,B‘

belongs t o

Therefore it makes sense t o d e f i n e t h e a c t i o n of ($

ke ) k , k l N

m

On

TX,A by

m

Then 8 i s a l i n e a r mapping from

T

X,A

into T

Y,B

11.4.8. Theorem. L

a. For each (8Q)k,LEm

TX,A

into

b. Let Y

:

I

T

YiB

TX,A

-+

E

TMT{(A,ILEm;

(pklkcm1

t h e l i n e a r mapping O from

as defined by M4 i s continuous.

Ty,E be

a continuous l i n e a r mapping. Then t h e matrix

(‘+‘ke)k,LEm defined by

belongs t o

TMTI ( A t ) e E

i

(u,)

ke

,J

and r e p r e s e n t s Y .

Proof. a . L e t s > 0 and f i x t > 0 such t h a t

Then t h e e s t i m a t i o n

shows t h a t e from

x

-sE

8 e

ttA

can be extended t o a Hilbert-Schmidt o p e r a t o r

i n t o Y . Hence O i s continuous by Condition d. o f Theorem 11.1.7.

b. Let s > 0. Then t h e r e e x i s t s t > 0 such t h a t t h e o p e r a t o r

-sE

(e

‘Y etA)* : Y

-+

x

i s bounded. I t follows t h a t

LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS

216

Based on the given matrix representations for the four types of continuous linear mappings a matrix calculus has been developed. The rather simple form of the characterizations is of great help in the construction of examples of continuous linear mappings, in the proof of the continuity of linear operators as mappings between analyticity spaces and trajectory spaces and in the construction of analyticity spaces on which a given set of operators acts continuously.

Some comments and bibliographical notes See also the notes on Chapter B.I. The proof of Theorem 11.1.6 as presented in [G 41, and, also, LEG 101, Ch.

I, is not correct. See Ichinose [Ich] for results on the closability of tensor products of unbounded closable operators (cf. Proposition II.2.1.d). For the proof of Lemma 11.2.3 see Weidmann [Weill. A

systematic approach to mixed topological tensor products of analyticity

and trajectory spaces is presented in CEG91. The matrix calculus of Section 11.4 is due to the first author, cf. [E 11 and CEG91. For generalizations of results of this chapter to a class of spaces which are both projective and inductive limits of Hilbert spaces see [EK] and [EGK]

.

217

CHAPTER I l l

I L L U S T R A T I O N S OF A N A L Y T I C I T Y SPACES AND TRAJECTORY SPACES

Chapters 1-11 contain the functional analytic introduction of the triple

If we consider a nonnegative self-adjoint operator

X

= L2(M,p),

where

(M,p)

A

in the Hilbert space

is a o-finite measure space, then

as a space of test "functions" on M and T tions on M. In particular, if nuclear analyticity space

S

(M,p)

X,A

S XrA

can be seen

as a space of generalized func-

is a Federer measure space, then any

L 2 ( M t u ) ,A

turns out to be a space of genuine

test functions. These functions obey a measure theoretical Sobolev lemma. Thus,

'L2(M,p) ,A

becomes a genuine test space if M is e.g. (an open set in)

lRn, a Riemannian manifold or a disjoint union of these. The precise formulations and proofs of these results are given in Chapter IV. Here we present some illustrations of analyticity spaces in terms of complex analysis.

Also,

we discuss the connection between the theory of analyticity

spaces and trajectory spaces and Lie group representation theory. The material of this chapter is composed of results gathered by the authors. It should be mentioned, that this chapter is a summary of these results and only serves as an illustration of the theory. Therefore, most proofs are omitted.

111.1. Analyticity spaces based o p the Laplacian operator For each Riemannian manifold M a so-called Laplacian operator A can be defined. The operator A is an elliptic differential operator. For the Riemannian measure

on M the operator A is an unbounded nonnegative self-

adjoint operator in L 2 ( M , p ) . Thus we obtain the triples

218

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

The elements of T

a r e s o l u t i o n s of t h e d i f f u s i o n equation

L ~ ( M , P ), A V

with t h e property t h a t f o r a l l t > 0 , u ( t )

E

L2(M,v).

I f M i s a compact r e a l a n a l y t i c Riemannian manifold, then s L2 (M,!J) ,A' can be i d e n t i f i e d with t h e space of a l l r e a l a n a l y t i c f u n c t i o n s on M. Here we consider

M

=

sq-',

M

= IR, (O,m), [ - n , n l

X = L2(m)

A

,

and CO,nl,

and v 2 3 ; f o r v = 4 we consider

mq.

t h e u n i t sphere i n

=

(--)d22 ' ,

A'=

v > 4 .

dx

The spectrum of AV e q u a l s t h e s e t of a l l nonnegative real numbers. For each i n i t i a l condition [ q ] (+)

L2(JR) t h e corresponding s o l u t i o n u of t h e equation

E

can be obtained by means of F o u r i e r transformation. I f Cql-

E

L2(lR)

denotes t h e Fourier transform of

C q l , then u i s given by

For each t > 0 , t h e c l a s s u ( t )

L 2 (IR) c o n t a i n s t h e e n t i r e a n a l y t i c re-

presentative u

E

t m

ut:

zI+-

1

dS

Ji;;

-m

which obeys t h e e s t i m a t e

-m

with

-B = (2V -1)

(t(2V)

2v

)

1

2v-1

Conversely, i f an e n t i r e function w s a t i s f i e s an e s t i m a t e of t y p e

(*),

then

ANALYTICITY SPACES BASED ON THE LRPLACIAN OPERATOR

f o r each t with 0 c t < ( 2 v ) grable function

x

-2v

2v- 1 (a)2v-1

,

219

t h e r e e x i s t s a square i n t e -

such t h a t

W e a r r i v e a t t h e following c h a r a c t e r i z a t i o n .

III. 1.1.

Theorem.

> $, can be i d e n t i f i e d with t h e space of a l l e n t i r e m e space S L2(W),AV' f u n c t i o n s w, which s a t i s f y

3 v 3A>0 B>O y6IR

I

Iw(x + i y ) 2 dx 5 Aexp

.

(2B 2%) Iy(

-m

i s not a n u c l e a r space. Since A' has continuous spectrum, sL 2( JR are t h e & f u n c t i o n s Examples of continuous l i n e a r f u n c t i o n a l s on s L2 (IR),dV with complex argument and t h e i r d e r i v a t i v e s , i . e .

where k

U

IN { O ) and a

6

C.

More general f u n c t i o n a l s can be obtained as follows: Take a Jordan curve and a bounded continuous f u n c t i o n g o n

w

g(z)

H

W(Z)

dz

c.

Consider t h e l i n e a r f u n c t i o n a l

.

C Examples of continuous l i n e a r mappings on

SL2 (IR), A v ' v >

$,

are the

following e

ZAV

,

z

E

C with R e z L 0 ,

220

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

with JI a multiplyer, i . e . an e n t i r e f u n c t i o n with t h e p r o p e r t y

I t follows from simple a n a l y t i c and geometric c o n s i d e r a t i o n s t h a t t h e

s L2 (IR), A V i n t o i t s e l f . S i m i l a r l y , t h e i r a d j o i n t s i n t o i t s e l f . Then t h e c o n t i n u i t y and e x t e n d i b i l i t y of t h e s e

mentioned o p e r a t o r s map map

sL2 ( IR) , A v

o p e r a t o r s follows from Corollary 1 1 . 1 . 4 The elements of t h e space T

can be regarded a s s t r o n g l y d i v e r g e n t

Lz(IR) ,AV

T

Fourier i n t e g r a l s of elements of

and Theorem 11.1.9.

A s usual

L,(IR),

2

denotes t h e

operator of m u l t i p l i c a t i o n by t h e i d e n t i t y f u n c t i o n i n L 2 ( I R ) . of

TL2(IR),(p)v

The elements

admit a simple c h a r a c t e r i z a t i o n .

T 2 can be i d e n t i f i e d with t h e space of L 2 ( I R ) , (2 ) a l l equivalence c l a s s e s Crpl of Bore1 f u n c t i o n s on IR with

The space

t h e property t h a t f o r a l l

E

> 0

c s H exp(-c(c2)v) r p ( ~ 1 1

L~(IR)

E

.

The F o u r i e r transform IF i s a u n i t a r y o p e r a t o r on L 2 ( I R ) , onto S 'Lz(IR) , A v L2 (IR), ( Q 2 ) V . l i n e a r mapping from T

Therefore IF can b e extended t o an i s o m e t r i c

L, (IR) ,AV

The space T

L2 (IR), A v

f u n c t i o n s G on C

X

onto T

L, (IR), ( P ) V '

can be i d e n t i f i e d with t h e space of a l l which a r e defined by

(O,-)

G(z,t) =

where any [ q ]

which maps

TL2 (IR) (22)" I

can be taken.

ANALYTICITY SPACES BASED ON THE LAPLACIAN OPERATOR

Heuristically speaking, the space T

L2 ( W , A possibly divergent Fourier integrals

221

v consists of all

Since e-tA" is not a Hilbert-Schmidt operator there are no kernel theorems valid in this case. That is to say, there exist continuous linear mappings from

S

L,(IR)

,AV

into T

which do not arise from a trajectory

L,(m

l ~ v

corresponding to the two-dimensional diffusion equation

%={(-$)

2 v

A

t

+(--;i>}K,

'a

at

0 .

aY

solution of the equation du _ dt-A 4

t > O

U ,

with initial condition [ c p ]

E

I

L2(IR),

is given by

m

u(t)

=

[x

I+ -m

Kt(x,S) rp(S)dS

1,

t > 0

,

where m

The kernel (x,S ;t) I+ K ( x , S ) is the Poisson kernel for solving the t Dirichlet problem on the half-plane t > 0:

This is not surprising, since at least formally

222

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

111.1.2.

Theorem.

The space

s L~ (IR),A$

can be considered as a space of f u n c t i o n s w , which a r e

a n a l y t i c on a s t r i p around t h e r e a l a x i s and which s a t i s f y m

Examples of e x t e n d i b l e l i n e a r mappings on

ezAa,

z c C , O s a < t ,

s L2(lR) ,A' e

za+

a r e t h e following:

,

Rez 5 0 , bEIR

Ra

,

a E m

,

Tb

,

ZA

,

/ I c IR\{O1

,

P

I

I

% . H e r e J, i s a f u n c t i o n which i s a n a l y t i c and uniformly bounded on a s t r i p

around t h e r e a l a x i s . Most remarks o f Example E l a l s o apply here.

In o r d e r t o make A a s e l f - a d j o i n t o p e r a t o r i n L2(0,-) boundary c o n d i t i o n s a t x = 0. Therefore w e d e f i n e A. operator

-$

w e have t o p r e s c r i b e as the differential

on t h e domain

regarded a s a subspace of L 2 ( ( 0 , m ) ) .

Here H2(0,-)

i s t h e Sobolev space de-

f i n e d by

u

E

H2(0,-)

:CI

u

E

c 1( O , - ) ,

u and u' are square

i n t e g r a b l e on ( O , - )

and

1

X

: u ' ( x ) = u'(0)

+ 0

rp(S) dg

.

ANALYTICITY SPACES BASED ON THE LAPLACIAN OPERATOR

The space

s

> 4,

-

L2((0,-))rAV'

we define A

Besides A

223

c o n t a i n s a l l odd f u n c t i o n s o f S

L2 (IR),A:.

a s t h e d i f f e r e n t i a l operator

-

0'

The reader w i l l n o t be s u r p r i s e d , t h a t t h e space

d2 -on dx2

sL2(0,a) ,A:'

t h e domain

'

'On-

t a i n s t h e even f u n c t i o n s of SL2 (IR), A v . W e observe t h a t

As an easy exercise, w e l e a v e t o t h e r e a d e r t h e c o n s t r u c t i o n of examples of continuous ( e x t e n d i b l e ) l i n e a r mappings and continuous l i n e a r f u n c t i o n a l s

i n t h e s p i r i t o f E l and E2.

The s e l f - a d j o i n t o p e r a t o r A

i s defined as t h e d i f f e r e n t i a l o p e r a t o r

P

P

) = {U E

H,(C-IT,~I) I ~ ( - 7 )=

d2 dx

on t h e domain

D(A

-

u(7)

A

u'(-7)

=

u'(n)1

2

,

where

111.1.3.

Theorem.

> 'r c o n s i s t s p r e c i s e l y of a l l 27-periodic L2 ( [-IT, 7 1) I A: ' e n t i r e f u n c t i o n s w, which s a t i s f y

The space S

The space

S

L 2 ( [ - n , r l ) ,A:

"

c o n s i s t s of a l l 27-periodic f u n c t i o n s which a r e

a n a l y t i c i n a s t r i p aroung the real a x i s .

n

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

224

Examples o f continuous l i n e a r mappings a r e t h e following

I f v > 5 w e have a l s o 1, with b

E

C.

An i n t e r e s t i n g c l a s s of continuous l i n e a r f u n c t i o n a l s on S

u > 5, i s given by

t v :W

J

H

L2 [ - T I ,

II

1, A':

wdv

C-n ,slx i m where

!J

denotes a s u i t a b l e Bore1 measure on C-n,nl XilR.

Special cases a r e

t h e &functions withcomplex argument and t h e i r d e r i v a t i v e s .

sL,(C-m,nl)

5 a r e represented by 2n,Ap p e r i o d i c hyperfunctions. L e t us e x p l a i n t h i s a s s e r t i o n a b i t . Each continuThe continuous l i n e a r f u n c t i o n a l s on

ous l i n e a r f u n c t i o n a l

1

on S

L2 (C-n,nl) ,A:

can be w r i t t e n a s

f o r a s u i t a b l e element G of T

L2 ([-n,n]) ,A' ' has a d i s c r e t e spectrum {E2 I n P malized eigenfunctions [en], n E Z,

The o p e r a t o r A

Each t r a j e c t o r y G i n IRx (0,m)

T

E

221 and corresponding nor-

-

L,(C-n,n])

,A:

can be represented by t h e f u n c t i o n G on

defined by

-

G(x,~)=

1 J21

1

cn e

- In I t e i n x

,

X E I R , t > O

neZ

with s u i t a b l e c o e f f i c i e n t s c (Cf. Chapter 1, appendix.)

s a t i s f y i n g sup Ic nEz

I e- I n i t

<

m

f o r a l l t > 0.

ANALYTICITY SPACES BASED ON THE LAPLACIAN OPERATOR

-

Y

225

Y

Now t o G we r e l a t e t h e f u n c t i o n s G+ and G- defined by

and

-

Due t o t h e growth condition on t h e c o e f f i c i e n t s c n , t h e f u n c t i o n G+ i s a Y

2n-periodic a n a l y t i c f u n c t i o n on I m z > 0 and G- i s a 2n-periodic a n a l y t i c f u n c t i o n on I m z < 0.

A straightforward calculation y i e l d s

Y

w(x - i t ) G+(x+ it) dx

<w,G> =

+

-n =

-n

#

W(;)HG(z)dz

I'

-

w(x + i t )G-(x- it)dx =

.

ct

Here t h e f u n c t i o n HG i s d e f i n e d by

and t h e path of i n t e g r a t i o n

ct

c o n s i s t s of t h e two d i r e c t e d l i n e segments

{z

E

C I I m z = - t , n 2 R e z 2 -n}

{z

E

C 1 I m z = t , -IT

,

and

W e observe t h a t

111.1.4.

S R e z 5 n]

HG(z) = - l i m lim Im z+-m I m-'2

H

G

(z) =

l c 2&

O '

Definition.

The space H(IR,2n-per)

c o n s i s t s of a l l 2n-periodic f u n c t i o n s @ which a r e

a n a l y t i c on 6\IR and f o r which t h e l i m i t

The space

TL2([-n,nI)

space H (IR,2n-per)

.

lim I m z+-

"(2)

=

-

lim O(z) e x i s t s . I m z+-m

3 can be c h a r a c t e r i z e d i n t e r m s o f t h e f u n c t i o n

,Ap

226

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

111.1.5. Theorem. G

E

TL2(C-n,nl),A; iff there exists 8

E

H(1~,2n-perlsuch that 8 = HG'

Consequently, we obtain the following representation theorem. 111.1.6. Theorem.

A linear functional 8

E

t on S

L2 (C-n,nl) ,A: H(1~,2n-per)such that

is continuous iff there exists

Here t > 0 has to be taken sufficiently small dependent on w.

-

H(~~,zn-per).Then 8 = H for some G E T Since G L,(C-n,nl) ,A: ' G(x,t) = G+(x +it) + G-(x -it) = 8(x +it) - 8(x -it) it follows that

Let 8

E

N

a . .

lim [8(x +ic) E+O

-

8(x -iE)1 exists in the distributional sense of

TL2(C-n,nl) . In hyperfunction theory 8(x +iO) - 8(x -iO) and it is called the

this limit is denoted by hyperfunction corresponding to 8 .

Abusively, 0 itself is called a hyperfunction, also. If 0(x + i O )

-

0 (x -iO) = h(x) , where h is an integrable function, then we have

Conversely, since every integrable function h can be regarded as an element to each integrable function h there corresponds a TL2 ([-r,n]) ,A$ ' hyperfunction 0 such that Of

In Section 111.3 we introduce a class of non-periodic hyperfunctions.

As in E3 the closed subspace of odd functions in the space is equal to

S

L2([Ofnl) ,A:,o

. Here the operator AP

s

L2 ( C - n , n l ) ,A; equals the differential

rO

ANALYTICITY SPACES BASED ON THE LAPLACIAN OPERATOR

-

o p e r a t o r - d2 dx2

221

with domain

+

The closed subspace of even f u n c t i o n s i n

.

sL2(C0,*l)

D(Ap,e)

Here A

= {v

equals Ple

can be w r i t t e n a s

on t h e domain

dx

D(A )

E

-

sL ~ ( [ - n , n l ) , A p

P

1 v'

( 0 ) = v' ( n ) = 0 ) 4 L 2 ( [ 0 , n l )

.

Examples of continuous l i n e a r mappings and continuous l i n e a r f u n c t i o n a l s

can be found by t a k i n g s u i t a b l e combinations of t h e o p e r a t o r s mentioned i n E4'

E6 Here Sq-l

X

=

L2(Sq-' ,w

q

)

,

'1 A = A LB

'

w

t h e normalized s u r f a c e measure 9 of Sq-' and ALB t h e Laplace-Beltrami o p e r a t o r . The o p e r a t o r ALB i s c l o s e l y 9 r e l a t e d t o t h e Laplacian A = .C In terms of t h e usual s p h e r i c a l 3 t h e o p e r a t o r A can be expressed by coordinates r , B 1 , 0 q- 1 denotes t h e u n i t sphere i n lRq,

(-

...,

3).

a w ,

j = 1 , . . . , q -1. j The eigenvalues of ALB a r e t h e e n t i r e numbers k ( k + q - l ) , k = 0.1.2, with

where

i s expressed i n terms of t h e d i f f e r e n t i a t i o n s

...

multiplicity

%.

The o r d e r of growth of t h e sequence (%IkcIN

The members of each %-dimensional

e q u a l s kq-2.

eigenspace corresponding t o t h e eigen-

value k ( k + q - l ) , a r e c a l l e d k-th o r d e r s p h e r i c a l harmonics. Next w e consider t h e o p e r a t o r

A simple c a l c u l a t i o n shows t h a t a s p h e r i c a l harmonic o f o r d e r k i s an Y

eigenvector of

,,A

with eigenvalue k2 and v i c e versa. Therefore, t h e opera-

228

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

a r e bounded. So w e have

k51 % e-kt

'

m

Since

<

OJ

f o r a l l t > 0 , t h e a n a l y t i c i t y space sL2(sq-1

t w q ) ,ALB

is nuclear.

111.1.7. a.

Theorem. 3

SL2 (sq- 1r wq) r ALB t i o n s on

can be seen a s the space o f a l l r e a l a n a l y t i c func-

W1.

can be seen a s t h e space of a l l f u n c t i o n s cp on Sq-l s ~ 2 ( ~ q - l , w q )$, ~ , , which can be extended t o a harmonic f u n c t i o n on an open neighbourhood of t h e c l o s e d u n i t b a l l Bq-l

= {€, E

lRn

1

16

lq

5

1).

Here

I * I 9 denotes

the

usual Euclidean norm.

0

W e p r e s e n t only some h e u r i s t i c arguments f o r t h e l a s t r e s u l t . I n t r o d u c t i o n -t of r = e transforms t h e Laplacian A i n t o

The expression between c u r l y b r a c k e t s can be f a c t o r e d i n t o two e v o l u t i o n equations

The second f a c t o r can be w r i t t e n as

a at

-$ +

'LB

The elements of SL (sq-l equation

2

-$

correspond t o s o l u t i o n s u of t h e e v o l u t i o n

~w~)~ALB

THE SPACES

saB

OF GELFAND AND SHILOV

229

Such a s o l u t i o n u i s a harmonic f u n c t i o n on t h e b a l l

and vice versa.

W e conclude t h i s s e c t i o n with t h e d e s c r i p t i o n of some c l a s s e s of l i n e a r

5 , which a r e based on a combination of a n a l y t i c

o p e r a t o r s on SL2 (sq-l

w q ) 8 ALB and geometric considerations.

- Mu1t i p l i e r s . L e t JI

SL

E

f i x e d . Then t h e l i n e a r mapping M

5 be (W1,%) ,ALB 4

s ~ 2 ( ~ q - l , w q,ALB )

JI

sends

i n t o i t s e l f because t h e prcduct of two ( r e a l ) a n a l y t i c i s again ( r e a l ) a n a l y t i c . Also,

f u n c t i o n s on Sq-l

i n the H i l b e r t space L2 (sq-',u

q

1, whence M

J,

maps

MJ,

i s a bounded o p e r a t o r

s~~ (sq-l tuq) ~ A5L B continu-

ously i n t o i t s e l f .

-

D i f f e r e n t i a t i o n operators.

Let

5

L lRq.

The o p e r a t o r u H (a,V)u i s defined a s follows. F i r s t extend u

to a harmonic f u n c t i o n

\.

Then c a l c u l a t e

Finally, r e s t r i c t t h i s function t o

sq-'.

I n s t e a d of c o n s t a n t s a .

I'

we can

consider a l s o m u l t i p l i e r s . W e g e t d i f f e r e n t i a t i o n o p e r a t o r s with v a r i a b l e c o e f f i c i e n t s . An i n t e r e s t i n g subclass of t h i s type i s o b t a i n e d i n t h e following way: Take a matrix A

4

'L2(Sq-'t%)

E

IRqxq. The o p e r a t o r

(A5,V) maps

i n t o i t s e l f . In p a r t i c u l a r (_x,V) is t h e normal d e r i v a t i v e .

If A i s antisymmetric, AT = -A, then t h e v e c t o r f i e l d s (A5,V) a r e tangent

to

sq-'.

They a r e l i n e a r combinations of t h e angular o p e r a t o r s i n quantum

mechanics. All mentioned d i f f e r e n t i a t i o n o p e r a t o r s a r e e x t e n d i b l e .

111.2.

The spaces

s!

of Gelfand and Shilov

One of Schwartz' d i s t r i b u t i o n t h e o r i e s is based on t h e t e s t space c o n s i s t s of

c

m

-functions of r a p i d decrease: A

c

m

s

which

-function rp on IR belongs

2 30

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

to S i f

The spaces functions

sB a r e subspaces of s. For f i x e d real a and B s B c o n s i s t s cp E s f o r which t h e r e are c o n s t a n t s A,B,C > 0 such t h a t

of

Because o f t h e c l a s s i c a l Sobolev embedding theorem, w e can t a k e t h e L 2 ( I R ) norm i n t h e D e f i n i t i o n 111.2.1 of t h e space

sB a'

i n s t e a d of t h e supnorm.

I f t h e c o n s t a n t 6 i s n o t g r e a t e r than 1 , every f u n c t i o n i n

sB

h a s an ana-

l y t i c c o n t i n u a t i o n i n t o t h e complex plane. I n p a r t i c u l a r f o r B < l , t h e space

sB

c o n s i s t s of a l l e n t i r e f u n c t i o n s cp with t h e following growth

behaviour.

The above i n e q u a l i t y l a y s such s t r o n g c o n s t r a i n t s on t h e f u n c t i o n cp t h a t

the space

'S

is t r i v i a l for a + B c I.

The space

s:

i s a union of Frechet spaces. Indeed, w e p u t

where

S B t Bdenotes t h e space of a l l f u n c t i o n s a,A

i > by i .

f o r a l l A > A and a l l replaced by

and B

cp E

sB with

B the i n e q u a l i t i e s 111.2.1

t h e property t h a t

are s a t i s f i e d with A

The Frechet topology i s determined by t h e

countable family of seminorms

,-,-,....

1 1 The topology of s B is t h e l o c a l l y convex i n d u c t i v e 2 3 l i m i t topology, induced by t h e Frechet spacaes .:s:

where 6 , p = 1

Examples of continuous l i n e a r mappings on

sB are

t h e t r a n s l a t i o n operators

THE SPACES Ta, a

E

s:

OF GELFAND AND SHILOV

IR, the phase shift operators

%,

b

231

IR, the dilation operators

E

z,, A E IR, the differentiation operator D and the multiplication operator 2. Further, we mention the Fourier transformation IF which maps sB bijectively o n t o

$. It

follows that the spaces

Saa'

a

_>

4, remain invariant under

Fourier transformation.

sB

A number of

spaces is of type S X , A . In this section, we present a candi-

date A a ,B for a = q/(p +q)p and B

p/(p +q)p with p,q

=

IN and 0 < p

E

5

1,

such that

We recall that in the definition (111.2.1) of the spaces take the norm 1 1 * 1 1 2 instead of 11 . l l m .

02 - @

( x ) = x cp(x) and

111.2.3.

Let cp

D

and

2. ( ( D c p ) (x) =

= 1.1

c

(IR) satisfy the following inequalities for fixed A,B,C > 0 and

fixed a,B > 0 with a + B 2 1,

Then for each n

IN and each r , s

E

( I N u {O))"

E

where C 1 = C ,

A 1 = 2uB+1 eaa

Is1 =

+

A ,

B1=

2ua+1 upB e

, u

and s1

... +

s

,

Irl = r1

For each plq E IN, we introduce the operator

111.2.4.

cp'(x),

a. a0

E

we can as well

The following result is based on

rather tedious manipulations with the operators (Qcp)

s:

8 PIq

"hen we have

=

(-

ST

+

.

( x ~ ) ~

+

... +

r

.

-1

= (a+B)

2 32

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

111.2.5.

e.

Let 0 < p cp

E

B scr.

5

1 and l e t p , q

Then f o r a l l n

E

E

IN.

Put a = q / ( p + q ) P . B = p / ( P + q ) P . Let

IN

Here t h e c o n s t a n t s K,L > 0 depend on cp.

Proof.

Let n

By assumption t h e r e e x i s t p o s i t i v e c o n s t a n t s A , B , C

IN.

E

where v . 3

Then we can w r i t e

(p2' , Q2')

i s a sum of (n) combinations of t h e form 3

,...,rb , s l ,...,sb

w i t h r1

such t h a t

E

IN

u to}, r l

+... +rb

Applying t h e previous lemma, w e o b t a i n

Thus we a r r i v e a t t h e following e s t i m a t i o n

=

j

,

s1

+...

+sb = n -j.

THE SPACES

sB OF

GELFAND AND SHILOV

233

where w e have p u t

,

A2 =

111.2.6.

.

0

B2 = ( ( 2 q ) B ~ 1 ) 2 q

e.

L e t A be a nonnegative s e l f - a d j o i n t o p e r a t o r i n a s e p a r a b l e H i l b e r t space

x.

Let

u > 0. Then w

SX,Au i f f w

E

ja>O 3b>0 'kcIN

*'

E

D"(A)

IIAkw IIx

and

(k!)'lvak b

5

.

Proof. -)

Let t h e r e be c o n s t a n t s a and b as i n d i c a t e d . Then w e have

It (A")

k w II

II A-l+vk-Cvkl

5

II II A' vkl+l w II

and

IIAC"kl+lwIIX

5

( ( [ v k l + l ) ! )l / v a [ v k l + l

The following i n e q u a l i t i e s a r e v a l i d

So t h e r e e x i s t s a l and b l such t h a t

II(A")kwIIX

5

which implies t h a t w *)

k!a k b 1 1 6

SX,AV.

Let w E

SX,AV. Then w

k

W e estimate as follows

E

IN.

= exp(-s A")f

for certain f

E

x

and s > 0. L e t

234

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

With b

=

Ilfllx and a = (l)’”

the assertion follows.

vs

A combination of Lemma 111.2.5 and Lemma 111.2.6 yields 111.2.7. Theorem.

In the following cases it has been proved (in different ways) that in Theorem 111.2.7 the inclusion sign

C

can be replaced by the equality sign =.

E7

s;/k

=

s

,

krlN

,

keIN

and by the Fourier transformation

1

=

s

d2

and by the Fourier transformation

111.3. Analyticity spaces based on classical polynomials A very transparent way to extend a separable Hilbert space

x

is by means

of formal series expansions with respect to some fixed orthonormal bases in

x.

We illustrate this in close connection with the set up of Chapter I.

ANALYTICITY SPACES BASED ON CLASSICAL POLYNOMIALS

235

L e t (Xn)nfINN,be a sequence of p o s i t i v e r e a l numbers with t h e p r o p e r t y t h a t

I: converges f o r a l l t > 0. Then we c o n s i d e r a v e c t o r m n= 1 of formal series anvn where t h e expansion c o e f f i c i e n t s (a ) n ncIN

the series space

T

nEl

satisfy

T

The topology f o r

i s t h e l o c a l l y convex topology induced by t h e seminorms

m

x

Z bnvn of n= 1 which s a t i s f y C lanbnl < = f o r a l l elements L a v o f T. The correspondn= 1 n=l n n = which c o n s i s t s of a l l t h e s e elements bnvn, i s denoted i n g subspace of A r e p r e s e n t a t i v e of t h e dual o f m

T

i s given by t h e elements m

x

.P1

by S. I t i s clear t h a t S c

X

T.

c OD

I t i s not hard t o prove t h a t

1 b v n=l n n

3T,0 : sup (Ibnl e h n T ) < = nc IN W e introduce t h e o p e r a t o r A i n

E

s

i f t h e sequence (bnInfIN s a t i s f i e s

.

X as follows.

Then A i s a nonnegative s e l f - a d j o i n t o p e r a t o r i n t h e H i l b e r t space X. We have S =

SX,A and T

=

TX,A.

I n Section 111.1 we have s t u d i e d t h e t r i p l e

L2(C-npn1)

where v

; I

%.The elements of T

d i v e r g e n t Fourier s e r i e s

L2 (C-v,vI) ,A:

'L2([-v,n])

,Av P

can be considered a s s t r o n g l y

236

ILLUSTRATIONS O F ANALYTICITY SPACES AND TRAJECTORY SPACES

1

XI+

inx cne

ne2Z with

Likewise, t h e elements of T

v can be considered as formal series L2 ( C - n t n ] ) ,Ap

expansions

with

where t h e f u n c t i o n s en are defined by

1

e(x)=-e

inx

6

,

x c m ,

n E m .

I n this s e c t i o n w e consider orthonormal bases which are r e l a t e d t o t h e c l a s s i c a l polynomials. E I 0 A n a l y t i c i t y spaces based on t h e Hermite polynomials

The Hermite polynomials H

n'

n

E

IN

u

{O},

a r e d e f i n e d by

Related t o t h e Hermite polynomials a r e t h e normalized Hermite f u n c t i o n s J,

n'

XEIR. n E IN u { O l , g i v e rise t an orthonorma b a s i s ( C t ~ ~ l ) ~ n' They a r e t h e eigenfunctions of t h e d i f f e r e n t i a l o p e r a t o r H,

The f u n c t i o n s J, i n L2(IR).

ANALYTICITY SPACES BASED ON CLASSICAL POLYNOMIALS

with t h e n a t u r a l numbers a s i t s eigenvalues. Regarding

H

s e lf-adj oi nt operator i n L 2 ( I R )

[$,I.

we have HI$,]

= (n + 1 )

237

as a positive As

already

sL2 ( IR 1/2a can be seen s: f o r a t 4. Thus we g e t

mentioned i n t h e previous s e c t i o n , t h e space

) t H

as a space of Cm-functions. I t e q u a l s t h e space

t h e following c h a r a c t e r i z a t i o n .

111.3.1.

Theorem.

L e t cp E C m ( l R ) .

([TI

, [$,I)

LZ

Then rq

E

s,:

4, i f f

a t

t h e r e e x i s t s t > 0 such t h a t

= O(exp(-t n1’2a)) . m e s e r i e s

converges uniformly on IR.

m

z

( C ~, [+,I) I

n=O

W e devote some e x t r a a t t e n t i o n t o t h e spaces

s44

and

s,.1

L2

g n ( x ) , x c IR,

F i r s t some h i s t o -

r i c a l comments. I n 1974, D e Bruijn published a paper on “a new theory o f g e n e r a l i z e d functions.

..”, t a k i n g

t h e space

s44 as

t h e t e s t space. With t h e a i d of complex

a n a l y t i c t o o l s he proved t h a t any element of cp E

s44

can be w r i t t e n a s

m

for s u i t a b l e 0 E

s 44

and a > 0. Since

i s t h e Hilbert-Schmidt k e r n e l of t h e o p e r a t o r e phrase d

4=

S

U e a>O

-aH

-aH

, his

r e s u l t can be r e -

4

(S4)

N o w a generalized f u n c t i o n i n De B r u i j n ‘ s theory i s r e p r e s e n t e d by a mapping

Q from t h e open i n t e r v a l (0,m)

Q ( ~ + B )=

e-aHQ(b)

into

s44 s a t i s f y i n g

,

a,B >

o

.

These mappings 0 a r e c a l l e d t r a c e s . Given a trace Q and a function J, Then t h e i r p a i r i n g i s d e f i n e d by

E

s44’

238

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

<$,Q> =

0 (x) 0 (x;a) dx -m

where $ = e-aH 0 f o r a s u f f i c i e n t l y small. One of t h e remarkable f e a t u r e s of D e B r u i j n ' s theory i s i t s t o t a l l a c k of f u n c t i o n a l a n a l y t i c and t o p o l o g i c a l c o n s i d e r a t i o n s . The whole theory i s

'i

presented i n terms of complex a n a l y s i s . The space Sk i s continuously embedded i n t o Schwartz' space

s.

tinuous l i n e a r f u n c t i o n a l s on

So a l l tempered d i s t r i b u t i o n s , i . e . a l l con-

s

can be represented by a t r a c e 0

It i s worth mentioning t h a t a l l e v a l u a t i o n f u n c t i o n a l s 6 z , z $

tinuous on

s$.

Consequently, t h e m u l t i p l i c a t i o n o p e r a t o r

complex p l a n e as i t s spectrum. The o p e r a t o r s Ta, a h

E

E

6,

T

L,(IR)

,H'

C a r e con-

E

2 has

I$, b

E

t h e whole E

6,

Zx,

IR, map S'i continuously i n t o S'i

'i-

$

Let us d e f i n e t h e l i n e a r o p e r a t o r A on

L 2 (IR) a s follows

where

For each Chl

E

L2(IR),

The transformation

t h e f u n c t i o n ACh] i s e n t i r e a n a l y t i c and s a t i s f i e s

A l e a d s t o t h e i n t r o d u c t i o n o f t h e Bargmann-Fock-Segal

space F. The space F can be considered a s a c l o s e d subspace of t h e H i l b e r t 2 2 space L2(m2,e-(x +Y ) dxdy). F c o n s i s t s of a l l e n t i r e f u n c t i o n s g f o r which the function (x,y)

H

e-'i(x

2

2 +' g(x+ iy)

i s square i n t e g r a b l e on IR2 with

r e s p e c t t o t h e Lebesgue measure. The o p e r a t o r A i s a u n i t a r y o p e r a t o r from L 2 ( ~ ) o n t o F. I n p a r t i c u l a r , (A[$,]) (2) = 2" , n E IN u {o}, z E 6 . Putting

G

=

A H A* we g e t

m

ANALYTICITY SPACES BASED ON CLASSICAL POLYNOMIALS

d + dz

I

= z

W e have

The space SF,-,

1

.

c o n s i s t s of a l l e n t i r e f u n c t i o n s w f o r which t h e r e a r e con-

s t a n t s c < 'I and D > 0 such t h a t Iw(z) of TF,fi

I

E

TF,;

CI

2

)

t h e r e e x i s t s an e n t i r e f u n c t i o n

Vc>+ 3D>o: l z ( z )

of t h e coherent s t a t e s i n

c

in

The space

L 2 (IR)

1

F

One of t h e advantages of t h e BFS-space

E

D exp ( c I z I

5

.

Also t h e elements

can be represented by e n t i r e f u n c t i o n s :

8

w,z

239

< D exp ( c I z I

- satisfying $

2

such t h a t

i s t h e very simple r e p r e s e n t a t i o n

-

by t h e exponential f u n c t i o n s ew : z H eWZ,

F.

s,1

.

is also an i n s p i r i n g example of an a n a l y t i c i t y space ( c f . a l s o

Section 111.4). The elements of

s 11

cm-functions

are

cp on IR which have an

a n a l y t i c extension t o a s t r i p I I m zI < r i n t h e complex plane and on t h i s strip satisfy

sup lerlXl q ( x + i y ) IIm z J < r

I

<

m

,

H e r e t h e width o f t h e s t r i p depends on cp E

z = x +iy

.

s,.1

An i n t e r e s t i n g class of continuous l i n e a r f u n c t i o n a l s on Let

Then we d e f i n e t h e l i n e a r f u n c t i o n a l

m-ic

Here

s11

i s t h e following:

be an a n a l y t i c function on Q\IR with t h e property t h a t f o r a l l r > 0

E

--ii

n

by

E

> 0 must be taken so small t h a t cp is a n a l y t i c on t h e s t r i p IIm z I < E

and s a t i s f i e s the mentioned e s t i m a t i o n on t h i s s t r i p .

240

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

E l l A n a l y t i c i t y spaces based on t h e Laguerre polynomials

The Laguerre polynomials L p )

Let

x

,v

> -1, a r e d e f i n e d by

denote t h e H i l b e r t space L2((0,-) , x

2V+l

d x ) . Then t h e f u n c t i o n s

LAv)

defined by

in X . give r i s e t o an orthonormal b a s i s (CL (') n I)nemutO) v The f u n c t i o n s L " ) ,

v = -$,$

,

a r e r e l a t e d t o t h e Hermite f u n c t i o n s

c o n s i s t s o f e i g e n v e c t o r s of t h e

The orthonormal b a s i s (If:) positive self-adjoint operator

Because of t h e R e l a t i o n s 111.3.2 we have t h e following c h a r a c t e r i z t i o n of t h e space S

111.3.3.

4 5 a < l .

X-+,

-.

The space

sX-5,

(A_+)

even f u n c t i o n s i n

sy

"

Y

5

5

a 5 1, can be considered as t h e space of a l l

with y = 1/2a.

The following r e s u l t is a much deeper one. 111.3.4.

a.

Let v > -1 and l e t % 5 a 5 1. Then we have

ANALYTICITY SPACES BASED ON CLASSICAL POLYNOMIALS

241

Lemma 111.3.3 and Lemma 111.3.4 y i e l d t h e complex a n a l y t i c c h a r a c t e r i z a t i o n of t h e spaces

S

'r

Xu, (A,)"'

a S 1, v > -1.

5

A s a consequence hereof we

obtain 111.3.5. Corollary. L e t rp be an even f u n c t i o n i n

exists coefficients c

a,n

sy

f

Y'

S y S 1. Then f o r each

( v ) of growth o r d e r O(exp(-t nu) )

u > -1 t h e r e for certain t > 0

and a = 1/2y, such t h a t

Next we introduce t h e f u n c t i o n s each u > -1,

t h e set

{[fA"'l 1 n

z(')

by r L v ) (x) = x"$1:

E

u~ { ON} } i s an orthonnormal b a s i s i n t h e

~

( x ) , x > 0. For

H i l b e r t space L2((0,m)). Put

u

d2 A = - - + x 2 dx

x v[.c"(')] n

Then

2

= (4n + 2 v + 2 )

+ -v2

-k 2

.

[2Au)1. A

c h a r a c t e r i z a t i o n o f t h e spaces

can be e a s i l y obtained from t h e preceding lemmas.

111.3.6.

m.

Let v > - 1 and l e t 'r

[(PI such

E

in

syY'

that x

5

H

a

5

1. Then [ p ]

E

sL2 (

rm)

t

(A,)"

i f f there e x i s t s

$ ( x ) , x > 0 , i s e x t e n d i b l e t o an even function

x-('+')

0

with y = 1/2a.

Now d e f i n e t h e Hankel transform M

V

by

m

where

J

denote t h e Bessel f u n c t i o n of o r d e r v . The following r e l a t i o n i s

Therefore we can extend M with

xv on sL2 (o,=),

*

'

t o a u n i t a r y o p e r a t o r on L a

> 0. I t follows t h a t I H

2

(O,m),

which commutes

242

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

a > 0, continuously and b i j e c t i v e l y i n t o i t s e l f . Consequently, t h e corresonding t r a j e c t o r y spaces T

can be regarded as Hankel i n v a r i -

L, (O,-), (&)a

a n t d i s t r i b u t i o n spaces. W e l i s t t h e following continuous l i n e a r f u n c t i o n a l s and l i n e a r mappings on

t h e spaces

sL2 (

The l i n e a r f u n c t i o n a l s 6 functionals 6

, i

,-),(&)a

1.

5 a

: w H w(x) with

X

: w H w ( z ) with

we g e t t h e l i n e a r f u n c t i o n a l

J

larg X I

v

21

<

x > 0 . For $

= MVGx,x > 0 ; we have

J z,v

For 5 < a 5 1 w e a l s o g e t t h e l i n e a r f u n c t i o n a l s larg(z) 1 <

6

= x2 6x and

which maps

Jx I V

w = ( M V w )( x ) .

= M 6 with v 2

TI.

The l i n e a r o p e r a t o r s

22

a < 1 the linear

h

Applying t h e Hankel transform

TI.

s L2 ( 0

22

and 8

BV Jx,v = x

I-)

Jx,v.

s L, ( to,-)

)

have

dx2 x Further, t h e l i n e a r operator

b i j e c t i v e l y onto

,(A,,) a

o p e r a t o r 0 , which maps

2

-d2 + '* ;" We

MVQ2 My =

=

S

L2 ( O , m ) , (Av+61a

into

, (Zv+l)a

-

S

26 ,

6 > 0,

The l i n e a r

'

u

L2 ( 0 , ~I )(A,)'

'

E I 2 A n a l y t i c i t y spaces based on t h e Jacobi polynomials

L e t a,B > -1.

Consider t h e H i l b e r t space

and t h e nonnegative s e l f - a d j o i n t o p e r a t o r A

Aa,B

=

"'2

a.6

in

x a,B d

.

has t h e d i s c r e t e spectrum { n ( n + a + B + 1 )

In

-(I - X

)

--

((8 -a)

dx

The o p e r a t o r A

a,B

An orthonormal b a s i s of eigenvectors of A

-

a,B

(a+8+2)x)

Jacobi polynomials R ( a '

n

where P (a'

2n+ a + % +1 p+B+1

denotes t h e polynomial

E

IN u {Ol}.

i s c o n s t i t u t e d by t h e normalized

T(n + 1 ) T ( n + a + B + 1)

r ( n + a + l ) r ( n + B + 1)

ANALrTICITY SPACES BASED ON CLASSICAL POLYNOMIALS

243

S p e c i a l types o f Jacobi polynomials a r e t h e Legendre polynomials P(O ' O ) and t h e Chebyshev polynomials P

(-4,-4)

The following r e s u l t i s very fundamental.

111.3.7.

Theorem.

Let a 1 B r y , 6 > -1 and l e t v 2

4. We have

Because of t h i s c l a s s i f i c a t i o n r e s u l t we can choose s u i t a b l e a and B i n o r d e r t o c h a r a c t e r i z e t h e a n a l y t i c i t y spaces

x = cos 8. Then t h e H i l b e r t space

s

-5.

i n complex a n a l y t i c terms. W e take a = B =

x -$, -4 t u r n s

Xa,i31 (Aa,B)

v f o r a l l a,B > -1

Consider t h e transformation into

Correspondingly, t h e Chebyshev polynomials s a t i s f y

and

The s p ce Example E

s

~ is the , same ~ ~a s

t h e SP -e S

L 2 ( [ 0 , n l ) , (Ap,e)v

described

in

5'

Next, w e consider t h e conformal mapping 5 = cos z . a n a l y t i c r e l a t i o n between ing characterizations.

sy,cv

and

sX,

(A_+,-+)"'

I t d e s c r i b e s t h e complex

Then w e o b t a i n t h e follow-

244

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

111.3.8.

Theorem.

a.

Ccpl

E

sX-$,-$I

5 i f f there e x i s t s a function

(A-f,-%,

N

cp

E

which can be

[cp]

extended t o an a n a l y t i c f u n c t i o n on an open neighbourhood o f t h e i n t e r v a l

c-1,ll.

v 7 f . Then ccpl E s i f f t h e r e e x i s t s a function L + , - + t (A+,-+)" Ccpl which can b e extended t o an e n t i r e f u n c t i o n , s a t i s f y i n g

b. Let N

cp

E

We l i s t t h e following examples of continuous l i n e a r f u n c t i o n a l s and continuous l i n e a r mappings. For u > 4, we have t h e l i n e a r f u n c t i o n a l s

z

E

6 and k E IN

u

{O}.

In

t h i s c a s e , a wider c l a s s i s obtained as follows: Consider a Bore1 measure p on IR2 with t h e p r o p e r t y t h a t

The d e f i n e t h e l i n e a r f u n c t i o n a l

eU

by

For v > 5, we have t h e l i n e a r mappings MJ,'

ZA, X

E

'

c, and D.

For u = b, we have t h e l i n e a r f u n c t i o n a l 6,): the l i n e a r operators The space

T

D, ZA, X

XOlO) (AO,O)

t h e i n t e r v a l [-1,11. T

E

[-1,11,

k

9' J,

E

s Lf ,

(A+,-$

- 5 8

E

E

4

as a space of hyperfunctions on

Therefore, we introduce t h e functions.

2ni

where we take 5 4 [-1,11.

1 , -?

x

E

c-1,11

,

Each f u n c t i o n 7 can be extended t o an a n a l y t i c 5 f u n c t i o n i n an open neighbourhood of [-1,11. So f o r a l l 5 E c\C-1,13, [TZl

E

s X0,Ot (Ao,o) 4 -

6,

IN u { o j and

x

c-1,11 and M

5 can be considered

: X H 1

5

E

sX-+,-+n A+,+)", la,a

ANALYTICITY SPACES RELATED TO L I E GROUPS

Now f o r each F

245

3 we d e f i n e t h e function HF by ~ O , O ~ ~ ~ O , O ~

E

The f u n c t i o n H i s a n a l y t i c on C\[-1,1] and H (-1 = 0. The converse s t a t e F F rnent i s t r u e , a l s o . Each f u n c t i o n 0 which i s a n a l y t i c on C\[-l,l] with

B(-)

= 0 can b e w r i t t e n as B = HF f o r c e r t a i n F

For each Cq1

E

s


-

X0,Of (Ao,o)

,F> =

where t h e contour (P

c

E

T

KO,,,

(AO,O)+

*

$ w e have

(p"-

C ( ~ ( 5 )HF(5) d5

m u s t be taken so small around t h e i n t e r v a l [-1,1],

that

i s a n a l y t i c within C.

The d i s t r i b u t i o n a l l i m i t x H l i m [HF(x + i E ) -HF(x - i c ) ] , €SO

x

E

[-1.11,

is

t h e hyperfunction corresponding t o F.

111.4. A n a l y t i c i t y spaces r e l a t e d t o u n i t a r y r e p r e s e n t a t i o n s of Lie groups I n Chapter I w e introduced t h e notion of a n a l y t i c v e c t o r f o r a s e l f - a d j o i n t

P i n a separable H i l b e r t space i s analytic f o r P i f operator

3a>o 3b,0 VkEm

:

The a n a l y t i c i t y domain, i . e . by

DW(P).

x.

We r e c a l l t h a t a v e c t o r w

I I f w I I x 2 a b n n:

OI

E

D (P)

.

t h e s e t of a n a l y t i c v e c t o r s f o r P i s denoted

I t can be r e a d i l y checked t h a t

DW(P) = s x , ( 7 + p 2 ) $ . Condition

(*)

does not e n t i r e l y explain t h e use of t h e a d j e c t i v e " a n a l y t i c " . A f u l l explanation i s i n t h e following e q u i v a l e n t d e f i n i t i o n . Since P i s a s e l f - a d j o i n t o p e r a t o r , i t generates a one-parameter group

x.

For a l l f of u n i t a r y o p e r a t o r s on (e )tern from JR i n t o i s continuous. Now a v e c t o r w E

x

v e c t o r f o r P i f t h e function t H e

itP

w,

t

E

E

x

x

t h e f u n c t i o n t H eitP f

i s s a i d t o be an a n a l y t i c

IR, i s ( r e a l ) a n a l y t i c . I t

means t h a t t h e r e e x i s t s r > 0 such t h a t t h e function t

I+

eitP w can be ex-

tended t o an a n a l y t i c X-valued function on t h e s t r i p 1 I m zI < r . Thus t h e a d j e c t i v e " a n a l y t i c " i s completely explained.

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

246

The s e t I R i s a Lie group i f w e consider t h e o r d i n a r y a d d i t i o n o f real

P in

numbers a s i t s group operation. Each s e l f - a d j o i n t o p e r a t o r a unitary representation t

H

e

itP

,

t

E

x. Conversely, by u ( x ) i s generated

u n i t a r y o p e r a t o r s on s e n t a t i o n of IR i n t o

x

generates

u(x)

IR, of (IR,+) i n t o t h e group

of

Stone's theorem each u n i t a r y repre-

x.

by a s e l f - a d j o i n t o p e r a t o r i n

Because of t h e second d e f i n i t i o n of a n a l y t i c v e c t o r f o r P w e can a s w e l l say t h a t tion t

I+

DW(P) c o n s i s t s e

itP

of a l l a n a l y t i c v e c t o r s f o r t h e u n i t a r y representa-

.

of (IR,+)

The i d e a t o introduce t h e a n a l y t i c i t y domain f o r u n i t a r y r e p r e s e n t a t i o n s of I R can be generalized t o any L i e group. W e present a s h o r t discussion.

Let

G be a f i n i t e dimensional

L i e group. A u n i t a r y r e p r e s e n t a t i o n

U

of G is

a homomorphism

,

g

G i n t o t h e group

u(x)

g from

vector w E

x

I+

u(g)

E

G

of u n i t a r y o p e r a t o r s on a H i l b e r t space

x.

A

is c a l l e d an a n a l y t i c v e c t o r f o r t h e r e p r e s e n t a t i o n U o f G , i f

the function g

I+

i s a n a l y t i c on

U(g) w

G.

( W e observe t h a t

t h e s e t of a n a l y t i c v e c t o r s f o r

G i s a real a n a l y t i c manifold.) We denote

c"(u).

by

The s e t

c"(u)

i s an a n a l y t i c i t y

space: Let A(G) denote t h e Lie a l g e b r a corresponding t o t h e Lie group l e t {p,, ...,p ,} be a b a s i s f o r

A ( G ) . For every p

E

G and

A ( G ) t h e o p e r a t o r valued

function s H U(exp(sp))

,

s

E

i s a one-parameter u n i t a r y group on

au(p)

generator denoted by

IR

x.

By S t o n e ' s theorem i t s i n f i n i t e s i m a l

i s skew-adjoint. Thus we o b t a i n t h e so-called

i n f i n i t e s i m a l r e p r e s e n t a t i o n aU of U which l i n k s t o any p

E

A ( G ) a skew-

a d j o i n t o p e r a t o r aU(p). Now w e p u t

A =

I

-

d

1

2

(au(pk))

.

k=l The o p e r a t o r A i s p o s i t i v e and e s s e n t i a l l y s e l f - a d j o i n t on the following r e s u l t .

cw(u). W e

have

247

ANALYTICITY SPACES RELATED TO L I E GROUPS

111.4.1.

Theorem.

The s e t of a n a l y t i c v e c t o r s

cw(u) equals sX , A $

0

'

There i s a n a t u r a l a c t i o n of G and A ( G ) on t h e space S X , A +

111.4.2.

.

Theorem.

a . The u n i t a r y o p e r a t o r s U ( g ) , g

E

G, map Sx,A$ continuously i n t o S X , A t

;

they a r e e x t e n d i b l e . b . The skew-adjoint o p e r a t o r s

Sx,A+

;

aucp), p

E

A ( G ) , map S x , A + continuously i n t o

0

they a r e extendible.

I n the remaining p a r t of t h i s s e c t i o n we work o u t some examples.

El,

The Heisenberg group

The Heisenberg group

IR x I R

x IRmod 2n

(a,b,c)

GH

i s the simply connected t h r e e dimensional L i e group

i n which t h e group o p e r a t i o n

0

i s d e f i n e d by

( a ' , b ' , c ' ) = ( a + a ' , b + b ' , c + c ' +ab')

0

.

Its L i e algebra i s spanned by t h e b a s i s {p,,p2,p3} which s a t i s f y t h e b r a c k e t

relation

Cpl.p21

=

p,

I

p, c e n t r a l

.

The u n i t a r y r e p r e s e n t a t i o n U of GH on L 2 ( I R ) i s given by

U(a,b,c)

[cpl

= Cx t+ exp(ibx + i c ) cp(x+ a ) ]

.

Correspondingly, aU(p ) = D, aU(p 1 = i Q and aU(p3) = i'1. fe have 2 1 ic = U ( a , O , O ) = exp(al)), R = U(O,b,O) = e x p ( i b 2 ) and u(O,O,c) = e '1. The b operator

T

-d2 + x2 i n L2 (IR). So t h e set of analydx2 t i c v e c t o r s f o r t h e u n i t a r y r e p r e s e n t a t i o n u i s given by

equals t h e s e l f - a d j o i n t o p e r a t o r

sL2

(W),

(- --&-+ x'>' s: d2

=

.

248

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRJAECTORY SPACES

This space is described in Section 111.2, ElO. An extension of the Heisenberg group can be derived by taking a semidirect product of

GH

and SL(2,IR). This results in the so-called SchrBdinger group.

The unitary representation of this group on L 2 (IR) is rather complicated; we do not present it here. The corresponding infinitesimal representation is SpaMed by the differential operators

i , ix, dx

'

ix2, x - d+ - x , d dx dx

-i- d2 2 dx

and the corresponding analyticity space equals Also

the spacek':S

sff

(cf. Elo).

is the analyticity domain of a unitary representation

of a Lie group on L2(IR). This Lie group is an extension of the Heisenberg group. The corresponding infinitesimal representation is spanned by the differential operators

z ,i -dxd ,..., -i(isy .

i , ix, d

E14 The Euclidian group The Euclidean group and a,b

E

E2

E 2'

is a three parameter group (O,a,b) with 0

E

[-n,nl

IR. The group multiplication is defined by (0,a,b)

o

(O',a,b')

= (0 +0'

,a cos0' +b sinO' +a'

,

,-a sin@' +b cos0' +b')

.

is a semidirect product of the abelian group of translations (a,b) and 2 the compact group of rotations ( 0 ) in the plane. So E

A well-known realization of

(0,a,b) +

The Lie algebra A ( E

2

[

E2

is the group of 3 x3-matrices

cos 0

-sin0

si; 0

0;c

81

.

is spanned by p 1, p2 and m with bracket relations

ANALYTICITY SPACES RELATED TO L I E GROUPS

[ p l ~ p 2 1=

0

, Cm.p,l

=

p2 , Cm,p21

.

s1

2 denote t h e u n i t c i r c l e i n IR 1 on L 2 ( s ) i s given by

Let

U ( @ , r , a ) Cpl = Cx

I+

249

-pl

=

Then a u n i t a r y r e p r e s e n t a t i o n

e x p ( i r cos (x - a ) ) p ( x - a ) ]

U of E

2

.

H e r e w e use p o l a r c o o r d i n a t e s ( r , a ) i n s t e a d of C a r t e s i a n c o o r d i n a t e s ( a , b ) . 1 For t h e i n f i n i t e s i m a l r e p r e s e n t a t i o n i n L 2 ( s ) we have

S 0 A = 2 - - d2

and t h e s e t

cw(f.l)

equals

dP2

This space i s described i n E

4'

E I 5 The r o t a t i o n group 0 3 The r o t a t i o n group 0, i s t h e group of a l l 3 x3-matrices with t h e p r o p e r t y

that

R RT

1 and d e t R

=

1. Here 1 denotes t h e 3 x 3 - i d e n t i t y matrix. 0,

=

is

a r e a l 3-parameter compact L i e group. The r e a l Lie a l g e b r a A ( 0 ) i s spanned 3 by j l , j 2 and j, with b r a c k e t r e l a t i o n s

Cjl,j21 Let

s2

=

j , , Cj3,jl1 = j,

.

3 denote t h e u n i t sphere i n IR

on L2(s2,w

3

)

, rj2,j31

=

jl

.

Then a u n i t a r y r e p r e s e n t a t i o n of 0

i s obtained as follows

s2 by

If we express t h e elements of

(0 ,O ) , 0 5 0 < n, 0 1 2 1 a r e of t h e form

a U ( j 1)

=

5

sine

0

2

means o f t h e s p h e r i c a l c o o r d i n a t e s

< 2 n , then t h e skew-adjoint o p e r a t o r s

a. a + 1

cos

o2 c o t

0

a

-

1 ao

2

a u ( jk )

3

ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES

250

aucj,)

= -cos

a + e2 ao 1

sin0 cotel 2

a , 3

The analyticity domain of this unitary representation is given by

with ALB the Laplace-Beltrami operator for the real analytic manifold

s2 .

In spherical coordinates we have

1

= _ -

sin el

'LB

a sin O1 ao a ao,

i

-

1 a2 2 2 sin el ao 2

The spherical harmonics Y F ) defined by

with

e

E

IN u {O} and k =

of the operator A

LB

-e ,-e + 1 ,...,e - 1 ,e

,

are the eigenfunctions

with

This space is also one of the spaces presented in E6'

Some comments and bibliographical notes We mention the following references for the illustrative examples. 1. Analyticity spaces based on the Laplacian operator: [G 11, CG 21, [G 51,

[EG 31, LEG 91, [Mil. 2. The Gelfand-Shilov spaces: [GS 21, [GZ],

[Go 21, [EGPI, [KO].

3 . Analyticity spaces based on classical polynomials: [Br],

CEG

11,

LEG

21,

LEG

[KO],

[E 21,

31.

The description of the Bargman-Segal-Fock space is to be found in [Ba 11 and [Ba 21.

ANALYTICITY SPACES RELATED TO L I E GROUPS

4 . A n a l y t i c i t y spaces r e l a t e d t o unitary representations of L i e groups: [Go

11, [Go 2 1 , [Go 31 and

W e observe t h a t example E15 f i t s i n t h e general d e s c r i p t i o n given i n

[Mil.

25 1

This Page Intentionally Left Blank

253

CHAPTER I V

THE CONCEPT OF D l R A C B A S I S L I F T E D TO TRAJECTORY SPACES

The t r i p l e S X I A

x

C

C

TX,,A introduced i n Chapter

I, can

be considered a s a

l i m i t of t h e Sobolev t r i p l e s

a s n tends t o i n f i n i t y . More p r e c i s e l y , S X , A = l o c a l l y convex topology f o r

s

n2m

1

exp(-;

A)

( X I , where t h e

i s t h e f i n e s t l o c a l l y convex topology f o r

X,A 1 SX,A f o r which a l l l i n e a r o p e r a t o r s exp(- - A )

:

x

- + S X , A a r e continuous. 1

(x).

I t means t h a t SX,A i s an i n d u c t i v e l i m i t o f t h e H i l b e r t spaces exp(- - A) 1 n Correspondingly, T ~ = ,ncm~ exp(; A ) ( X ) , where t h e l o c a l l y convex topology 1 A) : f o r TX,A i s t h e c o a r s e s t topology f o r which t h e l i n e a r mappings e x p ( :

TX,A

-+

x

aF

n

continuous. Thus T

X,A

is a projective l i m i t of t h e Hilbert

spaces exp( - A ) ( X ) . Being a l i m i t o f Sobolev t r i p l e s , we can reformulate t h e d e f i n i t i o n s and t h e statements of P a r t A, which a r e p u t i n terms of a Sobolev t r i p l e R(X) c

IV.l.

X

c R-l(X),

a l s o i n terms of t h e t r i p l e

S

X,A

c

X

c

TX,A.

A measure t h e o r e t i c a l Sobolev lemma f o r a n a l y t i c i t y spaces

L e t (M,p)

denote a a - f i n i t e measure space. In Chapter A . 1 1 w e have p o i n t e d

a t t h e problem how t o f i n d dense subspaces D of L 2 ( M , p ) , which can be regarded a s f u n c t i o n spaces. We mean, t h a t i n each c l a s s Cq11 a kind of canonical r e p r e s e n t a t i v e

[ql

H

E

D we can select

such t h a t t h e e v a l u a t i o n f u n c t i o n a l s

o ( x ) a r e continuous on 0, where D c a r r i e s a topology f i n e r than t h e

norm topology of

L2(M,p).

Definition A . I I . 3 . 3 )

and R a p o s i t i v e bounded Carleman o p e r a t o r on L (M,IJ)

(cf. Definition A . I . 2 . 1 ) , f u n c t i o n space.

I f (M,p) denotes a Federer measure space ( c f .

we have shown t h a t t h e space

D

2 = R(L2(M,p)) is

a

254

THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

In t h i s s e c t i o n w e prove t h a t a l s o the a n a l y t i c i t y spaces be considered a s genuine f u n c t i o n spaces, i f (M,v)

s

L2 (M,u) , A can i s a Federer measure

space and A t h e g e n e r a t o r of a one-parameter semi-group

(e- tA)t>o

which

c o n s i s t s o f Carleman o p e r a t o r s . Since each Hilbert-Schmidt o p e r a t o r on

L,(M,p)

i s a Carleman o p e r a t o r , it follows t h a t a l l n u c l e a r a n a l y t i c i t y

sL2 ( M , u ) , A

spaces

a r e genuine f u n c t i o n spaces whenever (M,p) i s a Federer

measure space.

Let A denote a nonnegative s e l f - a d j o i n t l i n e a r o p e r a t o r i n X and l e t (M,u) denote a Federer measure space. A l s o , l e t o p e r a t o r from X i n t o L,

(M,u),

v

denote a densely d e f i n e d l i n e a r

which maps S

i.e. f o r a l l t > 0, the operator v e

-tA

continuously i n t o ‘ I A

:

x

+

L

2

(M,p)

L2 (M,u),

i s bounded. On I7 w e

impose t h e following conditions.

IV.1.1.

- t A i s a Carleman o p e r a t o r .

a. For a l l t > 0, t h e bounded o p e r a t o r 0 e Let

kt

M

:

+

X denote t h e corresponding inducing f u n c t i o n of t h e

v e-*,

operator

t > 0.

b. The Borel f u n c t i o n s x I+

2 II k t (x) 11 x, t > 0 , a r e u - i n t e g r a b l e on

bounded Borel s e t s .

Remark. If f o r a l l t > 0 t h e o p e r a t o r e-tA i s Hilbert-Schmidt, then f o r a l l - t A i s Hilbert-Schmidt and hence Carleman. I t t > 0 a l s o the operator D e follows t h a t each f u n c t i o n Condition I V . l . l

Ilk (x)lI t X

XI+

-.

IV.1.2.

L e t t h e continuous l i n e a r mapping

V

:

SX,A

-+

L (M,u) s a t i s f y Condition 2

Then f o r a l l t > 0 and a l l T > 0 t h e r e e x i s t s a n u l l s e t

IV.l.l.

that for all x

E

Proof. L e t

v

(

x w (vn

,kt+T

t i o n x I+

(vn

Zn,t,T

i s square i n t e g r a b l e . I n t h i s c a s e

i s redundant.

(x)

M\Z

t, ‘c

~ denote ) ~ an ~orthonormal ~ basis i n

)x

is a r e p r e s e n t a t i v e of

,e-TA(kt (x) lx,n

such t h a t

such t # T

c

m.

U

x.

Then t h e f u n c t i o n

e-(t+T)Avn a s w e l l as t h e func-

It follows that t h e r e e x i s t s a n u l l set

A MEASURE THEORETICAL SOBOLEV LEMMA FOR ANALYTICITY SPACES

Now 2

=

t , T

z

u

e.

L e t t h e continuous l i n e a r mapping

IV.l.l.

0

s a t i s f i e s t h e requirement.

ncl" n , t , T

Taking R = e-tA i n Theorem A.I.4.5

IV.1.3.

255

we o b t a i n t h e following r e s u l t .

D

:

SX,A

+.

L,

(M.11)

s a t i s f y Condition

L e t t > 0 be f i x e d . Then t h e r e e x i s t s a n u l l s e t N

a r e p r e s e n t a t i v e (h) of Dw f o r each w

E

- t A (x) e

Nt and t h e r e e x i s t s

such t h a t t h e following

statements are v a l i d .

a. Let x

E

M\N t and l e t w

E

e

-tA

( X ) . W e have

(we c a l l the f u n c t i o n ( t h . ~the ) ~ canonicaZ r e p r e s e n t a t i v e of Vw.) b. For each x

M

E

t h e l i n e a r functional w

H i l b e r t space e

-tA

-

I+

( h ) (x) i s continuous on t h e

(x).

c. Suppose t h e f u n c t i o n x p o s i t i v e c o n s t a n t Ct.

Ilkt (x)IIx i s e s s e n t i a l l y bounded on M by a such t h a t Then t h e r e e x i s t s a n u l l set N 0.t I+

d. Suppose t h e X-valued f u n c t i o n

kt

i s continuous on

M. Then t h e representa-

w

t i v e s (Dw)

0

are continuous.

W e a r r i v e a t t h e following embedding theorem f o r a n a l y t i c i t y spaces. 1v.1.4.

Theorem.

L e t t h e continuous l i n e a r mapping

IV.l.l. of

D

:

SX,A +.

L2(M,p)

s a t i s f y Condition

-

Then t h e r e e x i s t s a n u l l s e t N and a canonical r e p r e s e n t a t i v e (Vw) f o r each w

a. For each x

E

E

S X , Asuch t h a t t h e following r e s u l t s a r e v a l i d .

M\N we have

2 56

THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

N

b. For each x

M

E

the l i n e a r functional x

I+

( x ) i s continuous on t h e

(&)

a n a l y t i c i t y space Sx, A.

Proof. L e t m,n

following Lemma IV.1.3 and f o r each w N

exp(-

E

1

-

N

(x) and (Ow),

sentatives

and NIIm, l/n A ) ( X I c exp(-, A) ( X ) repre-

IN, m > n. Then t h e r e e x i s t n u l l s e t s N

E

(x) such t h a t t h e s t a t e m e n t s a. - c .

of t h e

mentioned lemma a r e s a t i s f i e d . I t i s c l e a r t h a t

for all x

E

We put N =

M\(NIIn

kgIN

u N,,,).

NIIk.

E

sX , A

we d e f i n e t h e r e p r e s e n t a t i v e (&)

of Dw as follows. L e t w E SX,A. There e x i s t s n 1 w E exp(- ; A ) ( x ) . Let (Dw)" be defined by

(h)" does

Then t h e d e f i n i t i o n of

E

IN such t h a t

n o t depend on t h e choice of n

E

a. This a s s e r t i o n follows immediately from Lemma IV.1.3.a. b. By Lemma IV.1.3.b,

for all n

E

IN t h e l i n e a r f u n c t i o n a l w I+

continuous on t h e H i l b e r t space exp(Since

sX , A

-

Therefore t h e following d e f i n i t i o n s make sense.

For each w

IN.

-

(h) (x)

-1 A ) ( x ) .

is t h e i n d u c t i v e l i m i t of t h e H i l b e r t spaces exp(-

is

n1 A ) (x) 0

t h e r e s u l t follows. Let (M,p) denote a Federer measure space and l e t

B

denote a nonnegative

s e l f - a d j o i n t operator i n L 2 ( M , p ) . W e assume t h a t f o r each t > 0 t h e o p e r a t o r e-tB i s a Carleman o p e r a t o r with inducing f u n c t i o n bt

w e assume t h a t f o r a l l t > 0 t h e Borel f u n c t i o n x

H

: M -+ L, (M,p). Also . IIb ( x ) 11 2 is p-integrable

t

on bounded Borel sets of

M.

Following Theorem IV.1.4

t h e r e a r e continuous l i n e a r f u n c t i o n a l s

and a n u l l s e t N such t h a t f o r a l l Cql

-

E

S

L2 (M,P) ,B

N

cp : x

I+

L (Ccpl) i s a r e p r e s e n t a t i v e of Ccpl

-

q(x) = l i m u(B(x,r))-'( r+O

icy) du(y))

B (x. r )

. ,

x

E

M\N

.

ex, x

E

M,

257

A MEASURE THEORETICAL SOBOLEV LEMMA FOR ANALYTICITY SPACES

So the analyticity space

sists of the functions

sL2 (M,u),8 can be

regarded as a space which con-

By the representation theorem for continuous linear functionals on analyticity spaces, there exist elements 6 x

It follows that the trajectory space delta functions.

TL2(M,~),B'

TL2 (M,U) ,8

x

E

M, such that

contains a complete set of

A nonnegative self-adjoint operator 8 which generates a one-parameter semi-tB group (e )t>O of Hilbert-Schmidt operators satisfies all requirements. Therefore any nuclear analyticity space

sL 2 ( M , P ) , B

can be considered as a

genuine test function space. The corresponding nuclear trajectory space TL2 (M,P) ,B

contains a complete set of delta functions.

In almost all illustrations we presented in the preceding chapter the considered analyticity space is nuclear. So these spaces can be regarded as function spaces. Here we present two simple examples in which the analyticity space is not nuclear. IV. 1.5. E x q le ,

Consider the Dt

:

(XnY)t+-

operator -

d2 -

in L2(IR). For each dx2 operator with kernel

inducing function 2

6

dt

:

IR

-+

L2(IR)

is defined by

(*)

dt(x) = [y

I+

For all t > 0 and all x

Dt(x,y)1 E

,

x

IR we have

E

IR

.

IIdt(~)112 = (2nt)-'.

It follows that

d2

the operator - - satisfies the requirement. So from the consideration dx2 given above, it follows that s d2 is a genuine test function space.

L,

,-1 ax

This function space has been described-in Section 111.1, Example E l . The trajectory space

T

L,

d2 contains a complete set of delta functions dx

,-2

250

ISx

THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

Ix

E

IR).

jectories t

These g e n e r a l i z e d f u n c t i o n s 6x are r e p r e s e n t e d by t h e trab+

.

d t ( x ) d e f i n e d by ( * )

Example.

1v.1.6.

22

Consider t h e nonnegative s e l f - a d j o i n t o p e r a t o r o p e r a t o r of m u l t i p l i c a t i o n by t h e f u n c t i o n x

*

x

2

i n L 2 (IR). (Q2 is t h e

,

x

E

IR.)

Clearly t h e

o p e r a t o r e-t22 i s n o t a Carleman o p e r a t o r and t h e a n a l y t i c i t y space

sL,(IR)12

sL

2 i s n o t a f u n c t i o n space. Indeed,

(IR) 22 c o n s i s t s of equi-

2

'

s L,

(IR)4 2

valence c l a s s e s of Borel f u n c t i o n s w on I R such t h a t t h e f u n c t i o n tx2 x H e w(x) i s square i n t e g r a b l e f o r c e r t a i n t > 0. The F o u r i e r transformation IF on L 2 (IR) maps So f o r each Cw1

sL2 (TR)I 22 t h e r e

E

IF"w1.

( I F [ W I ) ~ of

onto S

d2 dx2

L 2 ( I R ) ,--

e x i s t s a canoncial r e p r e s e n t a t i v e

W e have f o r a l l x

-

IR

E

,

(IFCWl)-- ( x ) = < I F C W l , 6x> = < C w l

IV.2. D i r a c bases i n t r a j e c t o r y spaces Chapter A . 1 1 1 has been e n t i r e l y devoted t o t h e concept of Dirac b a s i s . We have introduced t h i s new mathematical concept i n t h e b a r e s e t t i n g of a Sobolev t r i p l e R(X) c

x

c R-'tX).

I n t h i s s e c t i o n we l i f t t h e notion of

Dirac b a s i s t o t r a j e c t o r y spaces. Let A denote a nonnegative s e l f - a d j o i n t o p e r a t o r i n t h e s e p a r a b l e H i l b e r t space

x.

We consider t h e t r i p l e SXIA

c

x

c

TX,A.

By (M,p) w e denote a a-

f i n i t e measure space. A

f u n c t i o n 8 : M + TX,A i s s a i d t o be a Borel f u n c t i o n i f f o r each w

E

t h e function x

H

<w , 8 ( x ) > i s a Borel function. In t h e space o f Borel

f u n c t i o n s from

M

t o TX,A w e i n t r o d u c e t h e equivalence r e l a t i o n

el

-

X

O2

:-

8 , ( x ) = B2(x)

We employ t h e usual n o t a t i o n respect t o

-.

f o r each w

E

p-almost

everywhere

S

X,A

-

.

C81 t o denote t h e equivalence class o f 8 with 0

:

SX,A t h e f u n c t i o n x

I+

A Borel f u n c t i o n

M

+

<w

TX,A i s c a l l e d weakly p-integrable i f

,@(XI>

i s p-integrable.

The Borel

DIRAC BASES IN TRAJECTORY SPACES

2 59

f u n c t i o n 8 i s s a i d t o be s t r o n g l y p-integrable i f t h e f u n c t i o n x

H

l e ( x ; t ) 11

X i s p-integrable f o r a l l t > 0.

IV. 2. I . Proposition. Let

e

:

M -+ T X , A be a s t r o n g l y p-integrable function. Then 0 i s weakly u-

i n t e g r a b l e and f o r a l l w E

sX , A

t h e r e e x i s t s t > 0 such t h a t

I t follows t h a t t h e l i n e a r f u n c t i o n a l

w

H

J

< w , e ( x ) > dp(x)

,

w

E

S

X,A

M i s continuous.

Proof.

Let w

E

SX,A. Then t h e r e e x i s t s t > 0 such t h a t w

E e

-tA

(x).

It

follows t h a t <w , 8 ( x ) > = (ed w , 8 ( x ; t ) ) x and so by Schwarz' i n e q u a l i t y

Thus w e o b t a i n

1v.2.2. Definition. Let 8 :

M

-+

T X , A be s t r o n g l y p-integrable. Then t h e i n t e g r a l expression

denotes t h e element % , p ( 8 ) of TX,A defined by

The e x i s t e n c e of I

M,P

(8)

E

TX,A follows from P r o p o s i t i o n IV.2.1

and Theorem

THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

260

1.3.2,

which s t a t e s t h a t any continuous l i n e a r f u n c t i o n a l on

I

O(X)

1s re-

TX,A. W e observe t h a t f o r a l l t > 0

presented by an element of

(

s X,A

J

dlJ(x))(t) =

M

e ( x ; t ) du(x)

M

where t h e l a t t e r i n t e g r a l i s a Bochner i n t e g r a l of t h e X-valued s t r o n g l y p-integrable function x

A s i n Section A . I I I . l

I+

O(x;t).

t h e following a u x i l i a r y r e s u l t i s needed i n o r d e r t o

a r r i v e a t a well-defined concept of Dirac b a s i s f o r t r a j e c t o r y spaces.

Lemu.

Iv.2.3.

There e x i s t s an orthonormal b a s i s (unInEm i n

x

which i s contained i n

and, a l s o , a Schauder b a s i s i n SX,A. I t means t h a t f o r a l l w

E

m

series

ngl

(w,un) u

sX , A

s X,A

the

a r e convergent i n the sense o f t h e i n d u c t i v e l i m i t

topology of SX,A.

Proof. Step 1. Assume t h a t A h a s pure p o i n t spectrum. Then t h e r e e x i s t s an orthonormal b a s i s ( f

E

x

u

~ and ) nonnegative ~ ~ ~ numbers A n ,

n

E

such t h a t f o r a l l

IN,

and a l l t 2 0 0)

e- t A f =

1

(f,un)Xun

n= 1 where t h e s e r i e s converges i n t h e sense o f t h e normtopology of

w

E

M,N

SX,A. Then there e x i s t s t > 0 such t h a t w E

E

e-tA(X).

IN

m

5

1 I ( et A w,

n=N+1 It follows t h a t t h e series m

x.

Let

So f o r a l l M > N ,

2 unlXI

.

DIRAC BASES I N TRAJECTORY SPACES

261

i s convergent i n t h e H i l b e r t space e-tA(X) and hence i n SX,*. shown t h a t t h e orthonormal b a s i s (

Thus w e have

sX , A '

~ is ) a Schauder ~ ~ ~basis i n

u

Step 2. L e t ( E X ) X E I R denote t h e s p e c t r a l r e s o l u t i o n of t h e i d e n t i t y belonging t o

A.

W e d e f i n e t h e mutually orthogonal p r o j e c t i o n s P,,

k

E

IN,

by

Y

and t h e o p e r a t o r A by m

1

N

A =

kPk

k= 1 with domain

I t is c l e a r t h a t

A has a d i s c r e t e spectrum. Both o p e r a t o r s e t X .-tA

Y

N

and

Y

etA e-tA a r e bounded with IIe tAe-tAll

=

e

t

11

and IIe

=

1. N

Now l e t ( u n ) n E I Ndenote an orthonormal b a s i s of eigenvectors of A. Then

un

E

e-tA(X)

u

f o r a l l t > 0, because

-tA

= e

tA (e

.-tX

txu )e n

and

F u r t h e r , f o r each w

E

I t follows t h a t (

u

e

-tA

(x),

t > 0 , and M , N

E

I N with M > N

~ is) a Schauder ~ ~ ~basis i n e

-tA

(x)

f o r a l l t > 0.

0

262

THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

Definition.

IV. 2 -4.

A denote a nonnegative s e l f - a d j o i n t o p e r a t o r i n a s e p a r a b l e H i l b e r t x, l e t (M,p) denote a o - f i n i t e measure space and l e t [ G I denote an equivalence class of B o r e 1 f u n c t i o n s G : M .+ TX,A. Then t h e q u i n t u p l e (CGI,M,!J,A,X) i s c a l l e d i n Dirac b a s i s i n T X,A' i f t h e r e e x i s t s an ortho-

Let

space

1

normal b a s i s ( each G

E

~ i n)

u

x, ~which ~

is ~ a l s o a Schauder i n

sX,A

such t h a t f o r

[ G I t h e following r e l a t i o n s a r e v a l i d

( I m p l i c i t l y , w e assume t h a t t h e f u n c t i o n s x I+ a r e square i n t e grable.)

The concept of Dirac b a s i s f o r t r a j e c t o r y spaces extends t h e concept of orthonormal b a s i s f o r H i l b e r t spaces.

CoroZZary.

Iv.2.6.

x

Each orthonormal b a s i s ( v ) in can be regarded as a Dirac b a s i s which n nEIN i s described by t h e q u i n t u p l e ( v , I N , T , B , X ) . Here t h e f u n c t i o n v : IN + is

x

given by v ( n ) = vn, n

E

IN,

T denotes t h e counting measure and

negative bounded l i n e a r o p e r a t o r on

Proof.

8 any non-

x. 0

The proof runs s i m i l a r l y t o t h e proof of C o r o l l a r y A.III.1.7.

L e t ([G],M,p,A,X)

6

denote a Dirac b a s i s . L e t

be any r e p r e s e n t a t i v e of

and l e t (unInEm denote an orthonormal b a s i s i n

x,

[GI

which i s a Schauder

b a s i s i n SX,A such t h a t t h e r e l a t i o n s IV.2.5 a r e s a t i s f i e d . For each k

E

IN,

1

by rp k w e denote t h e square i n t e g r a b l e f u n c t i o n x I+ . The R e l a t i o n s IV.2.5 express t h a t the s e t {[rp,] k E IN} i s orthonormal i n t h e H i l b e r t

I

space L2 ( M , p ) . L e t t h e l i n e a r o p e r a t o r V

:

m

Vf

=

1

(f,UkIX

k= 1

CVkl ,

f E

x

-+

L2(M,u) be d e f i n e d by

x.

Then V i s an isometry.

k21 ( W , I + ) ~ m

SX,A. Since t h e series each x E M w e g e t

Let w

E

\

i s convergent i n SX,A'

for

DIRAC BASES I N TRAJECTORY SPACES m

So

the function x

I+

263

m

<w , E ( x ) > i s a r e p r e s e n t a t i v e of t h e c l a s s m

Vw =

k= 1

(w,\Ix

:

x

(w,ukIx

k= 1

converges pointwise.

(pk

Theorem.

117.2.7.

Let ( [ G l , M , u , A , X )

V

The s e r i e s

[(p,].

-+

5

denote a Dirac b a s i s . Let

[ G I be f i x e d and l e t

E

i n t h e way described

L2(M,p) denote t h e isometry corresponding t o

above. a . For a l l w1,w2

SX,A w e have t h e Plancherel formula

E

(w1,w2)x =

M b. For each

E

I

[ G I and w

.

1

<wl , G ( x ) > <w2 , G ( x ) > dP(x)

s X,A

E

*

Y

t h e Bore1 f u n c t i o n x

r e p r e s e n t a t i v e of Vw. So t h e c o n s t r u c t i o n of

v

H

<w , G ( x ) > i s a

does n o t depend on t h e

choice of G. c. For a l l t > 0 , t h e o p e r a t o r

v

i s a bounded Carleman o p e r a t o r .

Proof. a . For each w

E

SX,A t h e f u n c t i o n x

I+

of Vw. It follows t h a t f o r a l l w1,w2

-

b. L e t G E [ G I .

.

<w , G ( x ) > , x

M, i s a r e p r e s e n t a t i v e

E

SX,A

E

-

Then w e can c o n s t r u c t an isometry V

:

x

* L2(M,u)

i n the

way described above. Since t h e f u n c t i o n s

x

H

<w

1

,G(x)>

Y

are e q u i v a l e n t f o r each w

w

E

x t+ <w

and

E

SX,A and hence t h a t V =

s v.

-X,A’

,G(x)>

i t follows t h a t Vw = ‘iiw f o r a l l

..

c. L e t t > 0. Define t h e X-valued function k

Then f o r a l l f

E

x,

t

on

M by

264

THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

So the Bore1 f u n c t i o n x ++ ( f

V e-tAf

E

L2(M,u).

.

, k t ( ~ ) ) Xi s a r e p r e s e n t a t i v e o f t h e c l a s s it follows t h a t V e-tA i s a

By D e f i n i t i o n A . I . l

0

Carleman o p e r a t o r .

There e x i s t s a one-to-one and i s o m e t r i e s V operator

1v.2.8.

v

e-tA

:

x

correspondence between Dirac b a s e s ( C G l , M , u , A , X )

L 2 ( M , p ) with t h e property t h a t f o r a l l t > 0 t h e

+

i s Carleman.

Theorem.

L2(M,u)

Let V denote an isometry from X i n t o

x.

adjoint operator i n

and A a nonnegative s e l f -

Suppose t h a t f o r a l l t > 0 , V e-tA

i s a Carleman

operator. Then t h e r e e x i s t s a Dirac b a s i s ( C G l , M , u , A , X ) , G

E

VW

.

[ G I and each w

Proof.

SX,A t h e f u n c t i o n x

E

-tA Carleman o p e r a t o r V e there e x i s t s a n u l l set

kt (x) NOW

kt

For each t > 0 , l e t

we put

Z

= e

. As zt ,

-(t-T)

zlln,

= n,mEDJ, u

:

A

M

+

x

such t h a t f o r each

1

I+

<w , G ( x ) > i s a r e p r e s e n t a t i v e of

denote an inducing f u n c t i o n of t h e

we have seen i n Lemma IV.1.2, such t h a t f o r a l l x

E

for t > T > 0

M\Zt

(kT(x)).

. Then z

l/m

i s a n u l l s e t . Next w e d e f i n e t h e

n>m function G

:

M

+

T X , A by

,

G(x;t) = 0

G(x;t) = exp(-(t

-

1

-m) A ) k l/m(x)

We prove t h a t G i s well-defined. L e t x 1 i n d t > - f o r c e r t a i n n,m E IN with n

m

= exp(-(t

- -1) A )

= exp(-(t

-

1 exp(-(n

-1) A ) k m

l/m(x)

M\Z

6

i f x E Z a n d t > O , x E M\Z and t > if w i t h m E DJ.

-1

1 and l e t t > 0. Suppose t > -

m. Then we have

- m-1) A )

1 (exp((n

- m-1) A )

hlln(x)) =

265

DIRAC BASES I N TRAJECTORY SPACES

So f o r x

M\Z

E

t h e d e f i n i t i o n of G ( x ; t ) does n o t depend on t h e choice of m.

I t is c l e a r t h a t t

G ( x ; t ) belongs t o

H

T

and t h a t x t+ G(x) i s a Bore1

X,A

function.

x

Let ( u n ) n E m denote an orthonormal b a s i s i n

SX,A.

Let w

SX,A. Then we have f o r a l l x

E

where m m u s t be chosen so l a r g e t h a t w

x

I+

<w

, G(x)> i s

a l l w1tw2

which i s a Schauder b a s i s i n

M

E

1 A) m

exp(--

E

a r e p r e s e n t a t i v e of Vw = V

1 exp(--

m

(x). A)

I t follows t h a t

1 m

(exp(- A ) w ) .

So f o r

SX,A

E

= (VW,

(W1'W2)X

,Vw2)L2 -

<wl , G ( x ) > <w2 , G ( x ) > d p ( x )

.

M I f [GI denotes t h e equivalence class o f t h e f u n c t i o n G, then

([~l,M,p,A,x) 0

i s a Dirac b a s i s with t h e wanted p r o p e r t i e s .

For each Dirac b a s i s ( [ G ] , M , p , A , X ) type r e s u l t =

M where w1,w2

SX,A and

E

So f o r each w

I

E

s X,A

we have proved t h e following Plancherel-

<wl , z ( x ) > <w2 , ; ( X I >

E

[GI.

.

dub)

.

t h e f u n c t i o n x H <w , G ( x ) > G(x) i s weakly p-integrable

and we o b t a i n t h e weak expansion

w =

<w , z ( x ) > t ( x ) d p ( x )

.

M

-t A )t,O

If t h e semigroup (e

c o n s i s t s of Hilbert-Schmidt o p e r a t o r s , t h e n t h e

above expansion converges i n call

(CGI,M,P,A,X)

a basis.

T

X,A-SenSe

. So now

it i s f u l l y j u s t i f i e d t o

266

THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

Theorem.

IV.2.9.

Let t h e a n a l y t i c i t y space b a s i s and l e t

E

E

SX,A be n u c l e a r . Let ( C G l , M , p , A , X )

[ G I be f i x e d . Then f o r each w

E

sX,A

denote a Dirac

t h e TXIA-valued

function x

I+ <w

, 2 ( x ) > E(x)

i s s t r o n g l y u-integrable.

w =

M

I

<W

So i n t h e expansion

, c ( x ) > E(x) dp(x)

t h e convergence i s i n

T

with ( a b s o l u t e ) convergence i n

Proof.

i.e. for a l l t > 0

X, A-sense‘

x.

As w e have shown i n Theorem IV.2.8,

induces t h e Hilbert-Schmidt o p e r a t o r t h e isometry from

([GI

,M,p,A,X)

. Now

x

v e-tA

t h e Bore1 f u n c t i o n x

H

f o r each t > 0 . Here

v

G(x;t) denotes

i n t o L2(M,p) corresponding t o t h e Dirac b a s i s Lemma A . I . 2 . 2

So f o r each t > 0 and each w

E

yields

sX , A

w e obtain

The following statements a r e similar t o t h e s t a t e m e n t s i n Lemma A.III.1.9 and Theorem A.III.1.11.

CANONICAL DIRAC BASES I N TRAJECTORY SPACES

267

Theorem.

IV. 1.10.

A denote a nonnegative s e l f - a d j o i n t unbounded o p e r a t o r i n t h e s e p a r a b l e H i l b e r t space x and l e t (M,u) denote a o - f i n i t e measure space.

Let

a. In TX,A t h e r e e x i s t s a Dirac b a s i s b. I f t h e measure space

(CGI,M,u,A,X)

(M,u)

(CGI,M,u,R,X).

i s purely atomic, then each Dirac b a s i s

i s an orthogonal b a s i s i n

X.

c. I f t h e a n a l y t i c i t y space SX,A i s n u c l e a r , then each isometry V

:

x

-+

L 2 ( M , p ) generates a Dirac b a s i s i n t h e described way.

Proof. a. The o p e r a t o r 1

+A

is unbounded. So 0 belongs t o t h e e s s e n t i a l spectrum

of ( 1 +A)-'.

Following Theorem A.I.3.6

X

such t h a t U ( 1 + A ) - '

U

:

-+

L2(M,p)

( 1 + A ) e-*

t > 0 , U e-tA = U ( I + A)-'

t h e r e e x i s t s a unitary operator

i s a Carleman o p e r a t o r .

SO

i s a Carleman o p e r a t o r .

f o r each

(Cf. Theo-

rem A.I.3.2.) b. The proof of t h i s statement runs similar t o t h e proof of Lemma A.IV.1.9 and t h e r e f o r e i s omitted.

c. For any isometry V

:

x

+

L2(M,p) t h e o p e r a t o r V e

- t A , t > 0, i s H i l b e r t 0

Schmidt and hence a Carleman o p e r a t o r .

IV.3. Canonical Dirac bases i n t r a j e c t o r y spaces A Dirac b a s i s ([GI,M,p,A,X)

i s an equivalence c l a s s of Bore1 f u n c t i o n s . I n

general t h e r e i s no p r i v i l e g e d r e p r e s e n t a t i v e of t h e c l a s s [ G I .

But, i f we

impose c e r t a i n c o n d i t i o n s on t h e measure space (M,p) and t h e o p e r a t o r A , then t h e r e e x i s t s a "canonical" r e p r e s e n t a t i v e of t h e c l a s s [ G I . Thus we a r r i v e a t t h e notion of canonical Dirac b a s i s . I V . 3. I .

(Cf. D e f i n i t i o n A . I V . 2 . 1 . )

Definition.

L e t (M,p) be a Federer measure space and l e t ( [ G l , M , p , A , X ) Y

be a Dirac b a s i s .

canonical Dirac b a s i s i f t h e r e e x i s t s SX,A and a l l x E M\W

A r e p r e s e n t a t i v e G E [GI i s c a l l e d a

a null s e t

R

such t h a t f o r a l l w

l i m p(B(x,r))-'( rCO N

E

/ B (x, r )

-

Y

<w , G ( y ) > d p ( y ) ) = <w , G ( x ) >

-

The canonical Dirac b a s i s G w i l l be denoted by (Gx)xcM.

.

268

THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

Throughout this section by (M,p) we mean a Federer measure space. So we can apply Theorem IV.1.4. IV.3.2. Let

X

A

e.

be a nonnegative self-adjoint operator in the separable Hilbert space

and let V be an isometry from

x into L2(M,p).

We assume that the operator

x)

V satisfies Condition IV. 1.1. Then the Dirac basis ([GI ,M,p,A,

induced by

N

V contains a canonical representative (Gx) xeM’

sX,A there exists a representative (Vw) of the class Vw with the following properties. For each x M the linear functional w (Vw) (x) is continuous on

Proof.

Following Theorem IV.1.4, for each w

(*)

E

I+

E

’X,A* N

(**)

There exists a null set N such that for all w

NOW for each x

E

M

E

sX.A

and all x

E

M\i

N

we define G x

TX,A by

N

clear that the function x

I+

G

sX,A

are in duality. It is X,A is a representative of [GI. The relation

This definition makes sense because

and T

N

(**)

indicates that (GxlxeM is a canonical Dirac basis.

0

1v.3.3. Theorem.

-

The Dirac basis ([GI,M,u,A,X) contains a canonical representative (G ) x XEM if for some ?i E [GI and every t > 0 the Borel function x I+ IlE(x;t)11; is integrable on bounded Borel subsets of

M.

x

Proof.

Following Theorem IV.2.7 there exists an isometry V : * L2(M,p) -tA such that for all t > 0 the operator V e is a Carlernan operator with the

property that for all f

,.

So for a fixed G

E

E

x

and each

6

E

[GI

[GI, the Borel function kt : x t+ G(x;t) induces the - tA Our assumptions imply that the family I k t I t > 01

Carleman operator V e

.

CANONICAL DIRAC BASES I N TRAJECTORY SPACES

s a t i s f i e s Condition 1V.l.l.b.

269

Now t h e a s s e r t i o n follows from Lemma

0

I V . 3.2.

Coro 1lary

I V . 3.4.

Let

.

SX,A be a n u c l e a r space. Then any Dirac b a s i s

(CG],M,u,A,X)

contains a

N

0

canonical r e p r e s e n t a t i v e (G ) x XEM' Each canonical Dirac b a s i s i s r e l a t e d t o a generalized eigenvalue problem. L e t us d e s c r i b e t h i s connection, here.

M which M i n L,(M,u)

L e t cp be a complex valued Borel function on

Bore1 s e t s . The m u l t i p l i c a t i o n o p e r a t o r

i s bounded on bounded

cp

i s defined by

and

M Cfl = Ccpfl cp

,

Cfl

E

D(Mcp)

.

E M and r > 0 t h e c h a r a c t e r i s t i c f u n c t i o n x B(x,r) belongs t o D(M ) , because t h e function cp i s bounded on bounded Borel sets. cp

W e observe t h a t f o r any x

S i m i l a r t o Theorem A.III.2.5

IV.3.5.

Theorem.

Let ( [ G ] , M , v , A , X )

V

:

w e prove t h e following fundamental r e s u l t .

denote a Dirac b a s i s induced by t h e isometry

X + L 2 ( M , p ) . W e assume t h a t V s a t i s f i e s Condition I V . l . l , i . e . t h e

c o n d i t i o n s of Theorem IV.3.3

a r e f u l f i l l e d . Then i n t h e c l a s s [ G I t h e r e N

e x i s t s a canonical D i r a c b a s i s (GXOxEM which possesses t h e following properties : a . There e x i s t s a n u l l s e t

lim r10

b. Let cp :

M

+

a

such t h a t f o r a l l x

Ilcx(t)- p ( B ( x , r ) )-1 ( e-4V *

E

M\N

and a l l t > 0 ,

x ~ ( ~ , I I~x ) =) 0

.

C be a Borel f u n c t i o n which i s bounded on bounded Borel s e t s .

Then t h e r e e x i s t s a n u l l s e t

N cp

such t h a t f o r a l l x

E

M\N

cp

and a l l t > 0

270

THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

Proof. a . Let k

E

IN. By Theorem A.III.3.5

for a l l x

-N

E

M\Nk

there exists a null s e t

-Nk

such t h a t

the X - l i m i t

exists.

Put

=

k2m

ik.Since

t > 0, f o r all x

So f o r each x

E

E

-

M\N

t h e o p e r a t o r e-tA i s continuous on and a l l k > m , k,m

M\N t h e X-valued f u n c t i o n

-

1

gx,l,m

Gx(t) = exp(-(t-;)A)

i s a well-defined member of Now l e t t > 0. Choose m

E

T

,

X,A'

E

s X.A

Ex on

and a l l x

E

M\a

for all

( 0 , ~ )given by

t > m' m

;.

I t follows t h a t

X

we have

IN,

IN so l a r g e t h a t t > 1

w e have

Further, f o r a l l w

E

E

m,

Then f o r all x

E

M\a

THE GENERALIZED EIGENVALUE PROBLEM REVISITED

271

-

Hence (Gx)xcM i s t h e wanted canonical Dirac b a s i s . b. The second a s s e r t i o n can be proved i n t h e same way a s Theorem A.III.2.5

n

has been proved. W e do n o t r e p e a t t h i s proof here.

Iv. 3.6. Corol law. A s s u m e t h e n o t a t i o n s and c o n d i t i o n s of Theorem IV.3.5. Also assume t h a t V i s a u n i t a r y o p e r a t o r and t h a t V* M V i s c l o s a b l e a s a l i n e a r mapping i n T

Then f o r a l l x

X,A'

E

cp

M\K

where V* M V denotes t h e c l o s u r e V* M V as a l i n e a r mapping i n TX,A. cp cp

Proof.

'* ('B

It i s c l e a r t h a t

(x,r)

) E

D(V* M

V)

=

'p

V* (D(M

)).

SO f o r a l l

'p

x

E

M\fi

D (V* M V ) . From Theorem IV.3.5 we g e t 'p

and

N

f o r a l l x c M\N.

IV.4.

--

Hence V* M V Gx = W

'p

(x)

Ex.

O

The generalized eigenvalue problem f o r s e l f - a d j o i n t o p e r a t o r s solved i n t h e s e t t i n g of t r a j e c t o r y spaces

In Chapter A.IV w e have presented a s o l u t i o n o f t h e g e n e r a l i z e d eigenvalue problem f o r a f i n i t e number o f mutually commuting s e l f - a d j o i n t o p e r a t o r s i n i t s rudimentary form. I n t h i s s e c t i o n w e study t h e same problem i n t h e s e t t i n g of t r a j e c t o r y spaces. W e r i g t h e r e s u l t s of Chapter A.IV, i n o r d e r t o a r r i v e a t a s o l u t i o n which can be l o o s e l y formulated a s follows. L e t t.here be given n mutually commuting s e l f - a d j o i n t o p e r a t o r s

...,P

Pl,

i n a s e p a r a b l e H i l b e r t space

nonnegative s e l f - a d j o i n t o p e r a t o r A i n tors

TX,A

Pe, 1 into T

=

1,. . . , n

X,A'

,

x . Then t h e r e e x i s t s a x such t h a t t h e opera-

a r e c l o s a b l e a s l i n e a r mappings from

Also, f o r almost each p o i n t A = ( A

I . .

., A n )

2 72

THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

i n t h e j o i n t spectrum o ( Pl,...,!n)

with m u l t i p l i c i t y mX t h e r e

N

e x i s t mX t r a j e c t o r i e s E A , l , . . . , E

The s e t

{ih. j

I A

o(P1,

E

...,P

a canonical Dirac b a s i s i n T

h , m X in T

1 5 j < m

),

~

A

such t h a t

,

~

+ I } constitutes

X,A'

F i r s t we r e c a l l some elementary r e s u l t s of Chapter A . I V .

m

Let

denote a countable set. On

m

w e impose t h e d i s c r e t e topology. L e t

m.

denotes t h e t o p o l o g i c a l space IRn x

M

On

I

.

.

Now In w e d e f i n e t h e metric d as

IRn c a r r y i t s usual Euclidean topology induced by some norm

M

follows

where x,y

E

lRn, p1,p2

E

m.

M.

The metric d induces t h e product topology of

E ID l e t w denote a f i n i t e Borel measure on IRn. Then ( I R n , v ) , P P i s a Federer measure space. We d e f i n e t h e measure v = 8 on M PEm

For each p p

ID,

E

by

where t h e Borel s e t i s o-finite.

B

= { ( x , p ) 1x

B

E

P'

p

E

I t can be r e a d i l y checked t h a t

ID}.

The measure space (M,w)

L 2 (M,w) can be i d e n t i f i e d with

8 L, ( d , w 1. L e t B( ( x , p ) , r ) denote the c l o s e d b a l l PE m P with c e n t r e ( x , p ) and r a d i u s r > 0 with r e s p e c t t o t h e m e t r i c d. Let

t h e H i l b e r t space in

M

B ( x , r ) denote t h e closed b a l l i n IRn with c e n t r e x and r a d i u s r > 0. We r e c a l l Theorem A.IV.2.1.

IV.4.1.

Theorem.

The measure space (M,u) admits a r e l a t i v e d i f f e r e n t i a t i o n theorem: L e t cp be

a complex-valued Borel f u n c t i o n on t h e f u n c t i o n cp

: x

I+

M

with t h e p r o p e r t y t h a t f o r each p

-

cp(x,p) i s i n t e g r a b l e on bounded Borel sets of

E

mn.

m

P Then t h e r e e x i s t s a r e p r e s e n t a t i v e cp of t h e c l a s s [cp 1, p E ID, and a u-null P P s e t N such t h a t f o r a l l ( x , p ) E M\N t h e closed b a l l B( ( x , p ) , r ) has posicp

t i v e w-measure and

cp

THE GENERALIZED EIGENVALUE PROBLEM REVISITED

x

Let

denote a separable H i l b e r t space and

u

273

a u n i t a r y o p e r a t o r from X onto

L 2 ( M , p ) . We f i x a nonnegative s e l f - a d j o i n t o p e r a t o r A such t h a t U e-tA Carleman o p e r a t o r f o r a l l t > 0. (E.g. t a k e A such t h a t t h e semigroup

-tA)

is a

c o n s i s t s of Hilbert-Schmidt o p e r a t o r s . ) For each t > 0 l e t X denote an inducing function of t h e Carleman o p e r a t o r U e- t A . On each function h we impose t h e condition t h a t f o r each p E ID t h e f u n c t i o n t x I+ Ilht(x,p) x E lRn, is u - i n t e g r a b l e on bounded Bore1 s e t s of mn.

(e

kt

:

t>O

M

+

IIi,

For each p

E

P

ID, x

E

lRn and r > 0 l e t Ap

B (x,r)

denote t h e f u n c t i o n on

M

defined by

A;(x,r)

x B ( x , r ) (')

(YPP') =

6pp'

'

p'

E

y

ID,

E

lRn.

e.

IV.4.2.

Let t > 0 be f i x e d . For each p

x

E

IRn,

E

ID t h e r e e x i s t s a v - n u l l s e t N and t h e r e e x i s t P P,t E supp(v ) \ N = 1, n and P P#t

e

such t h a t f o r each x

...,

zpx , t

and

l i m 11 x L P , r+O Here

Q1

v (B ( x , r ) -1 e - d U * qL AP B(x,r) " X = P

'

denotes t h e maximal s e l f - a d j o i n t o p e r a t o r of m u l t i p l i c a t i o n by t h e

coordinate f u n c t i o n idl

pFID

-

:

x

I+

xl i n each summand of t h e d i r e c t sum

L, (mn,vp)*

Proaf.

The statements follows from Theorem A.IV.2.3-4

replacing

R

by e

-tA

in t h e s e theorems. IV. 4 . 3 .

0

Theorem.

L e t p E ID.

There e x i s t s a

u -null set N

P P i n T X , A such t h a t f o r each x E supp(u )\Np, P

and t h e r e e x i s t t r a j e c t o r i e s a l l t > 0 and

l = l,.. .,n

Ep

THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

274

IIEp(t) -

lim r+O

v (B(x,r))-le-tA P

'*

'B(x,r)

L

= 1,

II X = o

and

I t means t h a t f o r a l l x

supp(u ) \ N and P P

E

...,n

and

Proof.

we use t h e n o t a t i o n s of Lemma I v . 4 . 2 .

Put N = u N L e t k , j E IN with k > j and l e t x E supp(v ) \ N P prl/k' P P' Since t h e o p e r a t o r e x p ( - ( y - -)A) i s bounded, Lennna IV.4.2 y i e l d s

3

k

1 ePx, l / j = exp(-(-j1 - -)A) k

'F x,l/k

ip E T a s follows. For each t > 0 we t a k e m X,A x t h a t 0 < - < t and n e x t we p u t m

Now w e d e f i n e

Due t o t h e r e l a t i o n ( * I , choice of m

IN.

E

t h e d e f i n i t i o n of

- t A f IIx

1

5 I I e x p ( - ( t - Tl) A )

IIe

E

x

and 0 <

1

J

P t h e r e a d e r t h a t t h e family E

sX,A

T

X,A'

and t h e simple i n e q u a l i t y 1

I1 IIexp(-r1 A) fll X

< t.

0

RetnUPk. For x E IRn\supp(v ) ,

So f o r each w

IN so l a r g e

does n o t depend on t h e

I t i s c l e a r t h a t Ep belongs t o

The wanted r e s u l t follows from Lemma IV.4.2

where f

Ep(t)

E

p

E

ID,

(ip) x (x,P)EM

the f u n c t i o n

-

w e p u t Ep

=

0. I t w i l l n o t s u r p r i s e

i s a canonical Dirac b a s i s i n

TX,A'

THE GENERALIZED EIGENVALUE PROBLEM REVISITED

is a canonical r e p r e s e n t a t i v e of t h e c l a s s Uw, i . e . f o r a l l x P

m,

E

e

For each

=

1,

-.., n

,

let

Pl

u* 2e u.

=

275

E

supp(v ) \ N , P P

Then t h e n-set c o n s i s t s of mutually

commuting s e l f - a d j o i n t o p e r a t o r s . From Theorem IV.4.3 it follows t h a t f o r all x

E

supp(v ) \ N and f o r a l l p P P

E

and

-

So each Ep can be considerd a s a candidate simultaneous ( g e n e r a l i z e d ) eigenv e c t o r f o r t h e n-set

( P I , ...,P

I f each o p e r a t o r

).

closed o p e r a t o r i n T X , A , then f o r t h e i r c l o s u r e -

P

e

PL

PL can be extended t o a w e have

.

u

Ep = xeiP

x

Thus we o b t a i n genuine simultaneous g e n e r a l i z e d e i g e n v e c t o r s f o r t h e n-set

-

-

(PI,...,?'

) . Of

course, we need n o n - t r i v i a l generalized e i g e n v e c t o r s .

Therefore w e prove t h e following lemma.

IV.4.4.

e.

Let t h e

gE

zp

be as i n Theorem IV.4.3.

such t h a t f o r a l l x

w.z Put

Let (

'X,A.

v

Put

P

= {x

E

E

Let p

E

15.:

supp(v ) P

=

k

"P $,(x) I t follows t h a t

Then t h e r e e x i s t s a n u l l s e t

,Ep>,

-

x

= = 0

E

IRn,

,

gp

is non-zero.

01.

~ denote ) ~ an ~orthonormal ~ basis in : x I+
ID.

supp(v ) \ z t h e t r a j e c t o r y P P

p

x

L

E

x

ID.

zP

which i s a Schauder b a s i s i n Then

( u vkIp

=

and

276

THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

where A

5

denotes t h e Borel f u n c t i o n on

M defined by

0

i f x i Z P

o r p Z p ' ,

1

i f x c z a n d p = p ' . P

A

( x ~ P ' )= ZP

t

i s zero v-almost everywhere. Hence

I t follows t h a t A

ZP

set.

zP

i s a v -null P

0

W e a r r i v e a t t h e main r e s u l t of t h i s s e c t i o n .

( Pl,...,P

Let

an n-set of mutually s t r o n g l y commuting s e l f - a d j o i n t

) denote

x.

operators i n

Then following Section A . I V . l

x

standard s p l i t t i n g of

t h e r e e x i s t s a so-called

i n t o d i r e c t summands

such t h a t t h e n-set a c t s i n v a r i a n t l y i n each summand t h e r e e x i s t f i n i t e Borel measures p m , p l ,

Also,

x

from

xm,

... and a

m = m,1,2,

... .

unitary operator

u

onto t h e H i l b e r t space

such t h a t

e

and f o r each

= 1,.

U Pe u*

=

.., n qe

m @ L2(IRn,pm) denotes t h e maximal o p e r a t o r of multij=1 p l i c a t i o n by t h e f u n c t i o n ide i n each summand L 2 ( I R n , p m ) .

qe

where

restricted to

Let ID denote t h e denumerable s e t { (m, j ) let

M

= IRn

ID.

e

PCID

v

. P

Im

=

m,

1,2,.

.. ,

1 5 j < m

Moreover, l e t t h e f i n i t e Borel measures v

~ =, urn~ f o r m =

by v

v =

x

W e have

a,1,2,

...

and l e t 1 5 j < m + l .

mr j

+ 1)

and

be d e f i n e d

As b e f o r e

we set

THE GENERALIZED EIGENVALUE PROBLEM REVISITED

So

u

i s a u n i t a r y o p e r a t o r from

a djoi nt operator

u Pe u*

x

o n t o L,(M,v).

For

e

277

= 1 , . ..,n

the self-

i s t h e maximal o p e r a t o r of m u l t i p l i c a t i o n by t h e

coordinate f u n c t i o n ide i n each summand L 2 ( I R n , v ) , p E ID. I t means t h a t P we can apply Theorem IV.4.3 and t h e r e s u l t s preceding t h i s theorem. We f i x a nonnegative s e l f - a d j o i n t o p e r a t o r

A in

x

such t h a t

u e- t A

is a

Carleman o p e r a t o r f o r a l l t > 0. We assume t h a t f o r a l l t > 0 t h e inducing

-t A

s a t i s f i e s Condition IV.l.l.b, f u n c t i o n k : M +. X of t h e o p e r a t o r U e t 2 i . e . t h e Borel function x I+ I1 k ( x , p ) II x , x E mn, i s v - i n t e g r a b l e on t P bounded Borel s e t s of IRn f o r each p E ID. We observe t h a t any nonnegative self-adjoint operator

A, which generates a semigroup ( e- t A

of H i l b e r t -

Schmidt o p e r a t o r s , s a t i s f i e s t h e s e requirements.

IV.4.5. Let m =

e. m,

1.2,.

.. and

let j

E

IN with 1

5 j <

each r > 0 w e d e f i n e t h e f u n c t i o n Arne' on B(x.r)

m + 1. For each x

E

IRn and

M by

and

-

-

and t r a j e c t o r i e s E ( m ) i n TX,A such t h a t Then t h e r e e x i s t s a 11 - n u l l s e t N m m, j XI j x = ( x l x ) and a l l = 1 n , w e have f o r a l l x E supp(um)\a m,j'

,...,

Proof.

e

,...,

-

Following Theorem IV.4.3 t h e r e e x i s t t r a j e c t o r i e s E ( m ) and a pm-null XI j s e t N m(1) , j such t h a t f o r a l l x E supp(u ) \ N ( ' ) t h e a s s e r t i o n s a. -b. a r e m m,j

THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

278

-(m) # 0 f o r a l l x Ex, j proof i s complete.

E

0

Remark. For a l l r > 0, x for

L

= 1,

E

,... and

lRn, m = m,1,2

1

_< j

...,n .

Under t h e c o n d i t i o n s imposed on t o almost each x = (x

(P1,.

Nr),

-

such t h a t there exists a null s e t (1) ''(2) then the supp(um)\Nf) I f we t a k e Nm, = Nm,j u Nm, j , ,I'

s a t i s f i e d . Following Lemma IV.4.4

,X

,...,x 2

1 with m u l t i p l i c i t y m

..,Pn)

-(s) ,

eigenvectors E

1 s j < m

x, j

< m + l , d m r J E D(Pe) x,r

A from t h e previous theorem

it follows t h a t

i n t h e j o i n t spectrum of t h e n-set

)

t h e r e correspond m

+1 in

X

candidate generalized

TX,A.

W e want genuine generalized eigenvectors. Therefore, w e prove t h a t t h e r e

e x i s t s a nonnegative s e l f - a d j o i n t o p e r a t o r A which possesses t h e a d d i t i o n a l

Pe can be extended t o a c l o s e d o p e r a t o r i n T X,A'

property t h a t each o p e r a t o r

We p r e s e n t t h e following lemma.

IV.4.6.

e.

Let ( P l,...,P

in

x.

) denote an n-set

of s t r o n g l y commuting s e l f - a d j o i n t o p e r a t o r s

Then t h e r e e x i s t s a nonnegative s e l f - a d j o i n t o p e r a t o r A i n

each o p e r a t o r

Pe,

e

= 1,

...,n , extends

TX,A and f o r a l l t > 0 t h e o p e r a t o r e

Proof.

By Theorem A.IV.3.4

l

t h a t each o p e r a t o r P l ,

-t A

x -such

t o a closed l i n e a r mapping

that

Pe i n

i s Hilbert-Schmidt.

t h e r e e x i s t s an orthonormal b a s i s (v ) = 1,. . . , n

,

k kdIN such has a column f i n i t e (and hence r o w

f i n i t e ) i n f i n i t e matrix with r e s p e c t t o t h i s b a s i s . L e t ( h k ) k E m be a se-Xkt L' e < m f o r a l l t > 0. We k= 1 d e f i n e t h e p o s i t i v e s e l f - a d j o i n t o p e r a t o r A by

quence of p o s i t i v e real numbers such t h a t

Avk = Xkvk

,

k

E

M

,

with usual s e l f - a d j o i n t extension. L e t t > 0 and l e t JI

D(JI(A) Pee-tA)

E

B+(IR). Since each vk, k

= 1,. ..,n

,

JN, belongs t o

because of t h e column f i n i t e n e s s of t h e matrix o f

-t A

follows t h a t t h e a d j o i n t (e

e

E

is closable i n

X.

Next w e prove t h a t t h e c l o s u r e

P it 'LtA PIJI(A),

PeJI(A))* i s densely defined. Hence e

Pl

in T

X,A

i s defined by

THE GENERALIZED EIGENVALUE PROBLEM REVISITED

279

and

7e

F i r s t , we observe t h a t with f

E

i s well-defined.

D((e-tAPe$(A))**)

I t is clear t h a t

pe

and g

extends PL i n

F i n a l l y w e prove t h a t

pL

E

Indeed, l e t F = $ ( A ) f = x(A)g

D ( ( e - t A P L ~ ( A ) ) * * f) o r a l l t > 0 . s i n c e

x.

i s closed. Therefore, l e t ( F n ) n c m be a sequence

Dam(?e ) with l i m i t F. Suppose t h e sequence (pe Fn ) ncIN i s convergent i n and a TX,A with l i m i t G. Then by Theorem 1 . 2 . 4 t h e r e e x i s t s J, E B -(IR) I in

in convergent sequence ( f ) n nElN n E IN, and F = $ ( A ) f . We have

x

with l i m i t f such t h a t F

= $(A) f n ,

and ( I f n - fllX + 0

as

n

+. m

.

So f o r a l l t > 0

and

-tA

G ( t ) = (e

Pe$(A))**f =

(pe F ) ( t ) .

0

C o l l e c t i n g a l l r e s u l t s of t h i s s e c t i o n we a r r i v e a t t h e main theorem.

Theorem. L e t (Pl,...,P ) denote an n-set of s t r o n g l y commuting s e l f - a d j o i n t o p e r a t o r s n IV.4.7.

x = xm Q X I tB x, ... denote t h e standard s p l i t t i n g of x and l e t p m , p , u ,... denote t h e a s s o c i a t e d d i s j o i n t m u l t i p l i c i t y measures. Then t h e r e 1 2

in

x.

Let

Q

e x i s t s a nonnegative s e l f - a d j o i n t operator A which g e n e r a t e s a semigroup (e-tA)t,O

c o n s i s t i n g of Hilbert-Schmidt o p e r a t o r s , such t h a t each o p e r a t o r

280

THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES

e e

...,

P , = I, n , is closable as a linear mapping in T X,A with closure ? (m) x E @, m = ",1,2,. . and Also there exists a canonical Dirac basis {E x.j and for each m = m , 1,2,. a pm-null set Nm 1 S j < m +1} in T m X,A ( = j21 Nm, j) with the following property: For all x E supp(pm)\Nm,

-

I

-

..

m

=

m,

1,2,.

.. and all a. =

1,.

..,n , E(m) x,1

E

Dom(P

e)

.

e-

-

with

Some comments and bibliographical notes Most concepts in this chapter are improvements and generalizations of ideas which came up earlier in [E 11. For the construction of Dirac bases in nuclear trajectory spaces see LEG 41, LEG 51, LEG 91 and LEG 101. In this chapter Dirac bases are also constructed in non-nuclear trajectory spaces. The remarks at the end of Chapter A.IV on the solution of the generalized eigenvalue problem also apply here. We note that Lemma IV.2.3 says that aZZ SX,A-spaces have a Schauder basis. Therefore all SX,A spaces are homeomorphic to sequence spaces. However, these sequence spaces are nice only in the Monte1 case.

28 1

A MATHEMATICAL INTERPRETATION OF

DIRA C '5 FOR MA1ISM The emission and absorption of energy in fixed quanta (photons), the wave and corpuscular character of light, both are phenomena, which physicists encountered at the end of the last century and for which they could not find an explanation in classical Newtonian mechanics. These phenomena required a new physical theory. In the beginning of this century theoretical physicists started the development of this theory known as QLcuntwn mechanics. The description of quantum mechanics demanded new mathematical concepts. At that time these concepts were not available. Eventually, the prominent mathematician John von Neumann introduced the sound mathematical Hilbert space theory. In terms of this theory von Neumann approaches the fundamentals of quantum mechanics. Each mathematician will admit that Hilbert space theory plays a significant role in today's mathematics. But von Neumann's Hilbert space approach to quantum mechanics is too rigid. This mathematical theory lacks the flexibility which this physical theory demands. One may even wonder whether there indeed exists such a flexible mathematical theory. About ten years before von Neumann presented his monograph Foundations of

quantwri mechanics, the famous physicist Paul Dirac introduced a mathemaLica1 formalism, in which he describes the principles of quantum mechanics. Looked at from a distance, Dirac's (bracket) formalism resembles the theory of finite dimensional vector spaces. However, finite sums are replaced by infinite sums (series) or continuous sums (integrals), discrete bases are replaced by continuous bases, the Kronecker delta used to express the orthonormality of a discrete basis, is replaced by a delta function indicating a

kind of orthonormality relations for continuous bases. In fact, Dirac did not worry too much about the lack of necessary mathematical rigour in his formalism. He is aware of this shortcoming in his theory. In the preface to the first edition of his celebrated book The p i n -

ciples o f quantum mechanics he observes (and here we quote). Mathematics i s the tool specially suited for dealing with abstract concepts o f any kind and there i s no l i m i t i n i t s power i n t h i s f i e l d . For t h i s reason a book on the new physics, i f not purely descriptive of experimental work, must be essentialZy mathematical. A l l the same the mathematics i s only a tool and one should learn t o hold the physical ideas i n one's mind without reference t o the mathematical form. A

bit further Dirac explains why he is in favour of the so-called symbolic

method to be the mathematical form in which the theory can be presented.

The symbolic method, however, seems t o go more deeply i n t o the nature of things. I t enables one t o express the physical laws i n a neat and concise way, and will probably be increasingzy used i n the future as i t becomes better understood and i t s own speciaZ mathematics gets developed. Indeed, during the past fifty years Dirac's formalism has become the common language of the theoretical physicists. Besides, several mathematicians have searched for a mathematical rigorification of Dirac's formalism. This formalism has been the source of inspiration in the development of distribution theory and no doubt distribution theory partially justifies Dirac's nonmathematical techniques. It partially solves the generalized eigenvalued problem and rigorizes certain formal computations with delta functions. Of course, one may be satisfied with a partial mathematical justification, arguing that it is impossible to develop a mathematical apparatus in which all features of the Dirac formalism fit. But the ideas provoked by Dirac are so appealing to mathematical intuition that it is worthwhile to seek for a mathematical justification which is as complete as possible. In the final part of our monograph we present a mathematical justification of the genuine Dirac formalism in terms of our theory of analyticity/trajectory spaces. We introduce the ket and the bra space both as trajectory

283

spaces. Each k e t i s i n one-to-one correspondence with a bra. So i n accordance with D i r a c ' s formalism w e do n o t d i s c r i m i n a t e between b r a s and k e t s . Dirac assumes t h a t t h e r e e x i s t s a p a i r i n g between t h e b r a space and t h e k e t space by introducing t h e b r a c k e t of a b r a and a k e t . Already, from D i r a c ' s monograph it becomes c l e a r t h a t t h i s b r a c k e t does not y i e l d a complex num-

e i n t e r p r e t e D i r a c ' s b r a c k e t notion by a d j o i n i n g t o each b e r , i n general. W couple c o n s i s t i n g of a b r a and a k e t , a complex valued f u n c t i o n on t h e open i n t e r v a l (0,m).

So i n our i n t e r p r e t a t i o n t h e b r a c k e t is not a number b u t a

f u n c t i o n . This new bracket h a s p r o p e r t i e s which a r e very s i m i l a r t o t h e p r o p e r t i e s of t h e ordinary i n n e r product of H i l b e r t space theory. Also,

i n our i n t e r p r e t a t i o n a c e n t r a l r o l e i s played by our new mathematical

notion of Dirac b a s i s . The mentioned concepts of Dirac b a s i s and of b r a c k e t t o g e t h e r c o n s t i t u t e t h e p i l l a r s on which we b u i l d a mathematical apparatus

that t o a g r e a t e x t e n t founds t h e bold claims on which Dirac has based h i s p r i n c i p l e s of quantum mechanics.

This Page Intentionally Left Blank

285

CHAPTER I

D I R A C ' S FORMALISM ACCORDING T O D I R A C AND ITS R E L A T I O N S

WITH L I N E A R ALGEBRA

This c h a p t e r c o n t a i n s a d e s c r i p t i o n of D i r a c ' s formalism according t o D i r a c ' s own o r i g i n a l i n t r o d u c t i o n . From our p o i n t of view Dirac claims t h a t s e v e r a l concepts of t h e theory of f i n i t e dimensional v e c t o r spaces remain v a l i d i n a c e r t a i n u n s p e c i f i e d type of i n f i n i t e dimensional v e c t o r s p a c e s , r e p l a c i n g sums by i n t e g r a l s , d i s c r e t e bases by continuum b a s e s , Kronecker d e l t a s by D i r a c d e l t a s , e t c . W e approach D i r a c ' s formalism having t h e s e rel a t i o n s with l i n e a r algebra i n mind. So it i s n a t u r a l t o summarize c e r t a i n well-known concepts of l i n e a r algebra p r e s e n t i n g them i n b r a c k e t n o t a t i o n .

1.1. Let

Some elementary concepts of l i n e a r a l g e b r a

v

isomorphism which sends

v. la,

v

denote a complex f i n i t e dimensional v e c t o r space. I t means t h a t

a complex v e c t o r space f o r which t h e r e e x i s t s a number d

v

where t h e l a b e l ( s ) a ( o r a l ,

...,a

v

a r e denoted by la> ( o r

) can be chosen s o t h a t c e r t 6

t a i n s p e c i a l p r o p e r t i e s of t h e considered v e c t o r a r e i n d i c a t e d . I f a and la> and Ib> belong t o

is

I N and a l i n e a r

o n t o Cd. The number d i s c a l l e d t h e dimension of

In the sequel we t a k e d f i x e d . The elements of

... a,>)

E

v,

then a l s o a l a > and la>

+

Ib> belong t o

v.

We

observe t h a t , i n g e n e r a l , it does not make sense t o w r i t e laa> o r la+ b > i n s t e a d of ala> o r la>

+

Ib> and even when it does, t h e v e c t o r s laa> and

l a + b > may n o t equal a l a > o r la> The a l g e b r a i c dual

v*

of V is t h e complex v e c t o r space, which c o n s i s t s of

a l l l i n e a r f u n c t i o n a l s on elements of

v*

+ Ib>.

v,

i.e.

are denoted by <mi

a l l l i n e a r mappings from ( o r <m l

... m

v

i n t o C . The

1 ) where t h e same remark

can be made on t h e choice of t h e l a b e l ( s ) . For each <ml

E

V* and la>

c

V

w e introduce t h e i r b r a c k e t <m I a > which i n d i c a t e s t h e complex number d <ml ( l a > ) . I f we f i x a l i n e a r isomorphism rp : v + d: , t h e Euclidean inner product

of Cd induces an i n n e r product i n

( * , a

)Cd

v

a s follows.

DIRAC'S FORMALISM

2 86

(Our inner products are linear in the first factor and anti-linear in the second.) So

v

is an inner product space. Let Ic>

induces the linear functional
:

la>

I+

(la>

v*

E

E

v.

Then the vector (c>

defined by

, Ic>lv ,

la>

E

V

.

It is a fundamental result of linear algebra that each element of

v*

arises

in the above described way. Thus there exists a one-to-one correspondence Ic> ++
v

and the elements of

spondence is anti-linear. So for Ic>,ld> ylc>

E

v

and y , 6

+ &Id> corresponds to the linear functional

v*.

This corre-

C the vector

E

7
the

following relations can easily be checked.

.

1.l.l.b.

VIakv



h 0

1.l.l.c.

Vla>cv :

= 0

:

-

la> = 0

Next, we consider linear operators on Let L

:

.

v.

v v be a linear mapping. For each couple /a>,Ib>EV we +

take the

bracket of Lla> and Ib>. This bracket depends linearly on la>. So we can represent it by for certain Ib'>E

V ,

i.e. for all la>

Now instead of have ) =
(Lla>)

E

v

and Ib>

V

E

E

v we

.

Therefore it makes sense to write being the product (in the right order) of . Besides the scalar product (la>

, Ib>)

=
I a > of the vectors la> and

lb> we can consider their sesquilinear tensor product
8

la> (not to be

287

SOME ELEMENTARY CONCEPTS OF LINEAR ALGEBRA

confused with t h e b i l i n e a r t e n s o r product Ib>

w e w r i t e la>
V

The t e n s o r product la>
into

V in

@



la>). I n s t e a d of

can be regarded as a l i n e a r

the following way

So t h e v e c t o r la> has t o be m u l t i p i i e d by t h e complex number . The t e n s o r product also a c t s from t h e l e f t on t h e elements o f v*

So t h e l i n e a r f u n c t i o n a l

L

:

v

+

v


v. v,

be a l i n e a r o p e r a t o r . Let Ib> E

t h e r e corresponds a unique element Ic> of

la> E

v.

The correspondence Ib>

+h

Ic>,

e x i s t s a l i n e a r o p e r a t o r K such t h a t Ic> t h e a d j o i n t of K and i s denoted by


Let

such t h a t


la>

=


Ia>,

induced by L , i s l i n e a r . So t h e r e = Klb>. T h i s o p e r a t o r

L*. Hence f o r each la>

f u n c t i o n a l corresponding t o f.*la> equals

I a>.

To t h e l i n e a r f u n c t i o n a l
E

v

K is c a l l e d the linear

< a l l . The following r e l a t i o n i s

obvious.

P u t t i n g L* i n s t e a d of L i n 1.1.4 we o b t a i n

Hence

L

**

= L

; t h e a d j o i n t of t h e a d j o i n t of a

l i n e a r o p e r a t o r on

o r i g i n a l l i n e a r o p e r a t o r . An o p e r a t o r L f o r which L = L * ,

v

is the

is called self-

a d j o i n t . W e mention t h e following property

Let

L

:

v

+

v

be a l i n e a r o p e r a t o r . Then f o r x E C we c o n s i d e r t h e eigen-

value problem

in

v.

I f t h e v e c t o r la>

E

v

i s a s o l u t i o n of t h i s e q u a t i o n , then la> i s

288

DIRAC'S FORMALISM

L

c a l l e d an eigenvector of operators

with corresponding eigenvalue x. For s e l f - a d j o i n t

P t h e eigenvalue problem induced by Equation

(*),

i s completely

solvable. W e p r e s e n t i t s s o l u t i o n here. Therefore w e introduce t h e concept

v.

of orthonormal b a s i s i n

v

In

a s e t of d v e c t o r s {la1> , . . . , l a d >) c o n s t i t u t e s an orthonormal b a s i s i f

f o r a l l 1 L k,e 5 d


k

l a

e

>

= 6

I n t h i s c a s e , each la>

E

ke

v

*

d

is represented by t h e sum

kEl

\ak > < a k 1 a > and,

correspondingly, each l i n e a r f u n c t i o n a l
.

d

I

1.1.6.

=

1

I%>
.

k= 1

we

observe t h a t t h e s e t {
Ik = 1,

...,d } i s

a b a s i s f o r v*. This b a s i s

i s orthonormal with r e s p e c t t o t h e i n n e r product i n v* d e f i n e d by

,
= . V* Now t h e s o l u t i o n of t h e eigenvalue problem f o r a s e l f - a d j o i n t o p e r a t o r (
v

on

can be formulated as follows.

P

-

The eigenvalues of

-

Two eigenvectors of P , which belong t o d i f f e r e n t eigenvalues, a r e ortho-

a r e r e a l numkers.

gonal.

-

There e x i s t s an orthonormal b a s i s i n of

which c o n s i s t s of eigenvectors

P.

For a s e l f - a d j o i n t o p e r a t o r and

v

vl, ...,v

P on

v,

let xlr...,x

denote i t s eigenvalues

t h e corresponding eigenspaces, i . e .

,...,r . Then t h e s e eigenspaces a r e mutually orthogonal and we can w r i t e

v

vl

v2

... vr.

v.

The dimension m . of is c a l l e d t h e m u l t i p l i c i t y 1 1 of t h e eigenvalue x . . Let t h e s e t Ix., k > k = 1 , . . , m . ) be an orthonormal 1 7 1 basis i n Then the s e t {lx,, k > j = 1 , . , r , k = 1,. , m . 1 i s an ortho3 1 1 =

@J

@

@

I

v..

I

..

.

..

SOME ELEMENTARY CONCEPTS OF LINEAR ALGEBRA

normal basis in

v.

289

The following relation is valid

or, equivalently, in terms of operators

v,

If two self-adjoint operators

P

P

and conversely. It follows that in

is also an eigenvector of

R

and R commute on

then an eigenvector of

v

there

exists an orthonormal basis of simultaneous eigenvectors of the couple of commuting operator (P,R). Now we consider tuples of eigenvalues (x,y) with x,y

E

IR for which the equations

are simultaneously solvable for certain la> c

V.

A tuple (x,y) c IRz for

which this equation is solvable is called an eigentuple of the pair ( P , R ) . Replacing the notion of eigenvalue for one self-adjoint operator by the notion of eigentuple for two commuting self-adjoint operators, we arrive at the following result. Let (P,R) denote a couple of two commuting self-adjoint operators on V, let

..., (xr,y

(xl,yl),

...,V

denote its eigentuples andVl,

)

the corresponding

eigenspaces, i.e.

v.1 =

{la>

E

V I

(Pla> = x.la>) 1

A

(Ria> = y.la>) 1 3

.

v..

The multiplicity m. of the eigentuple (x y , ) is defined by m . = dim 3 j' 1 1 1 Now there exists an orthonormal basis { Ix,, y . ,k> I 1 5 j 5 r, 1 5 k 5 rn, 1 in 1 1 1 V with the following properties m.

P

1.1.8. 2nd

=

f

j=1 k=l

x. Ix.,y.,k><x y ,kl j' j

DIFlAC'S FORMALISM

290

We can continue the above described process by considering a triple

(P1, P2,P3)

of mutually commuting self-adjoint operators, a quadruple, etc. 1 2 3 3 1 2 3 4 4 Correspondingly, there are eigentuples (x ,x ,x ) E IR , (x ,x ,x ,x 1 E IR , etc. However, starting with one self-adjoint operator many, say

P, ,...,Pn ,

such that the n-tuple (Pl,...,P

P, we )

can add finitely

consists of mutually

commuting self-adjoint operators with eigentuples (x1:x2,.

..,xn) each of

multiplicity one. Such an n-tuple is called complete. So for a complete n-

. .,P )

tuple (P1,. =

1,

...,d}

there exists an orthonormal basis

such that for k # 1 1 (5

I . . . ,

and, also, for i

Pk

=

xi) # (x; l,.. . , n

1

d

=

j=l

x. k Ixl J

1x1 I

... xn> Ij = 3

,...,x"e)

,

... X?><Xj 1 ... xnl . I

j

3

As we have seen, a self-adjoint operator sentation of the vectors of

v

P on v

induces a particular repre-

related to an orthonormal basis of eigenvec-

P induces also a particular representation of the linear operators from V into v. We show this in the special case that P is a complete 1tuple. So we assume that the eigenvalues of P have multiplicity 1. Then

tors.

...,xd and, correspondingly, an orthonormal

there exist real numbers xl,

..

basis { Ix.> I j = 1,. ,d} such that Pix.> = x .Ix.>. For a linear operator I I I 3 L : v + v the complex numbers <xkl L I x >, 1 5 k,e 5 d, determine L com-

e

pletely. We have

or in terms of operators d

DIRAC'S FORMALISM ACCORDING TO DIRAC

The numbers <\I

e> can be

L lx

<x21 LlXl>

291

stored in a matrix

...

<x21 LIX2>

<x21

LIXd>

Thus we arrive at the matrix representation of L induced by the self-adjoint operator P. We can develop the usual matrix calculus. The matrix of the adjoint L* is given by the Hermitean conjugate of the matrix of L , and the matrix of the composed linear mapping of the matrices of L

and L,.

L l L2 0

is given by the matrix product

In particular, the matrix of the self-adjoint

1 operator P equals the diagonal matrix

So far our summary of some concepts of linear algebra. The order in which

they are presented is the same one in which Dirac presents the various concepts of his formalism.

1.2. Dirac's formalism according to Dirac It is hardly possible to improve Dirac's own description of his own formalism. But we do not want to include the nearly seventy pages of Dirac's monograph which contain this description. Therefore we mix our presentation of Dirac's formalism according to Dirac with citations taken from Dirac's book. These citations are in italics. Although the physics in the formalism is not completely neglected, the main emphasis is on its mathematical connotations.

D I R A C ' S FORMALISM

292

A dynamical quantum mechanical system c o n s i s t s o f s t a t e s and dynamical v a r i -

a b l e s . A s we w i l l s e e t h e states and dynamical v a r i a b l e s are r e p r e s e n t e d by mathematical q u a n t i t i e s of a n a t u r e d i f f e r e n t from t h o s e used i n c l a s s i c a l Newtonian mechanics. The s t a t e s admit a s u p e r p o s i t i o n p r i n c i p l e , which i s a kind of a d d i t i v e p r o c e s s such t h a t states can i n some way be added t o g i v e new s t a t e s . I t follows t h a t t h e mathematical q u a n t i t i e s , which r e p r e s e n t t h e s t a t e s , can be added t o g e t h e r t o g i v e mathematical q u a n t i t i e s o f t h e same nature. Therefore i t i s n a t u r a l t o r e p r e s e n t t h e s t a t e s by t h e elements of

a v e c t o r space. I n g e n e r a l a f i n i t e dimensional v e c t o r space i s n o t s u f f i c i e n t l y l a r g e and a l s o i n f i n i t e dimensional v e c t o r spaces have t o be cons i d e r e d . The v e c t o r s which d e s c r i b e t h e s t a t e s of a quantum mechanical sys-

t e m are c a l l e d k e t Vectors or simply k e t s . They are denoted by I*>, where we can i n s e r t a l a b e l A i n t h e middle, t h u s IA>,

which s p e c i f i e s a p a r t i c u l a r

one of them. Being members of a v e c t o r space k e t v e c t o r s can be added and m u l t i p l i e d by a complex s c a l a r , i . e .

where IR> denotes t h e r e s u l t i n g k e t . A l s o more g e n e r a l p r o c e s s e s a r e allowed such as t h e a d d i t i o n of an i n f i n i t e sequence of k e t s o r t h e i n t e g r a t i o n of k e t v e c t o r s Ix> which a r e l a b e l e d by a parameter x with values i n a continuous range. The l a t t e r process r e s u l t s i n a k e t IS>

A set

of k e t v e c t o r s i s c a l l e d independent i f no one of them i s e x p r e s s i b l e

l i n e a r l y as a f i n i t e sum, a series, or an i n t e g r a l of t h e o t h e r s . Otherwise t h e s e t i s c a l l e d dependent.

We now assume t h a t each s t a t e of a dynamical system a t a p a r t i c u l a r time corresponds t o a k e t vector, the correspondence being such t h a t i f a s t a t e r e s u l t s from the superposition of c e r t a i n other s t a t e s , i t s corresponding k e t vector i s expressible l i n e a r l y i n terms of t h e corresponding k e t vect o r s of t h e other s t a t e s , and conversely. Because of t h i s assumption it makes sense t o speak of dependent and independent states. A s e t o f s t a t e s

i s c a l l e d (inldependent i f t h e corresponding s e t o f k e t v e c t o r s i s ( i n ) d e pendent.

DIRAC'S FORMALISM ACCORDING TO DIRAC

293

I f he superpose a s t a t e with i t s e l f we do n o t o b t a i n a new s t a t e . Let u s consider t h e k e t vector I A > and t h e complex numbers c1 and c 2 . I t i s c l e a r that c IA> 1

+

Now we may have c1

c21A> = ( c + c ) I A > 1 2

+

.

c 2 = 0 i n which case t h e corresponding s u p e r p o s i t i o n

of states means an i n t e r f e r e n c e by which t h e two components have c a n c e l l e d each o t h e r . I f t h i s i s n o t t h e case, then t h e k e t ( c + c ) I A > i n d i c a t e s t h e 1 2 same s t a t e a s t h e k e t IA> does. We conclude t h a t i f the k e t vector corre-

sponding t o a s t a t e i s m u l t i p l i e d by any complex nwnber, not zero, t h e r e s u l t i n g k e t vector w i l l correspond t o t h e same s t a t e . ( W e n o t e t h a t D i r a c ' s formalism would be more c o n s i s t e n t i f s c a l a r m u l t i p l i c a t i o n of k e t s were written as a r i g h t multiplication.)

Having introduced t h e vector space of k e t s w e can c o n s i d e r t h e space o f a l l l i n e a r f u n c t i o n a l s on t h i s k e t space, i . e .

i t s a l g e b r a i c d u a l . This space

is also a v e c t o r space and t h u s we have introduced a new space of v e c t o r s .

W e s h a l l c a l l the net) vectors bra vectors, o r s i m p l y bras, and denote a

general one of them by the symbol

< I,

t h e mirror image of the symbol f o r

a k e t vector. I f we want t o s p e c i f y a p a r t i c u l a r one o f them by a label, L

say, we w r i t e i t i n the middle, thus < L I . Next we i n t r o d u c e a s c a l a r

product, t h e b r a c k e t , of a b r a v e c t o r and a k e t vector. The b r a c k e t o f a bra


and a k e t IA> w i l l be denoted by < L

I A > where


I A > denotes

t h e complex number

i.e. t h e l i n e a r f u n c t i o n a l

.

The b r a c k e t expression

depends l i n e a r l y on t h e k e t s and t h e b r a s . T h i s i s expressed by t h e following relations

DIRAC'S FORMALISM

294

{
+



{ A

= X
=



I A>

+



.

It is clear that a bra vector is fixed by the values of its brackets with all kets. Expressed symbolically, 1.2.2.

If = 0 for all IA>, then

.

The bra vectors as they have been here introduced are quite a different kind of vectors from the k e t s and so f a r there i s no connection between them except f o r the existence of a scalar product of a bra and a ket. We n m make the assumption that there i s a one-to-one correspondence between the bras i s the sum of the and kets such that the bra corresponding t o IA> + IA'> bras corresponding t o (A> and IA'>, and the bra corresponding t o c (A> i s being the conjugate complex number t o times the bra corresponding t o IA>, C. We shall use the same label to specify a ket and the corresponding bra. Thus the bra corresponding t o IA> w i l l be written
c

c

symmetrical between bras and kets. For the kets IA> and IB> we can consider the bracket by taking the scalar product of the ket ]A> and the bra . The mapping (IA>, IB>) t+ is linear in IA> and antilinear in lB>. For !A> and IB> we can also consider the bracket
I B>. It is assumed that

the following relation is valid for all IA> and all IB>. 1.2.3.



= < B ~ A > .

Taking IB> = IA> it follows that is real. It is also assumed that for all kets IA> 1.2.4.



> o

except when IA> = 0. In terms of the bracket for kets and bras we can introduce a notion of orthogonality and of length. Two kets [A> and lB> are called orthogonal if the bracket is zero. Correspondingly, two states of the dynamical system are called orthogonal if their representing kets are orthogonal. The

295

DIRAC'S FORMALISM ACCORDING TO DIRAC

l e n g t h of a k e t IA> i s defined a s t h e square r o o t of t h e p o s i t i v e number
IA>.

As we have seen a s t a t e i s a l r e a d y determined by t h e d i r e c t i o n of

a ket, i.e.

a s t a t e i s f i x e d upto a numerical f a c t o r . W e can use t h i s f r e e -

dom by choosing t h i s numerical f a c t o r so t h a t t h e k e t h a s l e n g t h one. Then t h e v e c t o r i s s a i d t o be normalized. Even a normalized k e t i s not uniquely

e can always multiply by a complex number of modulus determined by a s t a t e . W 1 without changing t h e l e n g t h . Such a number eiy, y r e a l , i s c a l l e d a phase

factor.

I Linear o p e r a t o r s

We consider l i n e a r o p e r a t o r s on the k e t space. Let

L

denote a l i n e a r o p e r a t o r which maps t h e l i n e a r space of k e t v e c t o r s

i n t o i t s e l f . Then we can w r i t e t h e a c t i o n of

so LIA>.

L on

any k e t l A > as a product,

In t h i s product w e always put t h e k e t on t h e r i g h t of t h e l i n e a r

operator. The l i n e a r i t y of t h e o p e r a t o r

L

i s expressed by t h e following re-

lations.

1.2.5.

Let

L

1.2.6.

and

K denote two l i n e a r o p e r a t o r s . Then we d e f i n e t h e o p e r a t o r L + K by

(L+KHA>

= LIA>

+ KIA>

f o r any ( A >

.

Thus two l i n e a r o p e r a t o r s can be added. Two l i n e a r o p e r a t o r s can a l s o be m u l t i p l i e d . The product of t h e o p e r a t o r s

L

and K i s d e f i n e d by

In t h e sequel w e simply w r i t e LKIA> i n s t e a d of { L K ) I A > .

We observe t h a t LK

does not equal KL, i n general. So t h e commutative axiom o f m u l t i p l i c a t i o n does not hold f o r l i n e a r o p e r a t o r s . As w e s h a l l see, t h i s f a c t e x p l a i n s one of t h e e s s e n t i a l d i f f e r e n c e s between c l a s s i c a l Newtonian mechanics and quantum mechanics. To be more s p e c i f i c , t h e u n c e r t a i n t y p r i n c i p l e does n o t appear i n c l a s s i c a l mechanics.

296

DIRAC'S FORMALISM

Besides t h e product LIA> of a l i n e a r o p e r a t o r s i d e r a l s o t h e product of a b r a


L

and a k e t IA>,

w e can con-

L.

Therefore w e

and a l i n e a r o p e r a t o r

study t h e expression

i . e . the bra


a c t i n g on t h e k e t LIA>.

This expression is l i n e a r i n I A >

and y i e l d s a complex number. So f o r f i x e d
t h e mapping IA>

i s a l i n e a r f u n c t i o n a l . Hence t h e r e e x i s t s a b r a

The mapping
H
consider t h e expression


i s l i n e a r . Now i n s t e a d o f


H

)

such t h a t

w e w r i t e
We

< B I L a s t h e l e f t action of t h e l i n e a r operator L

on t h e b r a space. Thus a l i n e a r o p e r a t o r

L

which acts from t h e r i g h t on t h e

k e t space, a l s o a c t s from t h e l e f t on t h e b r a s p a c e , and conversely. Theref o r e , i t makes sense t o w r i t e

because no confusion w i l l a r i s e . There is a l s o another product which has a meaning i n t h e p r e s e n t scheme, namely t h e product of a k e t and a b r a with t h e k e t on t h e l e f t and t h e b r a on t h e r i g h t , such a s I A > < B l .

The product IA>
denotes t h e l i n e a r oper-

a t o r which a c t s on t h e k e t space a s

1.2.8.

and on t h e b r a space

1.2.8'

We now have a complete algebraic scheme involving three kinds of q u a n t i t i e s , bra vectors, k e t vectors and linear operators. They can be m u l t i p l i e d together i n the various ways discussed above and t h e a s s o c i a t i v e and d i s t r i b u t i v e axioms o f m u l t i p l i c a t i o n always hold, but the commutative axiom o f m u l t i p l i c a t i o n does not hold.

(...)

with regard t o the physical s i g n i f i c a n c e

DIRAC'S FORMALISM ACCORDING TO DIRAC

297

of the scheme, we have already assumed t h a t the bra v e c t o r s and k e t vectors,

o r rather the d i r e c t i o n s of these vectors, correspond t o the s t a t e s of a dynamical system a t a p a r t i c u l a r time. We now make the Jurther assumption t h a t the linear operators correspond t o the dynamical variables a t t h a t time. By dynamical variables are meant q u a n t i t i e s such as the coordinates and

the components o f velocity, momentum and angular momentum of p a r t i c l e s .

.. )

(.

The new assumption requires t h a t these q u a n t i t i e s s h a l l occur a l s o i n quantum mechanics, but with the s t r i k i n g d i f f e r e n c e t h a t they are now subj e c t t o an algebra i n which the commutative axiom o f m u l t i p l i c a t i o n does not hold. In general a linear operator corresponds to a complex dynamical variable. However, we are interested only in real dynamical variables. As we shall see the real dynamical variables correspond to a special type of linear operators. Therefore, we develop the notion of adjoint of a linear operator. Let

L

ket corresponding to the bra < A I L .

The mapping IA>

there exists a linear operator

1 such

there exists a linear operator

T

1.2.9.


i IA>

=

Since
f IA>

=

linear operators

L,

A linear operator

L.

H IA'>

that / A ' > = i l A > .

is linear. So

Put differently,

such that

LIB>


for all kets I A > and IB>.

-

let I A ' > denote the

denote a linear operator. Then for each bra < A \ ,

The linear operator


i IB>

and

L,


=

we have

L

IA>,

L,L,

=

5

is called the adjoint of

we have

L

-

=

i. Similarly, for

L. two

i2T1.

L is called self-adjoint or real, if L equals its adjoint

Multiplication by a complex scalar is a very special linear mapping. If

the linear operator a1 is defined by a l ( A > = alA> with a a complex number, then its adjoint

So if a is real, then a1 is a real linear

equals :I.

operator. Therefore it is reasonable to assume that each real linear operator corresponds to a real dynamical variable. We end this section with a computation of the adjoint of the linear operator I B > < A l . we have

IB>
I A > = IB>
and hence

IB>
=

IA>
.

R>

For each ket IR>

298

DIRAC'S FORMALISM

I f we t a k e

IA>

= IB> i n 1.2.10 it follows t h a t I A > < A I

is a r e a l l i n e a r

operator.

I

Eigenvalues and eigenvectors

I

Consider t h e following equation i n t h e k e t space

LIE>

1.2.11.

=

xlE>

where f. i s a l i n e a r o p e r a t o r a n d x a complex number. Such an equation u s u a l l y appears f o r a known o p e r a t o r L and unknown x and IE>.

W e ignore t h e t r i v i a l

s o l u t i o n IE> = 0. In 1.2.11 only t h e d i r e c t i o n of IE> i s important. So i f IE> s a t i s f i e s 1.2.11,

then a l s o clE> s a t i s f i e s t h i s equation f o r each com-

plex number c. Conjugate t o Equation 1.2.11 i s t h e equation i n t h e b r a space given by

where y i s a complex number. I f Equation 1.2.11 i s s a t i s f i e d by t h e p a i r ( x , I E > ) ,

then x is c a l l e d an

eigenvalue of t h e l i n e a r o p e r a t o r L and IE> i s c a l l e d an e i g e n k e t belonging t o t h e eigenvalue x. S i m i l a r l y , i f Equation 1.2.12 i s s a t i s f i e d by t h e p a i r (y

,
),

then y i s c a l l e d an eigenvalue and
bra.

Our main i n t e r e s t i s i n r e a l dynamical v a r i a b l e s . For a r e a l l i n e a r o p e r a t o r

P

t h r e e important r e s u l t s can be simply v e r i f i e d .

-

The eigenvalues of

-

The eigenvalues a s s o c i a t e d with e i g e n k e t s of

P a r e a l l r e a l numbers.

P

a r e t h e same as t h e eigen-

values a s s o c i a t e d with i t s eigenbras.

-

The b r a corresponding t o an eigenket of

P

i s a n eigenbra belonging t o

t h e same eigenvalue and conversely.

I t makes sense t o c a l l the s t a t e Corresponding t o an eigenket o f a r e a l

line ar operator by

P.

P an e i g e n s t a t e of t h e r e a l dynamical v a r i a b l e represented

Eigenvalues and e i g e n s t a t e s of a r e a l dynamical v a r i a b l e p l a y an

DIRAC'S FORMALISM ACCORDING TO DIRAC

299

e s s e n t i a l r o l e i n quantum mechanics. F i n a l l y we mention t h e following i m p o r t a n t property of a r e a l dynamical v a r i a b l e .

-

Two e i g e n s t a t e s of a r e a l dynamical v a r i a b l e belonging t o d i f f e r e n t e i g e n s t a t e s a r e orthogonal. (The proof i s standard. Let 1E l i n e a r operator

P with PIE

(E - E ) < E l I E2> 1 2

1

>=

1

> and IE2 > be two e i g e n k e t s of t h e E IE > and PIE2> = E21E2>. Then 1 1

real

= 0.)

a Observables

We have made a number of asswnptions about t h e way i n which s t a t e s and dynamical variabZes are t o be represented mathematically i n t h e theory. These assumptions are not, by themselves, laws o f nature, but become laws o f nat u r e when we make some f u r t h e r assumptions t h a t provide a physical i n t e r p r e t a t i o n of the theory. Such f u r t h e r asswnptions must take t h e form of e s t a b l i s h i n g connections between the r e s u l t s of observations, on one hand, and the equations of the mathematical formalism, on t h e o t h e r . An observation

i s a measurement of some dynamical v a r i a b l e . P h y s i c a l l y , such

a measurement must always y i e l d a r e a l number. So w e have t o r e s t r i c t our a t t e n t i o n t o r e a l dynamical v a r i a b l e s . However, it t u r n s o u t t h a t n o t every r e a l dynamical v a r i a b l e can be measured. The assumptions which a r e made for t h e p h y s i c a l i n t e r p r e t a t i o n of t h e theory l e a d t o f u r t h e r r e s t r i c t i o n s . For completeness we l i s t t h e s e p o s t u l a t e s here.

1. If t h e dynamical system i s i n an e i g e n s t a t e of a r e a l dynamical v a r i a b l e

P

belonging t o an eigenvalue p , then a measurement of

P will

certainly

g i v e a s r e s u l t t h e number p .

2.

If t h e system i s i n a s t a t e such t h a t a measurement of a r e a l dynamical

variable

P i s c e r t a i n t o g i v e one p a r t i c u l a r r e s u l t ( i n s t e a d of g i v i n g

one o r o t h e r of s e v e r a l p o s s i b l e r e s u l t s according t o a p r o b a b i l i t y law

as i s i n g e n e r a l t h e c a s e ) , then t h e s t a t e is i n an e i g e n s t a t e of P and t h e r e s u l t of t h e measurement i s t h e eigenvalue of

P t o which t h i s eigen-

s t a t e belongs.

3 . Any r e s u l t of a measurement of a r e a l dynamical v a r i a b l e i s one of i t s eigenvalues. Conversely, every eigenvalue i s a p o s s i b l e r e s u l t of a measurement of t h e dynamical v a r i a b l e f o r some s t a t e o f t h e system.

300

DIRAC'S FORMALISM

The s e t of eigenvalues of a r e a l dynamical v a r i a b l e c o n t a i n s j u s t t h e poss i b l e r e s u l t s of measurements of t h a t dynamical v a r i a b l e . Now t h e l a s t assumption i s the following. 4. I f a c e r t a i n r e a l dynamical v a r i a b l e

P i s measured with t h e system i n a

p a r t i c u l a r s t a t e , t h e s t a t e s i n t o t h e system may jump on account of t h e measurement a r e such t h a t t h e o r i g i n a l s t a t e i s dependent on them. Since a r e s u l t of a measurement of a real dynamical v a r i a b l e P i s an eigenvalue of

P,

i t follows from p o i n t 4 t h a t any s t a t e i s dependent on t h e

e i g e n s t a t e s of P . Therefore, we d e f i n e a complete s e t of s t a t e s t o be a s e t of s t a t e s such t h a t any s t a t e i s dependent on t h e s t a t e s i n t h i s set. We conclude t h a t t h e e i g e n s t a t e s of

P

should form a complete s e t . Thus t h e

notion of observable comes up. We c a l l a r e a l dynamical v a r i a b l e an observa b l e i f i t s e i g e n s t a t e s form a complete s e t . Any q u a n t i t y t h a t can be measured i s an observable. How does t h e condition look l i k e ? L e t

P

be a r e a l

l i n e a r o p e r a t o r . Then i t s eigenvalues may c o n s i s t of ( f i n i t e o r i n f i n i t e )

d i s c r e t e s e t s o f numbers, or a l t e r n a t i v e l y , they may c o n s i s t o f a l l numbers i n a c e r t a i n range, o r both t o g e t h e r . Let Ix,d> denote t h e e i g e n k e t s corresponding t o the d i s c r e t e s e t of eigenvalues and l e t Ix,c> denote t h e e i g e n k e t s corresponding t o t h e range of eigenvalues. The c o n d i t i o n f o r t h e eigenkets of

P

t o form a complete s e t

can be formulated as follows: For a l l k e t s IS> t h e r e e x i s t lx,d> and Ix,c> with IS> =

I

Ix,c> dx

+

1 Ix,d>

X

where t h e i n t e g r a l i s taken over t h e whole range of eigenvalues.

I t i s o f t e n very d i f f i c u l t t o decide mathematically, whether a p a r t i c u l a r real dynamical variable s a t i s f i e s the condition f o r being an observable or not, because the whole problem of f i n d i n g eigenvalues and eigenvectors i s i n general very d i f f i c u l t . However, we may have good reasons on experimental. grounds f o r b e l i e v i n g t h a t the dynamical variable can be measured and then we may reasonably assume t h a t i t i s an observable even though the mathemati c a l proof i s missing. In a n a t u r a l way Dirac concludes t h a t t h e eigenkets belonging t o t h e eigenvalues i n a range cannot have f i n i t e length. I f t h e two k e t s IA1> and I A Z > a r e defined by

DIRAC'S FORMALISM ACCORDING TO DIRAC

IA 1

>

=

!

Then f o r t h e bracket


IA2

I x , l , c > dx


I A2>

I

A

=

2

>

>=

1

ly,2,c> dy

30 1

.

we o b t a i n

<x,l,c

I y , 2 , c > dxdy

.

Consider now t h e s i n g l e i n t e g r a l

i

<x,l,c

I

y , 2 , c > dy

By t h e o r t h o g o n a l i t y < x , l , c <x,l,c
I

A

I

.

y,2,c> = 0 for a l l x

# y. I f t h e b r a c k e t

I x , 2 , c > were f i n i t e , t h e above i n t e g r a l vanishes and hence a l s o > would vanish, s o i n general < x , l I x , 2 > must be i n f i n i t e l y 2

great i n such a way as t o make the i n t e g r a l non-vanishing. The r e q u i r e d form of i n f i n i t y i s expressed by t h e Dirac d e l t a f u n c t i o n .

I n our work up t o the present i t has been implied t h a t our bra and k e t vect o r s are of f i n i t e length and t h e i r scalar products are f i n i t e . We see now the need f o r relaxing t h i s condition when we are dealing w i t h eigenkets (-bras) of an observable whose eigenvalues fopin a range. I f we d i d not relax i t , the phenomenon of ranges would not occur and our theory would be too weak f o r most p r a c t i c a l problems.

(.

.. )

The space of bra or k e t vectors

when the vectors are r e s t r i c t e d t o be of f i n i t e length and t o have f i n i t e scalar products i s called by mathematicians a Hilbert space. The bra and k e t vectors t h a t we now use form a more general space than a Hilbert space.

L I I I l Commutativity

P and P denote two observables. A s t a t e i s c a l l e d a simultaneous 1 2 e i g e n s t a t e o f t h e couple ( P , P ) i f it i s both an e i g e n s t a t e of PI and an 1 2 e i g e n s t a t e of P,. The e x i s t e n c e of a simultaneous e i g e n s t a t e i s r a t h e r ex-

Let

c e p t i o n a l . However, i f t h e observables

Pl and P2 commute, then t h e r e e x i s t s P, and P2 a r e

a complete s e t of simultaneous e i g e n s t a t e s . Conversely, i f

observables such t h a t t h e i r simultaneous e i g e n s t a t e s form a complete s e t , then

P, and P2 commute. The proof of both statements can be r e a d i l y obtained P, i n terms of t h e e i g e n s t a t e s of P2 and

by expanding t h e e i g e n s t a t e s of

then using t h e completeness of t h e s e t of e i g e n s t a t e s of

Pl and P2, respect-

i v e l y . Because of t h e previous r e s u l t s it makes sense t o speak of t h e eigen-

DIRAC'S FORMALISM

302

couples of the couple of commuting observables ( P 1 , P 2 ) . Each eigencouple

P 1 and an eigenvalue x2 of P, for which there exists a simultaneous eigenstate. We note that the notion

of (P1,P2) consists of an eigenvalue x1 of

of eigencouple does not appear in Dirac's monograph. If we apply the orthogonality theorem to simultaneous eigenstates, we derive that two simultaneous eigenstates of a couple of commuting observables are orthogonal if the eigencouples to which they correspond are distinct. The notion of simultaneous eigenstate can be extended to more than two observables. Then we consider a tuple of commuting observables together with their eigentuples. To each eigentuple there corresponds at least one simultaneous eigenstate. The totality of simultaneous eigenstates is a complete set. Simultaneous eigenstates with distinct eigentuples are orthogonal.

Thus one can give a meaning t o several commuting observables having vaZues a t the same time.

From the p o i n t of view of general theory, any two

( . .. )

or more commuting observabZes may be counted as a s i n g l e observable, the r e s u l t s of a measurement of which c o n s i s t of two or more numbers. The states to which this measurement is certain to lead, are the simultaneous eigenstates.

L Basic vectors

Consider a complete set

IB

of bra vectors. Then each bra can be expressed

as a linear combination of the bra vectors in

B (as a sum, a series or an

integral or possibly both together). Take any ket IA> and form its scalar product with each of the bras in

representative of

IA>.

B.

The numbers so obtained constitute the

These numbers determine

IA>

completely, since if an-

other ket I A ' > possesses the same representative then the scalar product of the ket IA> - I A ' > with any element of with any bra vanishes since

B

B

is zero. Hence its scalar product

is complete. This yields I A > = I A ' > .

Thus

the complete set 83 induces a representation of the ket vectors. The elements of

B

are called the basic bras of this representation.

We may suppose the b a s i c bras t o be labezed by one o r more parameters

...,xu,

each of which may take on c e r t a i n numerical values. The basic bras w i l l then be w r i t t e n <xl . x I and the representative of IA> w i l l xl,

be w r i t t e n <xl

..

... x

I A > . This representative w i l l now c o n s i s t of a s e t of numbers, one for each s e t of values t h a t xl,...,x may have i n t h e i r

DIRAC'S FORMALISM ACCORnING TO DIRAC

303

respective domains. Such a s e t of nwnbers j u s t forms a f u n c t i o n o f the variables xl,

...,xU'

Thus the representative of a k e t may be looked upon

e i t h e r as a s e t of nwnbers or as a f u n c t i o n of the variables used t o label the b a s i c bras. Although the basic bras of a representation do not have to be all independent, in practice they will satisfy the even more stringent condition that they are mutually orthogonal. The corresponding representation is called an orthogonal representation. Consider an orthogonal representation with basic bras <x1x2 by the real parameters x1,x2, tors

Ll,L 2,...,LU <xl

...,x . Then we

IA>

I labeled

define the real linear opera-

by

... xuI

Lj

x. 1 <x

IA> =

for any ket IA>. The operators ket

... x

... x

L1,...,LU

the representative of L.IA>, j 1

I A> ,

j = 1

,...,u

,

are well-defined because for each =

1,

...,u

, is determined. The

operators L l,...,LU mutually commute and each basic bra is a simultaneous eigenbra of the tuple ( L l,...,L

1 . Further, to each eigentuple ( x

I

-..,xu)

there corresponds only one simultaneous eigenbra. Thus we arrive at the following definition. A complete s e t of c o m t i n g observables i s a s e t of

observables which mutually commute and f o r which each eigent-uple corresponds t o p r e c i s e l y one simultaneous e i g e n s t a t e . Without proof we state the following results.

- The basic bras of an orthogonal representation are simultaneous eigenbras of a complete set of commuting observables.

-

Given a complete set of commuting observables we can set up an orthogonal representation in which the basic bras are simultaneous eigenbras of this complete set.

-

Any set of commuting observables can be made into a complete set by adding certain observables to it.

We call the kets corresponding to the basic bras of a representation the basic kets of the representation. The basic kets of a representation give rise to representatives of bras, etc. If we are dealing with an orthogonal representation it is natural to have some normalization convention for the basic kets (bras) of this representa-

304

DIRAC'S FORMALISM

tion. Here we introduce this convention in the special case of a single observable

P which forms itself a complete set. So each eigenvalue x of P

has multiplicity one, i.e. each eigenspace is of dimension one. If the eigenvalues xd and xi are taken from the discrete part of the spec-

trum, then we have the normalization

{

<Xd

I

xi> = 0

<Xd

I

Xd>

=

1

or combined <x

1.2.13.

d

I

xi>

=

6x x' dd

where the symbol 6 indicates the Kronecker delta. If the eigenvalues x x' of

P

kets Ix

and

are taken from the continuous part of the spectrum then the eigen-

> and

Ix'> have no finite length. The normalization is now taken

as follows.

where the symbol 6 denotes the delta function on IR at 0 . 6(x) i s not a

function of x according t o the usual mathematical d e f i n i t i o n of f u n c t i o n , which requires a f u n c t i o n t o have a d e f i n i t e value for each p o i n t i n i t s domain, b u t i s something more general, which we may c a l l an 'improper funct i o n ' t o show up i t s d i f f e r e n c e from a function defined by the usual d e f i n i t i o n . Thus 6(x) i s not a quantity which can be generally used i n mathematical analysis l i k e an ordinary f u n c t i o n , but i t s use must be confined t o c e r t a i n simple types of expressions f o r which it i s obvious t h a t no i n ( . ..) To get a p i c t u r e of 6(x), take a f u n c t i o n of

consistency can a r i s e .

the r e a l variable x which vanishes everywhere except i n s i d e a small domain, E say, surrounding the o r i g i n x = 0, and which i s s o large i n s i d e

of length

t h i s domain t h a t i t s i n t e g r a l over t h i s domain i s u n i t y . The exact shape of the f u n c t i o n i n s i d e t h i s domain does not matter, provided there are no w i l d variations ( f o r example provided the f u n c t i o n i s aZways of order E-'). Then i n the l i m i t

E -+

.

0 the f u n c t i o n w i l l go over i n t o 6 (x)

305

DIRAC'S FORMALISM ACCORDING TO D I R A C

P r o p e r t i e s of b a s i c k e t s W e only consider t h e case of a s i n g l e observable

P which forms i t s e l f a

complete s e t . The general c a s e of a complete set of commuting observables can be d e a l t with s i m i l a r l y . By t h e r e a l parameter xd we l a b e l t h e eigenvalues of

P

corresponding t o t h e d i s c r e t e p a r t of t h e spectrum and by x

we l a b e l t h e eigenvalues corresponding t o i t s continuous p a r t . To each xd (xc) belongs p r e c i s e l y one eigenket Ix

d

>

(Ix >) which w e t a k e normalized

a s i n d i c a t e d i n 1.2.13-14. Let x ' and x ' be f i x e d . Then it follows from 1.2.13 t h a t d

I.2.15.a.

[xu>=

d

1 1xd><xd 1 xh> xd

and from 1.2.14 t h a t

Since I.2.15.a-b holds f o r any b a s i c k e t I x ' > and Ix'>, d s i n c e t h e b a s i c k e t s form a complete s e t , we d e r i v e

1.2.16.

I

=

1

Ix ><xdl d

r e s p e c t i v e l y , and

+

Xd where

I

denotes t h e i d e n t i t y .

By means of Equation 1.2.16 we can expand any b r a o r k e t i n terms o f t h e b a s i c k e t s ( b r a s ) of t h e orthogonal r e p r e s e n t a t i o n induced by i n g Equation 1.2.16 from t h e r i g h t by t h e k e t I A > w e o b t a i n

and, s i m i l a r l y , from t h e l e f t by t h e b r a

1.2.18.


=

xd

<Xdl

+

F u r t h e r , multiplying 1.2.17 by t h e bra

I



C

>


dxc <xcl

.

P.

Multiply-

DIRAC'S FORMALISM

306

1.2.19.

1

=


I xd><xdI

A>

+

Xd Given an observable

p which

I


I xc> dxc <xc I A>

.

generates an orthogonal representation we can

represent each state by a set of numbers. Further, each state possesses an expansion in terms of the eigenstates of special case that

2

2 as

indicated by 1.2.17-19 in the

forms a complete set of commuting observables.

Given an orthogonal representation also the linear operators can be represented by a set of numbers. Again, we examine only the case of an observable

P which forms a complete set of commuting observables itself. In addition, we suppose that the discrete part of the spectrum of P is empty, i.e. we only have to deal with eigenvalues x

.

Let L be a linear operator. Then the numbers 1.2.20.

I;"<

Llxc>

where xc, x' run through the eigenvalues of

P,

determine the linear operator

L completely. The set of numbers 1.2.20 is called the representative of the linear operator

L

or of the dynamical variable L . The representative of L

is a function of two variables x

and x' which can vary continuously. Such

a function is called a matrix, although we cannot write it out as a twodimensional array like a discrete matrix. The identity operator is represented by 6(x - X I ) (= <x' I x > ) and this c c c c matrix is called the unit matrix. If L is real we have 1.2.21.

<xi1

L Ixc>

=

<xcl

L

IXi>

.

The matrix of a real linear operator is called Hermitean. The representative of

P is given by

1.2.22. Let L

1

1.2.23. and

<x'l P I X and L

2

>

= xc 6(xc- x;)

.

denote two linear operators. Then by 1.2.17-18 we have

<x;IL1

=

I C <x'l

L,

Ix"> dx:

<x"l

DIRAC'S FORMALISM ACCORDING TO DIRAC

L21xc> =

1.2.24.

Ix"> dx" c c

<x"I L2 Ix c

>

307

.

So by 1 . 2 . 1 9 we get

<xAl L1L2 Ixc>

1.2.25.

1, Ix"> dxp < x l l L (x

=

2

c

>.

The l a t t e r e x p r e s s i o n e s t a b l i s h e s t h e m a t r i x p r o d u c t of t h e m a t r i c e s formed by

< x ' l L Ix"> and <x"I c l c c

L2

Ix

c

>.

We now have a l i n e a r operator always represented by a matrix. The sum of

two l i n e a r operators i s represented by t h e sum o f the matrices representing the operators and t h i s together w i t h 1.2.25 means t h a t t h e matrices are subject t o t h e same algebraic r e l a t i o n s a s t h e l i n e a r operators. I f any algebraic equation holds between c e r t a i n l i n e a r operators, t h e same equation must hold betueen t h e matrices representing those operators. The scheme of matrices can be extended t o bring i n t h e r e p r e s e n t a t i v e s of k e t and bra v e c t o r s . The matrices representing l i n e a r operators are all square matrices w i t h the same number of rows and colwrms, and with, i n f a c t , a one-to-one

correspondence between t h e i r rows and columns. We may look

upon t h e r e p r e s e n t a t i v e o f a k e t s e t t i n g a l l t h e numbers < x c

I?+>

IA>

as a matrix with a s i n g l e column by

which form i t s r e p r e s e n t a t i v e one below

t h e other. The number o f rows i n t h i s matrix w i l l be t h e same as t h e number

of rows or columns i n the square matrices representing l i n e a r operators. Such a single-column matrix can be m u l t i p l i e d on t h e l e f t by a square matrix <xAI 1Ix

>

representing a l i n e a r operator by a r u l e s i m i l a r t o t h a t f o r

the m u l t i p l i c a t i o n of two square m a t r i c e s . The product i s another s i n g l e column matrix w i t h elements given by

From 1.2.16 t h i s i s j u s t equal t o <x;l

/A>, the r e p r e s e n t a t i v e of L I A > .

S i m i l a r l y we may look upon t h e r e p r e s e n t a t i v e o f a bra w i t h a s i n g l e row by s e t t i n g a l l the members


I

X I

>


as a m a t r i s

s i d e by s i d e . : ; u ~ h

a s i n g l e row may be m u l t i p l i e d on the r i g h t by a square matrix <x;I

1 Ixc>

t h e product being another single-row m a t r i x which i s j u s t the rspreseri/,/~liuf, of


The single-row matrix representing


may be muZLipZil;d Jrwm

t h e r i g h t by t h e single-coZwnn matrix representing IA>,

t h e produo/, b a i r q

308

DIRAC'S FORMALISM

a matrix w i t h j u s t a singZe element which i s equal t o

t h e single-row matrix representing < B I

I

A>.

Finally,

may be m u l t i p l i e d on the l e f t by

the single-colwnn matrix representing IA>, which i s the representative of I A > < B ~ .


the product being a square matrix

I n t h i s way a l l our abstract sym-

b o l s , linear operators, bra vectors and k e t vectors can be represented by matrices, which are s u b j e c t t o the same algebraic r e l a t i o n s as t h e abstract symbols themse lves.

Some comments and bibliographical notes This chapter contains a rewording of the formal aspects of the theory developed in Dirac's book [Dil.

309

CHAPTER I I

A MATHEMATICAL I N T E R P R E T A T I O N OF D I R A C ' S BRACKET FORMALISM

As

we have seen in the previous chapter, Dirac considers two kinds of vec-

tors, the kets and the bras. Dirac assumes a one-to-one correspondence between ket and bra vectors. Also certain algebraic relations are supposed to be satisfied by these vectors. However, the term algebraic seems to be misused because these relations are often expressed in terms of infinite sums and even integrals. Dirac does not bother whether they are convergent

or not. Yet, the (distributional) solutions of these convergence problems establish the import of our mathematical interpretation. Consequently, the ket space is taken to be a topological vector space, K say. In addition, we want to stick to Hilbert space theory as close as possible. For this we have the following reasons. From Dirac's set up we derive that there are kets to which a finite

length can be attached. Mathematically speaking, K must contain a normed subspace N. This subspace N must have infinite dimension. The vectors of

N may be called normalizable kets and the bras corresponding to these normalizable kets, normalizable bras. The scalar product of a normalizable bra and a normalizable ket is a complex number. So it is natural to assume that the norm of N arises from an inner product. Dirac supposed that the unnormalizable kets can, one way or another, be approximated by normalizable kets. Rephrased in mathematical terms, must be dense in

N

K.

In Dirac's scheme the observables are real dynamical variables with real eigenvalues and a complete set of eigenstates. The only mathematical concept that more or less resembles Dirac's interpretation of observables is the concept of self-adjoint operator in a Hilbert space. Regarded from a mathematical point of view, it is desirable and convenient

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

310

t o have a t o p o l i g i c a l v e c t o r space

K

with a r a t h e r simple t o p o l o g i c a l

structure. Because of a l l t h e s e c o n s i d e r a t i o n s t h e k e t space i s taken t o be a n w ? k Z P t r a j e c t o r y space

TX,A.

Thus two choices have t o be made: t h e choice of t h e

separable H i l b e r t space

A in

operator

x.

x

and t h e choice of t h e nonnegative s e l f - a d j o i n t

The choice of

x

i s i r r e l e v a n t , s i n c e a l l i n f i n i t e dimen-

s i o n a l separable H i l b e r t spaces are mutually isomorphic. The freedom o f choosing

A can be used t o take a space T

X,A

which i s b e s t adapted t o t h e

quantum dynamical system under c o n s i d e r a t i o n .

( A , X ) such t h a t T X,A A has d i s c r e t e spectrum { A d I d E ID] where

For t h e remaining p a r t of t h i s c h a p t e r we f i x a p a i r i s a nuclear space. I t means t h a t

ID denotes a countable index s e t . The eigenvalues hd which a r e counted t o

m u l t i p l i c i t y , a r e nonnegative. They s a t i s f y

By (

A,

~ w e )denote ~ ~a corresponding ~ orthonormal b a s i s of eigenvectors of

v

i . e . Avd = h v f o r a l l d d d

E

ID.

Throughout t h e whole c h a p t e r we employ D i r a c ' s b r a c k e t n o t a t i o n , which we consider a s one of t h e s t r i k i n g a s p e c t s of t h e formalism.

II.i.1.

Definition.

The elements of

T X , A a r e c a l l e d k e t v e c t o r s . We denote them by

IB> where B

i n d i c a t e s any l a b e l ( o r l a b e l s ) . So a k e t IB> i s an X-valued f u n c t i o n t

I+

IB>(t)

on t h e open i n t e r v a l ( O , = )

NuSa t i o n .

By e-t'AIB> we mean t h e k e t

which s a t i s f i e s

311

KETS

To each v e c t o r v d , d

E

Iv,d>(t) = e

II.i.2.

m,

we a s s o c i a t e t h e k e t Iv,d> defined by

-Xdt

vd ,

t > O .

We can extend t h e k e t Iv,d> t o an X-valued function on t h e whole r e a l a x i s . Therefore w e simply p u t

11.1.2'.

Iv,d>(t) =

d ,

t e r n .

Let t > 0 . Then we can w r i t e

and, hence,

Now t h e expression ( l B > ( t )

8,

= (IB>(t)

(*d)de ID

, Iv,d>(-t))X

, I v , d > ( - t ) ) X does

not depend on t , whence

i s a well-defined complex number. The sequence

satisfies

Employing t h e same n o t a t i o n as introduced i n t h e appendix t o Chapter B . 1 , we denote t h e s e t of a l l sequences (adIdcm which s a t i s f y I I . i . 3 by

TI (

~

II.i.4.

~

)

~

~

~

l

.

Theorem.

To each k e t IB> t h e r e corresponds a sequence (Bd)dcID

TI

lB> =

i.e.

i n t h e sequence space

such t h a t

1

de ID

Bdlv,d>

(convergence i n T X , A )

312

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

Conversely, each sequence (Cd)dEID

Proof.

of T{

defines the ket

Cf. Chapter B.1, Appendix.

The Hilbert space L,(ID) summable sequence (1,)

is contained in T{(Ad)dclD}.

dEID

So to each square

there corresponds the ket

It is clear that the limit lim (d:lD n,lv,d>(t)) exists in the sense of the tJ.0 x. This limit equals the vector d&, ndvd E Thus we

x.

normtopology of

arrive at the following definition. 11. i. 5. Definition. A ket

IB>

is called normalizable if the limit

lim IB>(t) tSO exists in X-sense. For a normalizable ket IB> we define

The length of the normalizable ket (B> is defined by

11 IB>(O) 11 X'

The following characterization of normalizable kets can be simply checked by the reader. II.i.6. Theorem. a. A ket 1B> is normalizable iff there exists g IB>(t) b.

A

=

E

x

such that

e-tA g, t 2 0.

ket lB> is normalizable iff there exists a sequence

such that

(nd ) deID

in L,(ID)

BRAS

313

0

The t e s t k e t s e s t a b l i s h a n o t h e r t y p e of s p e c i a l k e t s .

11.i . 7.

Definition.

A n o r m a l i z a b l e k e t IB>

is called a test ket i f

F o r a t e s t k e t IB> t h e r e e x i s t s u > 0 s u c h t h a t t h e k e t IB> c a n b e e x t e n d e d t o the i n t e r v a l [-or=-)

II.i.8.

with

Theorem.

a. A k e t IB> i s a t e s t k e t i f f t h e r e e x i s t s a n o r m a l i z a b l e k e t lX> and

u > 0 such t h a t

b. A k e t iB> is a t e s t k e t i f f t h e r e e x i s t s a n

t2(ID) - s e q u e n c e

(0

d dtID

and u > 0 s u c h t h a t

&mark.

A k e t IB> i s

= e-itAIB>($t)

is a l s o a n

As u s u a l ,

an X-valued f u n c t i o n on ( 0 , m ) .

w e conclude t h a t I B > ( t )

sX , A - v a l u e d

let

x'

E

sX , A

Since IB>(t)

=

f o r a l l t > 0. So a k e t IB>

f u n c t i o n on ( 0 , a ) .

denote t h e dual of

x,

i.e.

ous l i n e a r f u n c t i o n a l on the H i l b e r t space s p a c e w i t h t h e s u p norm,

x'

x.

i s t h e space of a l l continuW e note t h a t

x'

i s a Banach

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

314

IILIIXt

lecf, I

= sup

e x' .

,

€

f€X By t h e Riesz r e p r e s e n t a t i o n theorem, for each

x

fe i n

e E x'

t h e r e e x i s t s a unique

such t h a t

Conversely, each f

E

x

Lf de-

gives r i s e t o a continuous l i n e a r f u n c t i o n a l

f i n e d by

ef(d =

(g,f)x

-

W e thus a r r i v e a t an a n t i l i n e a r i s o m e t r i c isomorphism from a s s o c i a t e s t o each f

E

x

x

t h e continuous l i n e a r f u n c t i o n a l f '

onto E

x'

x' .

which

So we

have (crf + B g ) ' = i f '

+

= Ilfllx

.

Ilf'IIXI

In

x'

Bg'

,

we d e f i n e t h e i n n e r product (

-,

Since II f ' I l i u = ( f ' ,f')x o , t h e space

I

{vi d

E

*)

x'

x~

by

i s a H i l b e r t space. The set

ID} c o n s t i t u t e s an orthonormal b a s i s i n

adjoint operator

x'.

W e define t h e s e l f -

A' by

A'f' = ( A f ) '

=

c

xd(f',v;)xI

v'd '

d€m

f'

E

D(A')

.

We a r r i v e a t t h e d e f i n i t i o n of t h e bra space.

II.ii.1.

Definition.

The elements of t h e t r a j e c t o r y space T s h o r t l y , b r a s . We denote them by
a r e c a l l e d bra vectors o r , where t h e same remark can be make on

t h e l a b e l i n g as f o r k e t v e c t o r s . So a b r a < B l t h e open i n t e r v a l (O,-) which s a t i s f i e s

i s an Xl-valued f u n c t i o n on

BRAS

315

Notation. By < B l e - t ' A

Let d

E

m.

we mean t h e b r a

Then t o t h e l i n e a r f u n c t i o n a l v i

E

x'

we a s s o c i a t e t h e b r a


As i n Theorem I I . i . 4 we can prove t h a t each b r a
can be uniquely r e -

presented by a series


=

1

Bd
where t h e sequence (f3d)d6m

belongs t o

quence ( a d ) d 6 m i n T I

1

TI

TX ' , A ' )

'

Conversely, each se-

induces t h e b r a

ad
d€m

II.ii.2.

(convergence i n

d€m

: t P

1

ad e-ldtv;

.

de ID

Definition.

a. Let 1B> be a k e t ,

b r a
/B> =

ydlv,d>.

Then t o IB> we l e t correspond t h e

(same l a b e l i n g ) given by

b. L e t <XI be a b r a ,

<XI =

c d < v , d l . Then t o <XI

t h e r e corresponds

t h e k e t IX> defined by

The correspondence between t h e b r a and t h e k e t space introduced i n t h e pre-

3 16

A MATHEMATICAL

INTERPRETATION OF DIRAC'S BRACKET FORMALISM

vious d e f i n i t i o n i s a n t i l i n e a r . So t o t h e k e t alF> t h e bra


+


+

BIG> t h e r e corresponds

I t follows from t h i s correspondence t h a t we t r e a t

t h e k e t s and b r a s on t h e same l e v e l . F u r t h e r , w e observe t h a t =

(IG>(t))'

> 0.

Definition.

11.ii.3. A bra



i s c a l l e d normalizable i f i t s corresponding k e t IB> i s normaliz-

a b l e . For each normalizable b r a

e x i s t s . The length of

bra

the l i m i t

i s defined as

II
IIxt.

Corollary.

11.ii.4. A




such t h a t

i s normalizable i f f t h e r e e x i s t s a sequence (q ) in d deD = d:lD nd < v , d l .


e2(D) 0

CefinitiOn.

11.ii .5.

i s a t e s t k e t .

A bra

I t follows from t h i s d e f i n i t i o n t h a t each t e s t b r a


SXt,At.

E

A

t e s t bra



i s normalizable with

can be extended t o t h e i n t e r v a l

C - U , ~ ) for

c e r t a i n a > 0 with

II.ii.6. A bra

(nd)dcm

A bra

Coro~lary.

0 and a sequence E


t 2 ( I D ) such t h a t

i s an XI-valued function on (0,m).

So f o r each T > 0


317

BRACKETS

x.

a continuous l i n e a r f u n c t i o n a l on


on t h e a n a l y t i c i t y space


(w)

=:

Therefore we can d e f i n e an a c t i o n of

S X,A


,

where T > 0 has t o be chosen s o s m a l l t h a t w < B l ( w ) does not depend

on t h e choice of

T.

E

e

-TA

(x).

For i f 0 <

The d e f i n i t i o n of

?

< T , then w e have

W e now i n t e r p r e t e D i r a c ' s b r a c k e t notion a s follows.

1I.iii.l.

Definition.

The expression


I F>,

c a l l e d t h e b r a c k e t of

complex valued f u n c t i o n on ( 0 , m )

We note t h a t t h e b r a c k e t



and IF> denotes t h e

d e f i n e d by

I F > is

well-defined because I F > ( t >

E

sX,A

for

a l l t > 0. Let us examine t h e complex valued f u n c t i o n s of type


I

F>.

W e represent

a s follows.

and

where both

(Sd)dEID and

l a t i o n shows t h a t

( < d ) d E I D belong t o T{(id)dEID}.Thena simple calcu-

A MATHEMATICAL INTERPRETATION OF DIRAC‘S BRACKET FORMALISM

318

So we can extend the f u n c t i o n
I B>

t o a holomorphic f u n c t i o n e

on t h e open r i g h t h a l f of t h e complex p l a n e ; w e d e f i n e

where w e have p u t vd =

cdsd,

d

E

m.

We observe t h a t

We consider t h e l i n e a r space of a l l Bore1 f u n c t i o n s cp on IR f o r which t h e limit

N

i s f i n i t e . Taking equivalence c l a s s e s i n t h e usual manner we a r r i v e a t a H i l b e r t space P i n which t h e i n n e r product is given by N

The H i l b e r t space P i s n o t separable. The f u n c t i o n s ed : y t+ exp(-iAdyl, y

IR,

E

d

v

ID, c o n s t i t u t e an orthonormal system ( r e 1) i n P. By we d dcm d E I D } > i n P. We d e f i n e

E

denote t h e c l o s u r e of t h e l i n e a r span <{Led] the operator

c

in

v

I

by

and

CCcpl =

Then

TV,

c

c

1

d€m

Ad(Ccpl

,Cedi),, Cedl ,

is a nonnegative s e l f - a d j o i n t o p e r a t o r i n

and t h e a n a l y t i c i t y space

sv,c

a r e V-valued f u n c t i o n s on (0,m):

Ccpl

E

v.

a r e well-defined.

D(C)

.

So t h e t r a j e c t o r y space

The elements of T

v,c

319

BRACKETS

belongs t o T{(Xd)deID}. Each

where t h e sequence

[@I i n T

v,c is in correspondence with a holomorphic f u n c t i o n H C o 3 on t h e open r i g h t

one-to-one

h a l f of t h e complex plane

Thus we can (and w i l l ) regard TV,C

a s a space of holomorphic f u n c t i o n s on

Re z > 0. Each such holomorphic f u n c t i o n r e p r e s e n t s an almost p e r i o d i c distribution

I n a s i m i l a r way it can be shown t h a t t h e space

sv,c

can be regarded a s a

space of f u n c t i o n s HE@, which a r e holomorphic on t h e region Re z > -a

for

some a > 0 , where

kIDl w d l e-Xdt

W e note t h a t

11.iii.3.

<

m

for a l l t >

-0

Theorem.

For each b r a
and each k e t IF> t h e i r extended b r a c k e t


I

F>

belongs

to TV,C.

Proof.

As we have seen



can be w r i t t e n a s

where

Remark

We can regard t h e b r a c k e t

a s an almost p e r i o d i c d i s t r i b u -

t i o n on t h e imaginary a x i s . For each w'

E

S X g , A ~

w e can d e f i n e i t s a c t i o n o n t h e k e t space T

follows

w ' ( I F > ) := ( e T A 'w ' ) ( I F > ( T ) ) ,

X , A as

320

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

where T has to be taken sufficiently small. The above definition does not depend on the choice of T. 11.iii.4.

e.-tA

For all t > 0, {
}(IF>)

=

(t).

x. By definition we have for all t

> 0 and 0 < T < t

Remark. The above proof indicates the following relation

for all t > 0 and all T, 0

5

T 5 t.

11.iii.5. Theorem. The following relations are valid for any ket IF> and any bra

=

I F>

2

b.


< F I B > , i.e. Vt,o

:

(t)

0, i.e. Vt>O : (t) 2 0

=

(t)

.

.

Proof. a. Let t > 0 and let 0 < T < t. Then we have

II.iii.6.

m.

Let IF> be a normalizable ket and (t) exists and equals (\F>(O) , IB>(0))x. So our definition of tSO bracket is a generalization of the notion of inner product.

w.

This follows immediately from the definition of normalizable ket and

normalizable bra.

0

32 1

LINEAR OPERATORS

1I.iii.7.

e.

Let IF> be a t e s t k e t and
any bra. Then t h e r e e x i s t s u > 0 , dependent

on IF> such t h a t t h e complex valued f u n c t i o n < B holomorphic f u n c t i o n on t h e h a l f plane Re z > -u.

I

F>

can be extended t o a

I t s extension


I F>

i s defined by

Praof.

0

Cf. t h e d e f i n i t i o n of t e s t k e t and of b r a c k e t .

Remarks. a. In t h e previous lemma i n s t e a d of t e s t k e t and b r a w e can t a k e k e t and

t e s t bra.

(Cf. Theorem I I . i i i . 5 . )

b. For a t e s t k e t ( b r a ) IF> (

t

F>e

E

c. In t h e sequel we s h a l l w r i t e


I F>e


(IF>),

sv,c. < B I F > ( - T ) f o r 0 < T < u i n s t e a d of

(-7).

CIZIIJ Linear o p e r a t o r s

D i r a c ' s l i n e a r o p e r a t o r s have both a r i g h t and a l e f t a c t i o n . The i d e a t h a t a l i n e a r o p e r a t o r can a c t b t h from t h e r i g h t and from t h e l e f t i s a d i r e c t u p l i f t i n g from a r e s u l t o f l i n e a r a l g e b r a t o t h e b r a and k e t formalism. Indeed, i n l i n e a r algebra a matrix a c t s from t h e l e f t on columns and from t h e r i g h t on rows. Dirac claims t h a t t h i s p r o p e r t y can be extended t o g e n e r a l v e c t o r space theory. To our opinion t h i s c l a i m i s t o o bold. A t l e a s t , our i n t e r p r e t a t i o n does n o t supply t h e room t o i n t e r p r e t e t h i s a s p e c t of t h e theory t o t h e same extend as Dirac assumes. Therefore, w e c o n s i d e r l i n e a r o p e r a t o r s which only have t h e usual a c t i o n from t h e l e f t . Since w e have introduced t h r e e types of k e t v e c t o r s : t e s t k e t s , normalized k e t s and k e t s i n general, nine d i f f e r e n t types of l i n e a r o p e r a t o r s e n t e r our s e t up. Here we d i s c u s s f o u r of them.

II.iv.1.

Definition. A

The space TT XoX;Ael,IeAc o n s i s t s of a l l f u n c t i o n s 8 from (O,-) x (o,-) i n t o

x

@

x

which s a t i s f y ,

't

,T > o

vs,o>o

:

322

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

A

8 ( t + T ,s+u)

1

For each 8

E

= e

-aA

TTXeX;Ael,leA w e

--TA

B(t,s)e

.

where T > 0 has t o be taken so small t h a t I W > ( O ) makes sense, because

Iw>

d e f i n e its a c t i o n on a t e s t k e t

x0x

~ ( T , s ) E

and

IW>(-T)

e

E E

x

-TA

(x).

by

This d e f i n i t i o n

so t h a t (BlW>) ( s )

E

x.

Further, f o r a l l s > 0 and a > 0 w e have

I t follows t h a t 8 1 W > i s a k e t . We observe t h a t 8 1 W > is n o t a t e s t k e t i n

general.

Thus each element of

TTXeX;Ael,leA

induces a l i n e a r o p e r a t o r from t h e

l i n e a r space of t e s t k e t s i n t o t h e k e t space. For a b r a

w e d e f i n e t h e element

IF>
of

TTXeX ;AeI ,IeA

For each t e s t k e t IW> t h e a c t i o n of IF>

and a k e t IF>

by

on IW> i s given by

We note t h a t f o r each k e t IF> t h e l i n e a r o p e r a t o r /F>.

II.iv.2.

Definition.

SSXeX;Ael,lsA c o n s i s t s of a l l elements $ E TT XeX ;Ael ,IeA for which t h e r e e x i s t s to > 0 and so > 0 such t h a t f o r a l l t > 0 and s > 0 The space

1

W e can extend t h e f u n c t i o n 0 t o t h e r e g i o n (-to,-)x ( - s o , - ) ;

d e f i n e f o r -to < t 5 0 and - s

0

< s S 0

t h e r e f o r e we

323

LINEAR OPERATORS

Here any

For each

' E

> 0 can be taken.

S S X @ X ; A @ l , l @ A w d e f i n e i t s a c t i o n on a k e t IF> by

I

where

T

s > -so,

on

T.

must be taken smaller than to. Since @ ( - ? , s ) @ I F > ( s ) i s a member of

x.

x

E

@

x

for a l l

The d e f i n i t i o n of @IF> does n o t depend

W e observe t h a t @IF> is a t e s t k e t , because {@IF>}(O)

E

sA

D(e

)

for

a l l s , 0 < s < so. Each element of SSX@x;Aa,I,l@A

g i v e s r i s e t o a l i n e a r o p e r a t o r from t h e k e t

space i n t o t h e l i n e a r space of t e s t k e t s . For a t e s t k e t IW> and a t e s t b r a


Iw>
(t,s) = IV>(t)

E

t > -to , s > -s

,

0 IW>(S)

> 0 i s taken so s m a l l t h a t I V > ( O )

0 t h a t IW>(O)

II.iv.3.

SSXeX;AaI,laAby

E

E

D(etoA)

0

and so > 0 so small

D(eSOA). Its a c t i o n i s given by

Definition. 1

c o n s i s t s of a l l elements K o f TT

The space TSXeX;Ael,leA

t h e property t h a t f o r each s > 0 t h e r e e x i s t s t

vt>O

so given a f i x e d

:

i(t,s)

E

D(etSA

@

7)

. 1

s > 0 w e can extend t h e f u n c t i o n t l + K ( t , s )

(-ts,-).Therefore, w e d e f i n e f o r -t 1

X @ X ; k l , l @ Awith > 0 such t h a t

1

K ( t , s ) = K(E,s)

e

,

t o the interval

< t 5 0

E > 0 arbitrary

.

1

For each

' TSX@X;A@l, l @ A w e

d e f i n e its a c t i o n on a k e t IF> by

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

324

where T > 0 must be s m a l l e r than t

. The

d e f i n i t i o n o f t h e k e t KIF>

i s in-

dependent of T.

Each element of TSXeXiAeI,

induces a l i n e a r o p e r a t o r from t h e k e t space

i n t o t h e k e t space. A s p e c i a l c l a s s i n t h e space

TSXeX;AmI,IgA i s

by t h e elements IF><WI where IF> i s a k e t and <WI

IF><WI

where t

:

( t , s ) t+

IW>(t)

0

IF>(s)

> 0 i s taken so small t h a t I W > ( O )

0

,

t

Z

D(etOA).

E

formed

a t e s t bra. W e have -to , s > 0

The a c t i o n o f IF><WI

on t h e k e t space i s given by

The v e c t o r space TS

t h e s t r u c t u r e of an algebra. To

XmX;AeI,7eAp_ossesses t h i s end, we d e f i n e t h e product K 1 ' K 2 Of

where s

7

- - K-

2 We observe t h a t t h e o p e r a t o r product K

TSXeX;AepT,

E

in T S X g X ; A @ l , I g A by

with u > 0 , on i t s t u r n , s u f f i c i e n t l y s m a l l and de-

0, t > -t

p e n d e n t o n s . The d e f i n i t i o n of K,

il k2

2'

IeA,

does not depend on t h e choice of u .

OK,

of K L

symbolically (K,

0

K2)

=

1-

and K

K,

1

*

2

i s r e p r e s e n t e d by

K,

F i n a l l y w e introduce t h e v e c t o r space TS XeX;leA,AeI(mind t h e o r d e r ! ) . 11 . i v . 4.

Definition.

The space

TSXeX;IeA,AeI

c o n s i s t s of a l l elements L i n TT

t h e property t h a t f o r each t > 0 t h e r e e x i s t s s

So f o r each f i x e d t > 0 t h e f u n c t i o n s

t

XeX; AeI ,7eA with > 0 such t h a t

L

I+

L ( t , s ) , s > 0 , can be extended t o

t h e i n t e r v a l (-st,-). Therefore w e p u t 1

For each

-L

L(t,s) = e E

-sA+EA

L(t,E)

TSXex;IeA,Aerw e

,

E

> 0 arbitrary

.

d e f i n e i t s a c t i o n on t h e l i n e a r space of t e s t

325

LINEAR OPERATORS

k e t s by : S H ^L(T,S) {lW>(-T)}

LlW>

,

S > -S

where T > 0 must be taken so s m a l l t h a t I W > ( O )

E

e-TA(X).The d e f i n i t i o n of

L l W > does n o t depend on T .

Each element of TSXeX;

I@A,A@I

induces a l i n e a r o p e r a t o r from t h e l i n e a r

space of t e s t k e t s i n t o i t s e l f . For each t e s t k e t IW> d e f i n e IW>
with so so small t h a t I W > ( O ) IW>
Also

L1,L2

E

TSXgX;IgA,Aaz.

E

e

-soA

(x).


we

TSX@X;l@A,A@I.

where t > 0 and s > - s

I t is c l e a r t h a t

The a c t i o n of l W > < B I

i n t h e v e c t o r space TSX?X;l?A,AmI E

and each b r a

by

Then L ,

with

T

on a t e s t k e t IV> i s given by

we d e f i n e a product o p e r a t i o n . Let

L, is defined by

> 0 s u f f i c i e n t l y small. The d e f i n i t i o n o f

on t h e choice of T. Expressed symbolically we have

(L,

0

L2)

=

L1 ' L 2 .

Remark. With r e s p e c t t o Chapter B . 1 1 t h e following o b s e r v a t i o n s a r e u s e f u l and d e s i r a b l e .

-

The space

-

The space T XmX,AmA can be i d e n t i f i e d with

-

The space

-

The space T(SXeX,I@A

1

Dirac bases

Sx@X,AmA can be i d e n t i f i e d with

ss X @ X ; A @ I ,Z@A' TTXaX;AmI,

T(SXeXrAaI , 1

@

A ) can be i d e n t i f i e d with

,A

@

I)

TSX@X;A@I,IeA.

can be i d e n t i f i e d with T S X ~ , X ; I @ A , A @ ~ .

I

I n t h i s s e c t i o n we f i x a Federer measure space (M,!J).

W e recall the defini-

326

A MATHEMATICAL INTERPRETATION

OF DIRAC'S BRACKET FORMALISM

tion of (canonical) Dirac basis but now rephrased in terms of kets and bras 1I.v. 1. D e f i n i t i o n . A

set {lx> Ix

M}

E

in the ket space is called a (canonical) Dirac basis if

it possesses the following properties. a. For all d

E

b. For all d,;

ID the function x t+ E

<x I v,d> ( 0 ) is measurable.

ID the following relation is satisfied

Although we want to avoid too much repetition of former results given in Part A and B, we think it is wise to recall some of these results in the present special case. Let {lx> ( x E x

E

M

M} be a Dirac basis. Then for each test ket

IW> and for each

we have

where we have to take T 2 0 sufficiently small. For such following estimation

'I

we have the

DIWLC BASES

327

So t h e double series

converges a b s o l u t e l y , and

( W e note t h a t

<x

I v,d>(?) = e

-Ad?

<x I v , d > ( O ) . )

So i t i s allowed t o interchange summation and i n t e g r a t i o n .

It y i e l d s t h e following r e s u l t .

11.v.2.

Theorem (ParseVal).

Let { Ix>

Ix

E

M } be a Dirac b a s i s . Then f o r any p a i r of t e s t k e t s IV> and

328

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

lW> w e have ( 0 ) =

i

< V I x > ( 0 ) <x I W > ( 0 ) d p ( x )

.

0

M For a Dirac b a s i s { Ix>

Ix

E

M} t h e f u n c t i o n y

I+

11 l y > ( t ) 11

x,

y

E

M, is

square i n t e g r a b l e f o r each t > 0 , because we have t h e s t r a i g h t f o r w a r d c a l culation

So f o r each t > 0 t h e f u n c t i o n y B(x,r), r > 0 , x

E

M.

k+

11 l y > ( t ) IIx i s i n t e g r a b l e on each b a l l

The Dirac b a s i s { l x >

Ix

L

M I s a t i s f i e s t h e following

d i f f e r e n t i a t i o n theorem (which i n f a c t has been proved i n Chapter A . 1 1 1 ) .

11.v.3.

Theorem.

Let (Ix>

Ix

E

M) be a Dirac b a s i s and l e t N denote t h e corresponding n u l l

s e t introduced i n D e f i n i t i o n I I . i i i . 1 .

Proof.

L e t t > 0 and l e t x

F i r s t w e take n

E

E

Then f o r a l l x

E

M\N and a l l t > 0

M\N. we have t o prove t h a t

IN so l a r g e t h a t t >

a s follows: Fix a f i n i t e set IE

C

ID

W e estimate t h e t a i l of t h e s e r i e s ( * )

1 2

We s p l i t up t h e s e r i e s

329

DIRAC BASES

5 I1

Let

+

I11

> 0 . Then f i r s t t a k e I E so l a r g e and f i x e d t h a t I1 <

E

our assumptions on

E

2

. Because

of

N

w e can t a k e ro > 0 so small t h a t f o r a l l r , 0 < r < r 0' 2 I11 < 2~ and f o r a l l d E I E

I

(O)

-

p(B(x,r))-'(

B (x,r) Then I i s smaller than E*

with

E'

=

(3

dc ID

exp(-

2 n

I


>I

I y>(O) d u ( y )

<

E

.

f o r 0 < r < ro. I t follows t h a t

h ) d

+ IIIexp(- -1A ) 111).

0

An almost s t r a i g h t f o r w a r d consequence i s t h e following r e s u l t , which may be

recognized a s a s p e c i a l p a r t of t h e Sobolev lemma p r e s e n t e d i n S e c t i o n B.IV.l.

W e omit i t s proof.

11.v. 4 ,

Corol law.

L e t IW> be a t e s t k e t . Then f o r a l l x

The functions system

(pd

: x I+ <x

I v,d>

E

(O), d

M\N

ID,

e s t a b l i s h an orthonormal

i n t h e H i l b e r t space L2(M,p).

We denote t h e c l o s u r e of

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

330

t h e l i n e a r span

Id

<{crpdl

e ID}> by Y . So Y i s a c l o s e d subspace o f L,

(M,p).

Let 8 denote t h e nonnegative s e l f - a d j o i n t o p e r a t o r i n Y d e f i n e d by

and

For each k e t IF> t h e Y-valued f u n c t i o n

belongs t o

Ty,8.

belongs t o

Y. Put d i f f e r e n t l y , t h e l i n e a r o p e r a t o r

In p a r t i c u l a r , i f IF> i s normalizable, then

from

x

into

Y defined

by

Uf

=

1

de rn

(f,vd)X CVdl

i s u n i t a r y . W e can extend U t o a l i n e a r isometry from

we have U ( S X , A ) =

sy,8.

Thus it follows t h a t

f u n c t i o n space, i . e .

s Y,B

T

XtA

onto

T

Y,B'

Also

can be considered as a

can be regarded a s t h e space of a l l f u n c t i o n s

'Y,B I W > (0) where IW> i s any t e s t k e t . Outside a n u l l set N which only depends on A ( o r B ) , t h e f u n c t i o n s s a t i s f y a d i f f e r e n t i a t i o n - l i k e r e s u l t x

I+

<x

a s expressed by Corollary I I . v . 4 . Let Q

E

Ty,8.

Because Q ( t E) S

Y,B

f o r a l l t > 0 , each O ( t ) i s a f u n c t i o n on

M. Therefore we can consider t h e elements of T v a r i a b l e s x and t with x

E

Y,B

as f u n c t i o n s o f t h e

M and t > 0. Put d i f f e r e n t l y , Ty,8 c o n s i s t s of

a l l functions

where IF> denotes any k e t . Now w e are i n a p o s i t i o n t o i n t e r p r e t e D i r a c ' s

orthogonaZity r e l a t i o n s f o r b a s i c k e t s .

331

DIRAC BASES

Let y

M be f i x e d . Then w e d e f i n e 6

E

6 (x,t) = Y

(i.e. 6

Y

1

E

Y

Ty,B by

( t ) <x I v,d>(O)

de rn

= Uly>).

So f o r each x

M and

E

I t i s c l e a r t h a t f o r each f u n c t i o n 'p E

t > 0 we have

sY.6

Ty,g a c t s a s an e v a l u a t i o n f u n c t i o n a l on Sy , B . Popular-

SO t h e f u n c t i o n

6y

l y speaking, 6

i s a Dirac d e l t a function. Now t h e g e n e r a l i z a t i o n o f t h e

€

Y orthonormality r e l a t i o n s f o r orthonormal b a s i s a s suggested by Dirac, i . e .

admits t h e i n t e r p r e t a t i o n

II.v.5.

<x

I

y > ( t ) = 6 (x,t)

Y

.

I n t h e next s e c t i o n we i n t e r p r e t e D i r a c ' s expansion claim, with r e s p e c t t o a s e t of b a s i c k e t s , with t h e a i d of our notion of Dirac b a s i s . A t t h e end of t h e p r e s e n t s e c t i o n we show t h a t any isometry from

x

i n t o L 2 ( M , p ) gener-

a t e s a Dirac b a s i s i n t h e sense of D e f i n i t i o n I I . v . 1 .

Let

J be an isometry from

x

i n t o L 2 ( M , p ) . Then f o r each d

E

m

Jvd i s an

denote a r e p r e s e n t a t i v e of J v f o r each d E ID. 2 Zd {exp(- ; A d ) I ' p d j 1 a r e i n t e g r a b l e on The f u n c t i o n s (pd, I ' p d j z and, a l s o , d€m bounded Bore1 s e t s f o r a l l d E ID and n E I N . Since (M,u) i s a Federer element of L 2 ( M , p ) . L e t

(pd

measure space, t h e r e e x i s t r e p r e s e n t a t i v e s set

N

such t h a t f o r a l l x

E

M\N, d

E

-

(pd

ID and n

of Jv,, E

IN

d E ID,

and a n u l l

332

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

-

For convenience we put cp (x) = 0 for x d For each x

E

N.

M we define

E

Then the set {lx>

Ix

E

M } satisfies the requirements of Definition II.v.1.

It follows that there exists a one-to-one correspondence between Dirac bases and isometries

J

from X into L 2 ( M , p ) .

I Representatives of kets and bras 1 It is useless to introduce the notion of Dirac basis if such a basis does

not admit expansion and representation theorems. In this section we prove such theorems. It turns out that the notion of bracket and the notion of Dirac basis fit beautifully together. We remark that, to some extent, results from Part B are proved again. But they are put in a new and less tied

jacket. Let {Ix> Ix

E

M)

be a Dirac basis in the sense of Definition II.v.1. For a

bra , we have by Theorem II.v.2 for t > 0 and all T, O < T < t (t) = ) =

REPRESENTATIVES OF KETS AND BRAS

333

We emphasize t h a t t h e previous i n t e g r a l expression does not depend on t h e choice of T , 0 < T < t. Thus w e a r r i v e a t t h e following i n t e r p r e t a t i o n of D i r a c ' s Parseval i d e n t i t y .

II.vi.1.

Theorem.

The formula
I x > < x IF> d u ( x )

M a s suggested by Dirac, can be i n t e r p r e t e d a s

where T , 0

T

< t , may be a r b i t r a r y for each t > 0 . I f


is a test bra

then we may t a k e T = 0 and i f IF> i s a t e s t k e t then we may t a k e T = t .

Proof. Let

0

The proof follows from previous c o n s i d e r a t i o n s .

J denote t h e isometry from

Dirac b a s i s {lx>

Y = J(X)

Ix

E

M}

into

L 2 (M,p)

which i s connected t o t h e

a s explained i n t h e previous s e c t i o n . P u t

and 8 =

c L2(M,p)

x

JU*.W e

have t h e following r e s u l t on represen-

tatives.

11.v i . 2 .

Coro Zlary

For each b r a


. and each k e t IF> t h e r e e x i s t r e p r e s e n t a t i v e s 0 and

Y in

Ty,8 such t h a t
I F>

In p a r t i c u l a r , i f

: t I+


I-

O(x,T) ' Y ( x , t - ? ) d p ( x )

.

i s a t e s t b r a , then we can t a k e

expression. So t h e r e e x i s t s cp

E

sY88 such

? =

0 i n t h e above

that

I f on t h e o t h e r hand IF> i s a t e s t k e t , then we can t a k e T = t. So t h e r e e x i s t s J,

E

s Y,8

such t h a t

334

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM


IF>

m.We observe 0

( x , t ) t+

( t )=

I-

M

O ( x , t ) $(XI du(x)

.

t h a t t h e f u n c t i o n s P : ( x , t ) H <x

I F > (t) and

I B > ( t ) a r e members of T Y,B' 1 B > ( 0 ) belongs t o sy,g and, if IF> i s a t e s t k e t then J, : x I+ <x I F > (0) belongs t o sy,B.

If

:


<x

i s a t e s t b r a , then cp : x I+ <x

Remark. I f


0

i s a t e s t b r a , w e have

I f IF> i s a t e s t k e t , w e have

Here T > 0 has t o be taken s u f f i c i e n t l y small.

Let t > 0 . Then we can w r i t e

with convergence i n

x.

Moreover,

f o r each T , 0 < T < t. Thus we o b t a i n

We want t o interchange i n t e g r a t i o n and summation. Therefore, w e show absol u t e convergence.

335

REPRESENTATIVES O F KETS AND BRAS

Interchanging i n t e g r a t i o n and summation y i e l d s

II.vi.3.

w.

Let t > 0 . Then f o r any k e t IB> and x

<x

T ,

0 < T < t , t h e X-valued f u n c t i o n

I B > ( t- T ) I x > ( T ) i s s t r o n g l y i n t e g r a b l e and

Moreover, t h e i n t e g r a l does not depend on T .

prooS. We only have t o show a b s o l u t e convergence. Therefore, consider t h e following e s t i m a t i o n

B o t h i n t e g r a l s on t h e r i g h t hand s i d e are f i n i t e .

II.vi.4.

0

Corollary.

Let IW> be a t e s t k e t . Then t h e r e e x i s t s to > 0 s u f f i c i e n t l y small such that

IW>(o)

=

i

<x I W >

(-T)

Ix>(T) d p ( x )

,

0 < ‘I < t

0 ’

M

Proof.

Since IW> i s a t e s t k e t t h e r e e x i s t s t

0

> 0 such t h a t IW> = e

-t A

IF>

3 36

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

for a certain ket IF>. From the previous lemma we obtain for 0 < u < t,

and hence IW>(O)

=

e-(tO-t)AIF>(t)

=

I

<x I F >

(0)

Ix>(tO-ul dp(x)

.

M Now take u

=

to -

T

with 0 <

T

< to and observe that

Remark. Corollary II.vi.4 expresses the usual expansion result that we derived in Section B.IV.2. From Lemma II.vi.3 it follows that IF> can be expressed as

So the following interpretation naturally comes up. II.vi.5. Theorem. The expansion

as suggested by Dirac can be interpreted as

where the integrals exist in strong sense and do not depend on the choice of T, 0 < T

t.

If IF> is a test ket, then we can take 7 = t in the above integral ex-

pression. Conjugate to the above interpretation we have

-U

REPRESENTATIVES OF KETS AND BRAS

11.vi.5'.

337

Theorem.

The expansion

a s suggested by Dirac, can be i n t e r p r e t e d a s

where t h e i n t e g r a l s e x i s t i n s t r o n g sense and do n o t depend on t h e choice of T , 0 < T < t .

If
0

pression.

II.vi.6.

Corollary.

L e t IF> be a k e t . Then t h e r e e x i s t s a r e p r e s e n t a t i v e '4

T

E

Y,B

such t h a t

IF> can be w r i t t e n a s

IF> : t b

Y (x,t

- T) Ix>(T)

dp(x)

,

t > 0

.

M Let IW> be a t e s t k e t . Then t h e r e e x i s t s a r e p r e s e n t a t i v e J,

II.vi.6'. Let


E

sY,B

such t h a t

Corollary. be a b r a . Then t h e r e e x i s t s a r e p r e s e n t a t i v e 0

E

T

Y,B

such t h a t


Let


: t t+

I

cp(x) < X I

( t ) dp(x)

.

E

sY,B

such t h a t

0

338

A

MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

Due to Corollary II.vi.6-6' it makes sense to call T

Y,B

the representation

space of the kets (bras) corresponding to the Dirac basis (representation) {lx>

I x E MI.

In particular,

sY,Bis the representation space of

the test

kets (test bras).

Representatives of linear operators Given a basis in a finite dimensional vector space E every linear mapping on E can be represented by a matrix with respect to this basis. The product of two operators B1 and B has the matrix [ B l o B21 which is derived by 2 matrix multiplication of the matrices [ B 1 and [ B 1. In his formalism Dirac 1

2

suggests that a matrix notion can also be defined for a set of basic kets and that the resulting matrix calculus has similar properties as the usual matrix calculus of linear algebra. Let {lx> Ix

E

M } be a Dirac basis and

J

denote the corresponding isometry

from X into L2(M,p). We put Y = J (x) and B = JAJ*. AS we have seen the space S y , B can be considered as a space of functions on M, i.e. sY,g contains precisely all functions x

H

<x I W > ( 0 ) where IW> is a test ket. Eack test ket

lW> has a representative cp : x

H

<x

I W > ( 0 ) in

sY,B

such that

The space T y , g can be considered as a space of functions on ket IF> has a representative @

:

M

x

(0,m).

Each

(x,t) H <x I F > (t) in Ty,g such that

In this section we derive the same kind of representation theorems for the four types of linear operators introduced in Section II.iv. For each pair (x,y) E M

x

M the sesquilinear tensor product ly> m Ix> is a E SXeX,AsA we can M given by

member of TXeX,AmA (cf. Section B.II.2). So to each @ link the function on M

x

3 39

REPRESENTATIVES OF L I N E A R OPERATORS

syey,BmBcan

In f a c t t h e space

be regarded a s t h e f u n c t i o n space which con-

sists of a l l f u n c t i o n s on M x M a s described by I I . v i i . 1 .

In D e f i n i t i o n

I I . i v . 1 we have introduced t h e double t r a j e c t o r y space TT

'

-

each TTXeX ;&I, IeA and a l l t , s > 0 we have 8 ( t, s ) t h e following d e f i n i t i o n makes sense.

1

E

TTXeX;AeI,Ie,A.

Dirac b a s i s { Ix>

E

II.vii.3.

=.

1

E

Then we d e f i n e i t s matrix

Ix

E

[el

with r e s p e c t t o t h e

M} by

b! and t , s > 0 .

where x , y

Let 0

XeX; Ae,7,leA* For SXeX,AteA. Therefore

Definition.

11.v i i . 2 .

Let 8

E

TTXeX;AeI,IeA

and l e t t , s > 0. Then we have

where t h e i n t e g r a n d is a s t r o n g l y i n t e g r a b l e X@X-valued function. The i n t e g r a l is independent of t h e choice of T , 0 < T < t and

0,

0 <

U

<

p r o o f . Let -

t > 0 , s > 0 , 0 < T < t and 0 <

U

belongs t o

Syey,BmB ( q L 2 (M x M , P 8 u ) ) .

we can e s t i m a t e a s follows

SO

< s . The f u n c t i o n

It follows t h a t t h e i n t e g r a n d

( x , y ) fi

c e ~ ~ ~( t (x >~< y, ~ ~( t)- T , s - u )

S .

340

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

is strongly integrable.

MxM

J

[exyl

I n a d d i t i o n we have

(Ix>
(T,O)

( t - T

.

, s - u ) du(x) dp(y)

The interchange ( * ) of i n t e g r a t i o n and summation follows from t h e a b s o l u t e convergence of dcE m

MxMl (. .. )

d , Ec I D

dp ( x ) du ( y )

. This

can be proved s t r a i g h t -

0

forwardly.

Replacing

x

Y

by

TTyay;wr,l@B. (O,-)

and

A

B

by

So each 0

E

i n Definition 1I.iv.l

TTy@yy;Br,raB

In p a r t i c u l a r Q ( t , s )E

Syay,mB,

be seen a s a function on

M

x

M.

t , s > 0. So f o r each t , s > 0 , n ( t , s ) can I t follows t h a t each 0 i n

a complex-valued f u n c t i o n on t h e space Q ( x , y ; t , s ) , x,y

onto

X

@

X

-+

Y

@

M

x

(0,m)

TT Y@Y;B@I,l@B

J*)Q

is

(0,~)W .e write

x

t h e function u n i t a r y mapping

Y i s e x t e n d i b l e t o a l i n e a r mapping from TT X@X;A@I,I@A

TT Yay; B l , l@B'

Hence each element Q @

x

M, t , s > 0 . For each 8

E

( J ~ J J: )( t~, s )I+

(J*

-

M

E TTXmX C ~ l x y ( t , sbelongs ) t o TTyayjB7,1aB. The

(x,y ; t , s ) t+ :

i s an yay-valued f u n c t i o n on

(O,-) which s a t i s f i e s

x

J o J

w e g e t t h e space

E

~ 6 ( t ,J*~ ,)

TTyay;Bl,laa

t,s >

o

.

d e s c r i b e s t h e m a t r i x of

TTXaX:Aal,I@A'

We a r r i v e a t t h e following r e s u l t s and conclusions.

REPRESENTATIVES OF LINEAR OPERATORS

11.vii.4.

Let

6

Theorem.

TTXaX;Amr,rmA.

E

34 1

Then t h e expression

a s suggested by Dirac, m u s t be i n t e r p r e t e d a s :

I

8(t,s) =

a.

[elxy(T,O)

( I X > < y l ) ( t- T

,S

-0)

dp(x) dp(y)

MxM

where the i n t e g r a l s converge s t r o n g l y i n choice of T ,

0 <

?

x

@

x

and do n o t depend on t h e

< t and u , 0 < u < s. &

For each t e s t k e t IW> t h e a c t i o n 0 of 0 on IW> can be expressed as

eIw>

b.

: s I-+

MxM

I

[elxy(T,U)


I W>(-T)

I x > ( s - u ) dp(x) d u ( y )

where T > 0 m u s t be taken s u f f i c i e n t l y small and 0 < converge s t r o n g l y i n

x.

u < s . The i n t e g r a l s

Thus we i n t e r p r e t D i r a c ' s h e u r i s t i c formula

C.

where t h e i n t e g r a l s do n o t depend on t h e s u f f i c i e n t l y small chosen T > 0 .

Proof.

Assertion a . has a l r e a d y been shown.

Let ( W > be a t e s t k e t . The ( s t r o n g ) i n t e g r a b i l i t y of t h e f u n c t i o n s (x,yj

H

) I W>(-T) [elXY ( 1 , ~
I x > ( s - a ) and y

W

[el XY ( T , u )

6

M


I

W>(-T)

follows from t h e observation t h a t

vs>o

: x

I+

II I x > ( s ) IIx

v t , s > 0 : ( x , y ) I+

,

IC81xy(t,s)I ,

x

x,y

E

M

,

i s square i n t e g r a b l e

,

i s square i n t e g r a b l e

342

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

37>0

: y -+


,

I W>(-'I)

y

E

,

M

i s square i n t e g r a b l e

and from a s t r a i g h t f o r w a r d a p p l i c a t i o n of Holders i n e q u a l i t y .

we have defined t h e a c t i o n of

6

by

where t h e expression does n o t depend on T. Applying a. we o b t a i n f o r 0

C

T C 'I'

and 0

C

u

s

C

(81W>) (s) =

J MxM

I

=

L8l

(T

,u) ( Ix> < y l ) (T'-T,

S-0)

~ W > ( - T ' )du (x)du (y) =

XY

[ B l x y ( ~ , u )
W>(-T)

Ix>(s - u ) dv(x) d u ( y )

.

MxM

From t h e d e r i v a t i o n it i s c l e a r t h a t t h e i n t e g r a l expression f o r ( 8 1 W > ) ( s ) does not depend on T and Finally, f o r x'

E

U.

M we have

W e r e w r i t e the r e s u l t s of t h e preceding theorem i n terms of r e p r e s e n t a t i v e s .

II.vii.5.

Corollary.

Let €IE TTXeXjAa7 ,leA and l e t I W > be a t e s t k e t . Then t h e r e e x i s t s

2

TTy@y;BeI,lgB and

Cp

'Y,B

such that

343

REPRESENTATIVES OF LINEAR OPERATORS

<XI

9 IW>

: s I+

M

i

g ( x , y ; 7 , s ) ( e T Ec p ) ( y ) d u ( y )

Here T > 0 must be taken s o small t h a t e

Proof.

We t a k e g ( x , y ) ( t , s ) = C9l XY

TE cp

E

.

sy,B.

( t , s ) and cp(y) = ( O )

w i t h x,y

M

E

0

and t , s > 0.

I n t h e remaining p a r t of t h i s s e c t i o n w e prove s i m i l a r r e s u l t s f o r t h e o t h e r t h r e e types of l i n e a r o p e r a t o r s a s presented i n Section 1 I . i ~ .

spaces SSX@X;A@I J@A' TSX@X;A@I, T @ A and TSX@,X;I@A,A@I a r e subspaces of So t o t h e elements of each of t h e s e spaces we can l i n k a

TTX@X;A@I,r@A.

matrix with r e s p e c t t o t h e Dirac b a s i s { l x >

Ix

MI.

E

From t h e c h a r a c t e r i z a -

t i o n s of t h e s e spaces we o b t a i n t h e following p r o p e r t i e s .

-

Let

0

SSX@x;AgI,T@A.

E

each x,y

E

M

Then t h e r e e x i s t s t

can be extended t o a f u n c t i o n on

SSX@X;A@T,leA

belongs t o

-

Let

the function

SSy@y;EgI,rmpB. The

SSY@Y;E@1,7eB'

Onto

^K

E

> 0 and so > 0 such t h a t f o r

2 ( t , s ) E IR

I

t,s > 0

,

t > -to, s > - s

2

u n i t a r y operator J

We have ( J @ J ) z =

TSX@X;A@T,I@A.

,

< i ( t , s ), ly> $ I x > >

C0lXY : ( t , s )

E

0

the function

@

E

1.

F o r each

J maps SS

X@X; A@T ,T@A

2.

Then f o r each p a i r ( x , y )

0

M

x

M the function

[ K l x y can be extended i n t h e following way Vs,o

3ts>0 V t>-ts

:

CKlxy(t,s)

=


s ) e t S A , ly>

@

T

Ix>>

For each s > 0 t h e choice of ts does n o t depend on ( x , y ) . The u n i t a r y operator J

@

J maps TSX@X;A@T, l @ A onto TSY@Y,E@I,r@E. we have

(J@J)k : -

Let

L

E

( x , y ; t , s ) t+

TSX@X;I@A,A@l.

CKI

XY

( t , s ) = _K(x,y; t , s )

Then f o r each p a i r ( x , y )

E

M

x

.

M t h e function

.

A MATHEMATICAL INTERPRETATION OF DIRAC’S BRACKET FORMALISM

344

[Llxy

can be extended i n t h e following way

:

V t > O 3st>0 vs>-st

CLI XY ( t , s )

=

s A

<e t

L(t,

s+ st),ly>

@

T

Ix>>

.

For each t > 0 t h e choice of s t does n o t depend on ( x , y ) . The u n i t a r y

J

operator

@

J

maps TS

X@X;l@A,A@I

TS’i@Y; l@B,B@l’ We have

Onto

The above defined e x t e n s i o n s make sense. Therefore w e r e c a l l t h e following definitions.

:@(t,s)

E

SX@x,ABA

3t0 >o,s 0>o V t > - t O , s > - s 0

.

W e a r r i v e a t t h e following r e s u l t s .

II.vii.6.

Theorem.

Then t h e r e e x i s t s to > 0 and s > 0 such t h a t f o r a l l ~ e t E SSxmx;~@i, i@~0 0 < T < t o , 0 < a < so and t > -T, s > -a

a.

-

@ : ( t , s ) I+

J

[@lw(-T,-U)

(Ix>
( t+ T

,s + o )

dp(x)dp(y)

MxM

where t h e i n t e g r a l s converge s t r o n g l y i n

x

a

x.

For each k e t IF>

REPRESENTATIVES OF LINEAR OPERATORS

345

where 0 < T < t o , 0 < a < so and s + a > 0 . The i n t e g r a l s converge s t r o n g l y

in

x

and a r e independent o f a and

<XI

C.

8 IF> :

S H

T.

J [@I

M

The x-th component of


(-T,S)

I

F > ( T ) dp(y)

XY

where s > -so. The i n t e g r a l s a r e independent of

Proof.

There e x i s t s t

for certain

6

E

TT

0

T,

0 <

T

< t 0'

> 0 and so > 0 such t h a t

X@X iA @ l ,l @ A '

G(t,s) =

IF> i s given by

I

So f o r t > 0 and s > 0 f i x e d , w e have

~Blw(T,U)

( I x > < y l ) ( t- T

,s - 0 )

du(x) dp(y)

MxM

where t h e i n t e g r a l converges s t r o n g l y i n 0 c

T c

t and a , 0 < u

I

=

C

x

@

x

and i s independent of

T,

s . So we g e t

.

~81xy(-t0+T,-~0+U (Ix>
MxM

Herefrom t h e wanted r e s u l t a . can be derived. By t h e usual arguments it can be shown t h a t t h e f u n c t i o n s

a r e i n t e g r a b l e f o r a l l p e r m i t t e d values of t h e parameters.

.

The a c t i o n of 8 on IF> i s defined by

where t h e expression i s independent of

T',

0 < T' < to. Because o f a . we

346

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

have f o r a l l s >

-u, 0

where w e t a k e 0 C Let x '

t

SSXeX;AeI,IeA and

11.vii.8.

i?

c

exists t

a.

C T C

to. Thus b. follows.

Corollary.

t a t i o n s . There e x i s t

Let

u < s 0'

M. Taking t h e b r a c k e t of < x ' l and @IF>w e g e t

E

11.vii.7. Let

T'

C

2

E

l e t IF> be a k e t . We have t h e following represen-

SSyey;BeI,ZoB and

H

6

T

such t h a t

Y.8

Theorem.

TSXeX;AeI,IaA.

Then f o r each s > 0 and each

> 0 such t h a t f o r a l l T , 0 <

R(t,s) =

MxM

I

CKlxy(-T,U)

?

< t

( I x > < y l ) ( t+ T

H e r e each i n t e g r a l converges s t r o n g l y i n

x

Q

u, 0

< a < s, there

and a l l t , t + ?> 0

x

,s

-U) dpod dp(y)

.

and does n o t depend on t h e

347

REPRESENTATIVES OF LINEAR OPERATORS

choice of

T

and a. 1

For each k e t IF> t h e a c t i o n KIF> of K on IF> i s given by

Here f o r each s > 0 , u with 0 < u < s must be f i x e d and then T with

0 < T < to. The i n t e g r a l s i n b. converge s t r o n g l y i n

x

and do not depend

on T and u . The x-th component of KIF> i s given by

C.

Here w e t a k e 0 <

T

< t

. The

i n t e g r a l s i n c . do not depend on

T.

Strong i n t e g r a b i l i t y follows from t h e observation t h a t t h e f u n c t i o n s

and

are i n

s

In order t o prove Relation a . , l e t s ' > 0 and l e t u with 0 <

(J

of t h e

< s ' be

f i x e d . Then f o r a f i x e d T, 0 < T < t , t h e mapping

belongs t o

ssXeX;AeI,IeA.I t

follows t h a t

J

1

K ( t - T

,S'+U)

=

[Klxy(-T,U)

(Ix>
(t,s'I dp(X)du(y)

MxM

f o r a l l t , s ' > 0 ( c f . Theorem I I . v i i . 6 ) .

Hence f o r a l l t , s > 0

348

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

Thus Relation a . has been shown.

K

The a c t i o n o f

where 0 < T'

C

on IF> i s given by

tS. The d e f i n i t i o n of

KIF>

does n o t depend on t h e choice of

T'. For each s > 0 and u, 0 c u < s we have

Corollary.

II.vii.9.

2

~ e t E ~sXeX;Aei,ieA t a t i o n s . There e x i s t s

and l e t IF> be a k e t . W e have t h e following represen-

5

E

TS

Yay; bl ,l @ B

Here having f i x e d s > 0 and 0

-

II.vii.10. Let L

E

C U C s,

and H

E

we take 0

T

Y ,B

C T

such t h a t

C

t

and

t + T

> 0.

Theorem.

TSXeX;IeA,AeI.

Then f o r each t > 0 and each T , 0 c

T

< t , there

0

349

REPRESENTATIVES OF LINEAR OPERATORS

exists

s

T

> 0 such that for all 0 < u < sT and s , s + u > 0 ,

Here the integrals converge strongly in

x

Q

x.

I

For each test ket IW> the action of

Here first

T

L

on IW> is given by

> 0 must be chosen sufficiently small, next u with 0 < u < s

and, finally, s > -U. The integrals in b. converge strongly in not depend on

T

x

T

and do

and U.

The x-th component of LlW> is given by

C.

Here we take

T >

0 sufficiently small and

s

> -sT, The integrals in c. are

independent of T.

Proof.

We leave the proof of this theorem to the reader. It is a repetition

0

of previous arguments. II.vii.11. Corollary. Let

. L

E

TSXQX;IQA,AaI

and let IW> be a test ket. We have the following re-

presentations. There exist cp

E

sy , B

and

r ' T S y ~ y ; I @ B , B g lsuch that

Here first T must be chosen sufficiently small, next 0 < u < sT and finally s >

-u.

Ll

A MATHEMATICAL INTERPRETATION O F D I R A C ' S BRACKET FORMALISM

350

We r e c a l l t h a t t h e v e c t o r spaces TS X ~ X ; A ~ IZ , ~ and A T S ~ a ~ ; ~ are e ~ , ~ a ~ where t h e corresponding product o p e r a t i o n s are d e f i n e d as follows.

-

-

Let

1

K 1 ,K 2

TSXeX;Ael,leA. Then

E

1

-

K, * K 2

w e define

by

Here t h e product on t h e r i g h t hand s i d e i s j u s t t h e o p e r a t o r product i n

x

t h e H i l b e r t space

0

x

of Hilbert-Schmidt o p e r a t o r s . F u r t h e r , f o r each

s > 0 we have t o t a k e 0 < T < t:') I

follows t h a t t h e product K 1

(K1 K,)

*

I

=

-

and next any t > - t ( 2 ) Then . it

1

-K2

belongs t o

TSXeX;A~l,18A.W e

have

.

K, K,.

c, .I2

:

( t , s )H

12(t,-u) .

+,S)

Here f o r each t > 0 w e f i x o with 0 < o < sL2) and then any s > - s . . I

The product

L l L, 9

belongs t o

TSX8x;18A,A81 and ( L , L,)

I

=

L,

(1)

- 0

1

*

L,

.

.

I t i s a n a t u r a l question whether t h e r e e x i s t s a mathematical i n t e r p r e t a t i o n

of Dirac's h e u r i s t i c formula ( c f . Ch. I , Formula 1 . 2 . 2 5 ) .

The next theorems y i e l d such an i n t e r p r e t a t i o n .

Theorem.

II.vii.12. -

Let

I

K 1,K 2

E

TSX8X;A81,18A.

Then f o r a l l x . y

Here f o r each s > 0 w e m u s t t a k e 0 <

prooj. The o p e r a t o r J

8

J

maps TS

'I

M w e have

E

< t ( l ) and n e x t t >-t

XeX;A@I , I @ A

Onto

1

( ( J @ J ) K j ) ( x , y t; , s ) = C K . 1

3 XY

(t,s)

,

(2)

.

TSy8y;B81, 1eB' where j = 1,2

.

351

OBSERVABLES

I

2) so small that K (-T,s) E S x @ , x , A ~ A . For a l l t > - t( T 1 SXmx,AmA. Now t h e wanted r e l a t i o n e a s i l y follows, because

L e t s > 0. W e f i x T >

.

w e have K 2 ( t , r )

E

z1.z2

t h e product

o

of t h e e l e m e n t s z 1 p

z

E

s

i s defined by

The following r e s u l t can be proved s i m i l a r l y .

Theorem.

II.vii.13. .

Let

.

I

11 , L 2

e

TSXoXjI@A,A@7. Then f o r a l l x,y

E

M w e have

Here f o r each t > 0 w e must t a k e 0 < u < s ( ~ and ) then s > -sA1), t

Proof.

0

The proof i s l e f t t o t h e reader.

I Observables I According t o Dirac a real dynamical v a r i a b l e i s n o t n e c e s s a r i l y a n observa b l e . Only r e a l dynamical v a r i a b l e s with a t o t a l s e t of e i g e n s t a t e s can be measured. Each measurement of an observable f o r c e s t h e system i n t o an eigens t a t e of t h i s observable and t h e r e s u l t of t h e measurement i s t h e eigenvalue corresponding t o t h i s s t a t e . So an observable i s completely determined i f w e give i t s set of e i g e n s t a t e s with t h e corresponding eigenvalues. Theref o r e , i n s t e a d of considering an observable a s a s p e c i a l r e a l dynamical v a r i a b l e , we can as w e l l consider an observable a s a t o t a l s e t of orthogonal s t a t e s {la>

IxCl l a

E

Ia

E

/A)

t o g e t h e r with a s u i t a b l e s e t of r e a l numbers

4).

We have i n t e r p r e t e d D i r a c ' s notion of t o t a l s e t of orthogonal s t a t e s by means of t h e notion of (canonical) Dirac b a s i s . A s w e have seen each Dirac b a s i s i s f i x e d by a Federer measure space (M,u) and an isometry

J from

x

i n t o L, ( M , p ) .

In p a r t i c u l a r , i f J i s a u n i t a r y o p e r a t o r (J*J = JJ* = l ) , then t h e c o r r e sponding Dirac b a s i s (15>)

r e p r e s e n t s a commutative a l g e b r a of observCEM

a b l e s : Consider a r e a l valued Bore1 f u n c t i o n cp on M which i s bounded on

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

352

bounded Bore1 sets. Then to each 15> we link the number cp(S),

6

E

M. We can

represent this observable by a self-adjoint (unbounded) operator T

in

cp

x.

T Q is formally defined by

A

sound mathematical definition is given in Chapter B.IV. In fact, T

J MPJ*, where M denotes the maximal operator of multiplication by L, (M, p )

.

cp

From Chapter B.IV it also follows that each self-adjoint operator

cp

P

cp

equals in

in

x

I X E o(P), 1 5 j < m x +l} which consists of candidate eigenkets. (Here m X denotes the multiplicity of X E o(P).) It admits a Dirac basis {IX,m,,j>

follows that each self-adjoint operator in

x

represents an observable; it

induces a Dirac basis which is labeled by its spectral points counted according to multiplicity. Instead of one self-adjoint operator

P we

strongly commuting self-adjoint operators

can consider a finite number of

( Pl,...,P 1.

A l s o , in this case

there exists a Dirac basis

...,Pn).

which consists of joint candidate eigenstates of the n-set (PI, particular, if the whole spectrum a(?,,

...,P

)

then the observables represented by the n-set Pl,...,P set. It means that if X = (Al,

...,A

)

In

has joint multiplicity 1 , establish a complete

is the result of a measurement, then

the system is with certainty in the state lA>. In the current mathematical descriptions of quantum mechanics it is assumed

that the set of observables is in one-to-one correspondence with the set of self-adjoint operators in a Hilbert space

x.

But if we consider a specific

quantum dynamical system we do not need all self-adjoint operators. To each quantum system one can link a class of relevant observables, e.9. energy, momentum and spin. Many times this special class of observables can be seen as the infinitesimal representation of a Lie algebra corresponding to a unitary representation of a Lie group. Then also the algebraic relations of the observables are involved. Now we use the freedom in choosing the non-

THE HARMONIC OSCILLATOR

353

negative self-adjoint operator A in order to get a ket space T

X,A

on which

the self-adjoint operators representing the relevant observables, act continuously. So by Kernel theorem K4 they are part of the algebra

TS~m~;~mi,~a~* In the next section we discuss two illustrations of the above described phenomenon.

The harmonic oscillator The quantum system which is known as the harmonic oscillator is described by the dynamical variables q and p. Here q is called the position and p its conjugate momentum. The Hamiltonian operator, i.e. the dynamical variable

of the energy of the system, which determines its dynamics, is given by

Here m denotes the mass of the oscillating particle and w 2n-times its frequency. For simplicity we take the constants normalized, m =

W

=

1.

Besides the variables p and 4 we consider the complex variables =

1 (p+iq) and

A

nn

=

-

I- =

1 -

A

(p-iq). We have

12. (p+iq)(p-iq)

=

T1 ( p2 + q 2 +

i(qp-pq)) =

where we have inserted the commutation relation pq - q p = -i. Similarly we obtain

?n

=

1

ff + 2

. Hence -n q - n q-

= 1

and, also, nff-ffn

=

n.

By induction it can be proved that

-nrl n

nn-1 - q n = n n

Let X be an eigenvalue of

H

. with corresponding eigenket I X > .

Then we have

A MATHEMATICAL INTERPRETATION OF DIRAC‘S BRACKET FORMALISM

354

is positive. So A t 4 and A

The variable n:

=

5 if and only if ;IA>

=

0.

In addition we have the relation

So starting with the eigenket I A > with eigenvalue A, we arrive at the -2 eigenket nlA> with eigenvalue A - 1 , next 0 lA> with eigenvalue 1 - 2 , etc.

Thus we obtain the eigenvalues A,A-l,...,A-n with n 1 A-n 2 2

X

and

-

(n+l)

5

-21

IN such that

E

.

by n and arguing similarly we obtain the eigenvalues

Replacing

X,A+l,X+2,... with corresponding eigenkets I A > , r l l X > , q

2

[A>,

... .

Now we get

an infinite number of eigenvalues. So if we assume that there is a ket Ic> with the property that ilc>

1

2

’

2

’

2

2 2

’

=

0, then we see that the half-odd integers

--

are eigenvalues of ff with corresponding eigenkets

We have nnnlc> kets nnlc>, n

n- 1 nn Ic>. This shows that

=

I-

or

applied to one of the

IN u { O } , yields a ket which is dependent on these kets.

E

Since we assume that the harmonic oscillator is completely fixed by the

-

operators n , n and ff, all dynamical variables for this system are expressible in terms of n and

17.

So the kets nnlc>, n

E

IN u { O } establish a total set,

i.e. ff itself is a complete set of commuting observables. We normalize the ground state, = 1. Then we obtain


-n n

n Ic>

=

n!

1 nnlc>,

n

=

II

Jn!

... are normalized basic kets of

0,1,2,3,

representation diagonalizing ff. For any ket n

E

IN

u {O} such that

IX>

the

there are numbers x n’

355

THE HARMONIC OSCILLATOR

n

m

c

Thus t h e k e t IX> corresponds t o a formal power s e r i e s variable

n.

x n=O n

rl

Jnl

in the

I t means t h a t we may c o n s i d e r t h e k e t Ic> a s a s t a n d a r d k e t .

The approach by means of formal power s e r i e s i s due t o Fock.

The usual mathematical approach t o t h e above described harmonic o s c i l l a t o r i s t h e following. Consider a separable H i l b e r t space

b a s i s (v,) nE ~

" ).1

x

with an orthonormal

Define t h e unbounded l i n e a r o p e r a t o r s

lnzo

m 2 ) = D(L-) = {f E X n (f,vnIXI < 17 17 Next we d e f i n e t h e p o s i t i v e s e l f - a d j o i n t o p e r a t o r L H by

where w e t a k e

D(L

where w e t a k e

D(LH)

= If

E

X

I

nzo m

n2

I

I ( f , v n ) X 1 2<

L

n

and L- by 17

-1.

-1.

Then t h e following a l g e b r a i c r e l a t i o n s a r e v a l i d

L-1

nrl

=

L L - = L rlrl

L-L

rlrl

,

LH + $1

H

,

-$l

- L QL-n

=

7

.

Also we i n t r o d u c e

A s t r a i g h t f o r w a r d computation y i e l d s

L L - L L =-iz. P Q

ClP

NOW f o r t h e k e t space w e t a k e t h e t r a j e c t o r y space

TX,L+

H

. The corresponding

356

A MATHEMATICAL INTERPRETATION O F DIRAC'S BRACKET FORMALISM

a n a l y t i c i t y space

L

s X , LH5

i s t h e j o i n t a n a l y t i c i t y space of t h e o p e r a t o r s

and 7 . (Cf. S e c t i o n B . I I I . 4 . ) From Chapter B . 1 1 it follows t h a t L 4 P a r e e x t e n d i b l e l i n e a r mappings. (Compute t h e matrices!) Moreover, 4 T X , L t i s a nuclear space. So T 'r c o n t a i n s t h e e i g e n k e t s of L and L We x, LH P 4'

P'

and

L

L

compute t h e eigenvalues and eigenkets of

L ci

be an eigenvalue of we g e t

I-. a.

L

P

and

L

Q

a s follows. Let x

1

1

-

-an

x

IR

1 a Iv,n>. Then n=O n

= o ,

al

+

E

m

with eigenket Ix>. We w r i t e Ix> =

an+ 1 + e a n + 2 = 0 ,

n=0,1,2

,....

We compare t h i s t h r e e term recurrence r e l a t i o n with t h e one s a t i s f i e d by t h e Hermite polynomials. To t h i s end w e p u t

a

n

AT) a .

= (2'"

This y i e l d s f o r a l l n

E

IN

U

(01

o r , equivalently,

2(n+l)

[al

-

gn -

-

2xa

2xao = 0

-

n+l

+ a n+2

= o ,

.

The l a t t e r t h r e e t e r m recurrence r e l a t i o n i s s a t i s f i e d by t h e Hermite polynomials. Thus we o b t a i n

with

Here t h e f u n c t i o n x H c ( x ) can be taken such t h a t t h e k e t s Ix>, x E IR a r e 1 e-+x2, x E IR. Then w e have

normalized. Therefore w e t a k e c ( x ) =

J;;

THE HARMONIC OSCILLATOR

357

m

]

a ( x ) a d x = 6 m nm

.

The eigenket Ix,q> is given by m

Ix,q> =

for each x

E

1 an(x) Iv,n>

n=O

IR. The set {lx,q>

'n,m<m

4 .


I

Ix

E

IR) is a Dirac basis,

x,q>(O) <x,q I v,m>(O)dx

=

6nm

.

IR

Similarly, we can prove the eigenvalue equation L lk> = klk> for each k

E

P

IR. We obtain

Ik,p> =

1

n=O

in an(k) Iv,n>

It is clear that the set {lk,p>

Ik

E

. IR) is a Dirac basis also. We arrive

at the following representations.

Position representation Starting from the Dirac basis { Ix,q> I x

IR) which consists of eigenkets

E

of the self-adjoint operator Lo, each ket has a representative in the space

,($(--$ + .2))+ . This can be seen as follows. The linear isometry

TL2 (IR) U from

x

onto L (IR) is given by 2

Since the orthonormal set {[a

1 In

E

IN u { O ) ) is total In L2 (IR), the op-

(-5 .

erator U is unitary. In addition U L,,U*Ca the differential operator

+ x2)

of multiplying by the identity function x d the differential operator -i dx

.

1

=

(n+$)[a

1. So U LHU* equals

" *

nFinallyf f.l Lq k+

x and

u L u* P

= =

2

the operator

P where P denotes

358

A

MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

Momentwn representation The linear isometry

v

from

x

into L2 (IR) corresponding to the Dirac basis

{lk,p> ( k E IR] is given by m

Vf

1

=

in(f,vn)X [an]

,

f c

n=O

Now let IF IFCa

1

:

L2(IR)

1, n

= in[a

-f

E

x .

L2 (IR) denote the Fourier transformation. Then IN u {O}.

It follows that

v

= IFU.

So the momentum

representation can be derived from the position representation by means of the Fourier transformation. We have

V L v* = I F Q I F *

Q

=

P,

V L v*=P'pIF*=o,.

P

We mention a third elegant representation which is not based on the choice of a particular Dirac basis. The Bargmann-Segal-Fock space F. The Hilbert space F consists of all entire functions f for which

dxdy <

m

.

2 IR Now we define L f rl

monomials u n have

:

z

I+

:

1 z W z f(z) and L-f : z I+ f'(z). So LH = z d + 7 . The rl n z , z E Q: establish an orthonormal basis in F. We

-

Jn:

LH un = (n+$)un

,

n

E

IN u I01

.

Forthetrajectory space of representativeswetake T space consists of entire functions, also. Indeed G E T

F.(z

2 + i)

F,(z

2

+

iff 4 is an entire function

m

4(z) =

1

n=O

n a Z-

Jn:

with Vt,o

$) . This ,

:

a

=

tn

O(e

)

.

ANGULAR MOMENTA

359

The k e t s Ix,q> a r e represented by

,

Q x : z t + - e1x p ( - $ z 2 + f i x z )

z E (I:,

J;; f o r each x

E

IR and t h e k e t s Ik,p> by : z t+

Yk

f o r each k

E

1 exp(-$z2 + i f i k z ) ,

J;;

z t C

We observe t h a t t h e s e t s { e x ( x

IR.

canonical Dirac bases i n t h e t r a j e c t o r y space T of Section B.IV.2.

(t,z) t+

E

F,(z

, IR) and {Yk / k d +

+)

E

IR) a r e

i n t h e sense

The corresponding t r a j e c t o r i e s a r e given by

1 exp(-$e -2t

z 2 t f i e - t xz)

Jr; and

Consider a p a r t i c l e described by t h e t h r e e C a r t e s i a n c o o r d i n a t e s X and t h e i r conjugate momenta p l , p 2 ,

1' x 2 ' x3

p,. Their angular momenta m 1' m 2 ' m3

a r e defined by

The dynamical v a r i a b l e s

ml, m2, m3 s a t i s f y t h e following commutation re-

lations

Cm,,mll

=

m2 ,

[ml,m2]

=

m

I

Cm2,m,l

=

m1

.

We introduce t h e q u a n t i t y 2

2

B = m 1 + m2

t m

2

3 '

m l , m2 and m 3 . Let u s assume t h a t our quantum system i s completely f i x e d by t h e dynamical v a r i a b l e s ml, m2 and m3. Then f3 com-

Then B commutes with

mutes with any observable and t h e r e f o r e 6 i s a number.

360

Put

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

n

=

-

ml - i m 2 , :q

m 1 + i m 2 . We have

q =

- m2 + m

= @

3

3

and, s i m i l a r l y ,

-

2

n r l = @ - m3 - m 3 -

I t follows t h a t

A l s o we assume t h a t

m3 is

an observable. Let m;

be a n eigenvalue of

m 3'

Then we have

Since !q

i s a p o s i t i v e observable w e o b t a i n

(mil2

-

+

mi

o

t

or

2

B+kt(m;-LI) Thus we s e e t h a t @ t k > -$.

k + %= (6

+ 5)'

=

( m 2l + m 22 + m 32 + h ) ' so

that

I t follows t h a t

mi

k t

with e q u a l i t y m;

and

m3 So i f

-+. Put

.

m i - 1 -#

0

t -k

-

= 0 and m;

= k i f qlm'>

3

Im;>

-k,

k t 0

and

= :(

m3+

);

Im;>

=

= -k i f rllm;>

(mi + 1 )

Im;>

= 0. W e observe t h a t

.

> i s an e i g e n k e t of m 3 with eigenvalue m 3' - 1 , and, i f 3 i s an e i g e n k e t of m3 with eigenvalue m;+1, e t c . Gathering qlm

m' + 1 # k , q l m > 3 3 a l l r e s u l t s we s e e t h a t t h e eigenvalues of m3 a r e t h e r e a l numbers

36 1

ANGULAR MOMENTA

-k,-k+l ,-k+2

,..., k - 1

,k

.

From considerations based on the symmetry of the problem, it follows that the eigenvalues of m 1, m2 and m3 are the same. Let Imax> be an eigenket of m3 belonging to the maximum eigenvalue k so that

-

qlmax> =

o

We form the sequence of kets Imax>, qlmax>

,..., 11 2k Imax> .

These kets are eigenkets of m3 with corresponding eigenvalues k ,k-1

,...,-k + l

,-k

Since we have assumed that q and

.

n are the dynamical variables which de-

scribe the quantum system completely, it follows that m3 itself establishes a complete set of commuting observables. Next we introduce the magnitude of the angular momentum vector

m

=

(ml,m2,m3) to be k

=

( p + & ) ' - 4. From the previous discussion we de-

rive that the only possible values of k are the half integers 0

, T1 , 1 , 3~ , ,.... 2

(we note that it is more convenient to take k instead

of B4 because the possible values for 6'

are a more complicated set of

numbers. ) In general, more dynamical variables than q and

are involved in this sys-

tem. Then 8 and hence k is no longer a number but an operator. We assume that 8 and hence k is an observable. We call k the observable of the magnitude of the angular momentum m. Hence {k,m3} is a set of commuting observables with eigenstates Ik' ,m'> where 3 k' = 0

, T1 , 1 , T3 ,.. .

and

m' = -k' 3

, -k' + 1

,. . ., k' - 1 ,k .

We have :lk',m;>

=

Ik',m;+ 1 >

if m'3 < k'

qlk',m;>

=

Ik',m;- 1>

if mi > -k'

,

and

.

-

In addition, nlk',k'> = 0 and nIk',-k'> = 0. The observables k and m

3

are not independent of each other but partly func-

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

362

tions of each other. The above described quantum system, which consists of the angular momenta

of one particle, is closely related to the rotation group 0,.

0, is the

compact Lie group of all 3 x 3 orthogonal matrices. The corresponding real Lie algebra is spanned by the vectors

jl,j2 and

j3

which satisfy the

bracket relations

Cjl,j21

=

[j2,j31 = j l ,

j3 ,

[j3,jll= j 2

.

Usually, one considers the following unitary representation of the group 0 3 s2 denote the unit sphere in IR3 Then for each R E 0, we put

.

Let

Then U(R) is a unitary operator on L spherical coordinates (O,cp),

0

2

(s2 , sin BdOdq) , where we

use the

2n. Correspondingly the

S 0 5 ~ l , 0 5 ~p 5

infinitesimal representation of its Lie algebra is spanned by the skewadjoint operators sinrps a + coscpcotQ-a

au(jl)

=

aucj,)

= -COST-

au(j3)

=

,

am

a + ao

a -ae

a

sincpcoto-

am

*

As our ket space we take the trajectory space T

a A O = - - - sin O a0 i.e. A

sino-a

ao -

L2(S2) ,A;

'

where

sin2 O- a L

am2

0 is the Laplace-Beltrami operator of s2. The corresponding analyti4 is just the analyticity domain of the representation SL2 (S2) ,A0

city space

u.

(Cf. Section B.III.4.)

The kets Ik' ,mi> are represented by the spherical harmonics Y(2mi) ( 0 , ~ ) where mi

=

0 ,+ , 1 ,

3

, . ..

and k'

=

-mi , -m' + 1 3

,. . ., mi.

k'

We have

ANGULAR MOMENTA

The set {Y (2m;) 12m; k' 2 ma1 basis of L 2 ( S 1.

E

IN u { O } , k' = -m;

36 3

, -m'3

We want to involve also the coordinates X1,

+ 1 ,...,mi} is an orthonor-

x2

and X3 of the particle in

our set of observables. so we want to consider the observables

x1 , x2 , x3 , m l m2 m3 I

.

These observables satisfy the following commutation relations.

[ml,m21

=

m

,

Cm 2' m 3l = m

Cxl,mll

=

o

,

Cx1,m2]

C X 2 .m 11

=

ix

,

C X 2 ' m2 l = o

,

C X 2'

Cx3,m1l

=

-ix

Cx3,m21

,

Cx3,rn,l

= ix

=

ixl

, I

Cm3,m1l

=

m2

[xl,m31

=

-ix2 '

'

m 3l = - i x1 ' =

o

.

This system of observables is related to the Euclidean motion group E,, i.e. 3 the group of translations and rotations of IR This group can be repre-

.

sented by the 4 X

A

4

matrix group

unitary representation of

E3

3

on L (IR ) is given by 2

The corresponding infinitesimal representation of the Lie algebra A ( E spanned by the skew-adjoint differential operators

3

)

is

364

A MATHEMATICAL INTERPRETATION OF DIRAC'S BRACKET FORMALISM

These operators represent the observables x 1, for the ket space we could take T

AE But A

E

+ =

2

X2,

X3,

4 where

ml, m 2 and m3. Now

L2 ( m 3 ),AE

- +Q + +M - +M .

has continuous spectrum, whence T

4 is not a nuclear vector

L, ( m 3 )I AE

space. In order to obtain a nuclear trajectory space, we extend the Lie algebra to a ten-dimensional Lie algebra spanned by iQl I iQ2 , iQ3 ,M,

, M, , M, , iP1 , iP, , iP3 , i7

I

This ten-dimensional Lie algebra is the infinitesimal representation corre3 sponding to the unitary representation of the Lie group H 8 0, on L 2 ( )~.

(H,

8

0, is the semidirect product of the Heisenberg group and the ortho-

gonal group.) This unitary representation is given by

where

a,b

3 IR , R

E

E

0

3

and c

E

IR.

The analyticity domain of the unitary representation T equals

where

So in spherical coordinates we have

3

The normalized eigenfunctions of the operator AHo in L (IR 2

where n,2m

E

IN u { O }

and k

=

-m

, -m + 1 ,. .., m

)

. Here we put

are given by

365

ANGULAR MOMENTA

with L y '

t h e n-th Laguerre polynomial of o r d e r k + $ . We have

'HO

( 2 m ) = (4n En,k

+

I t follows t h a t sL2(IR3) , A 4

t r a j e c t o r y space T

HO 5

L2(m3) AHO I

4k

+

3

+

Zk(2k + 1 ) )

.

EI(1):

i s n u c l e a r . So a s u i t a b l e k e t space i s t h e '

Some comments and b i b l i o g r a p h i c a l notes For some i d e o l o g i c a l motivation of t h i s c h a p t e r and t h e next one w e r e f e r t o t h e Preface. Because of explanatory purposes t h e n o t a t i o n of t h i s c h a p t e r i s r a t h e r heavy. B u t , once t h e mathematical

j u s t i f i c a t i o n has been understood D i r a c ' s

e l e g a n t o r i g i n a l b r a c k e t n o t a t i o n can b e maintained. I t i s a minor flaw t h a t Dirac denotes s c a l a r m u l t i p l i c a t i o n of k e t s a s a l e f t m u l t i p l i c a t i o n . I t would be more c o n s i s t e n t t o w r i t e IA>a i n s t e a d of alA>. The f i r s t i d e a s regarding o u r mathematical i n t e r p r e t a t i o n of t h e genuine Dirac formalism appeared i n [E 11. In LEG 4 1 ,

[EG 51 and LEG 91 t h e theory

g o t f u r t h e r developed. E a r l i e r r e f e r e n c e s on D i r a c ' s formalism a r e t h e following: [An],

LBO],

[ J a l , [GV 41, [Me11 and [Ro].

For H i l b e r t spaces of almost p e r i o d i c f u n c t i o n s we r e f e r t o [Wei 11 and [AcG].

In LAG 1-22

so-called P a r t i a l Inner Product spaces a r e introduced. In such

a s e t t i n g t h e b r a c k e t of a k e t IB> and a b r a
exists i f the

i n t h e sense of some b i n a r y r e l a -

tion. A p a r t i a l i n n e r product s t r u c t u r e can be imposed on

e n t ways. For example F,G E T l i m (F(t) t40

,G ( t ) ) X

exists.

XrA

T

X,A

i n several differ-

can be defined t o be compatible i f

This Page Intentionally Left Blank

367

CHAPTER I l l

THE FREE F I E L D OPERATOR FORMALISM

In Quantum mechanics one mostly deals with systems consisting of a number of identical particles, e.g. a number of electrons, of photons, e t c . Since the particles cannot be distinguished the observables remain the same when two of these particles are interchanged. These observations have led to an adaptation of t h e physical theory, which we shortly discuss here. The considerations of this section are taken from Dirac's P r i n c i p l e s of q u a n t m

mechanics. 1

2

Suppose we have a system of d identical particles. Let lal>, lal>,

...

be

the basic kets describing the states for the first particle, considered as a dynamical system itself. Correspondingly, there will be basic kets for the second particle and so on. A basic ket for the assembly of d particles is represented by

This ket represents the state in which particle 1 is in state all, particle 2 in state a12 and so on. We may interchange the kets for the first two particles and get another ket

This transposition of kets is a linear process. Similarly, the process of interchanging any pair of kets corresponding to two different particles, is linear. Repeated transpositions lead to permutations. Hence any permutation of the ket la;1>

,..., laLd> d

in (1) is a dynamical variable for the assembly

of d particles. A permutation is called even (odd) if it is built up from an even (odd) number of interchanges. A ket IX> for the assembly of d particles is called symmetrical if it is un-

THE FREE FIELD OPERATOR FORMALISM

368

changed by any permutation P, i.e. if

for any permutation P. A ket IX> for the assembly of d particles is called anti-symmetrical, if

+

where the

or - sign has to be taken according to whether P is even or odd.

States described by symmetrical kets are called symmetrical states; states described by anti-symmetrical kets are called anti-symmetrical. In nature only two different types of particles occur: bosons and fermions. An assembly of identical bosons is described by symmetrical states; an assembly of identical fermions is described by anti-symmetrical states. Let us consider the special case of two identical particles which can be only in two independent states a and b. Then the following symmetrical states occur: lal> la2>, Ibl> Ib > and lal> Ib > + IbZ> lal>, i.e. both 2 2 particles in state a, both particles in state b, or one particle in state a and the other in state b. If both particles are fermions only one anti-

> Ib2> - Ib1> la2>, i.e. one particle 1 in state a and one in state b. Two fermions cannot occupy the same state.

symmetrical state occurs, namely la

More generally, the states occupied by an assembly of fermions, must be independent. This rule is known as Pauli’s exclusion principle. Next we consider an assembly of d particles. 1 2 Let the kets la > , la > ,

...

establish a discrete basis for the one-particle

ket space. Then the various products of these basic kets

...,id

where il,

E

m, establish

the basic kets for the assembly. Now suppose

the particles are bosons. Then only symmetrical states occur. We introduce the symmetrizing operator V

where the sum ranges over all d! permutations. We obtain the basic symmet-

369

INTRODUCTION

r i c a l k e t s by p u t t i n g

on t h e b a s i c k e t s ( a l l .

v

1

. . aid> as d

follows

where t h e labels o f t h e p a r t i c l e s are o m i t t e d s i n c e t h e y a r e no l o n g e r relevant. The d i s t r i b u t i o n o f t h e d bosons o v e r t h e s t a t e s a how many b o s o n s occupy e a c h s t a t e a

i

,

i

E

IN.

i .

1 s s p e c i f i e d by saying

So we c a n d e n o t e a d i s t r i b u -

t i o n of t h e bosons by a sequence ( n l , n 2 , . . . ) which i n d i c a t e s t h a t n1 bosons 1 are i n t h e s t a t e a , n2 bosons i n t h e s t a t e a2 and s o on. Of c o u r s e , we

s t i l l have the a d d i t i o n a l r e s t r i c t i o n m

Thus w e o b t a i n o b s e r v a b l e s N1,N2,...

with eigenvalues n . E I N U { O } , m 1 0 2 n . 2 d and N . = d. Each e i g e n v a l u e n . of N . i n d i c a t e s t h a t t h e num1 j=l J 1 1 b e r o f bosons i n t h e s t a t e a’ i s n , . The o b s e r v a b l e s N . j E IN, a r e c a l l e d 3‘ 1 t h e o c c u p a t i o n number o p e r a t o r s . They m u t u a l l y commute. A complete s e t of

..

s i m u l t a n e o u s e i g e n k e t s i s g i v e n by ( 7 ) where any ( i l , . , i ) d taken,

E

INd

may b e

F i n a l l y , we a l s o want t o v a r y t h e number d of p a r t i c l e s . T h e r e f o r e w e i n t r o duce t h e o b s e r v a b l e 0,1,2,

... .

N , t h e s o - c a l l e d number o p e r a t o r , w i t h e i g e n v a l u e s

Each k e t i n ( 7 ) i s a n e i g e n k e t of N w i t h e i g e n v a l u e d . We ar-

range t h e e i g e n k e t s of

Here t h e k e t

I >

N as follows

r e p r e s e n t s t h e vacuum s t a t e i n which t h e r e a r e no bosons

p r e s e n t . The k e t s la11> r e p r e s e n t t h e b a s i c s t a t e s i n which t h e r e i s one

.

.

boson p r e s e n t . The k e t s Vlall al*> r e p r e s e n t t h e b a s i c s t a t e s i n which t h e r e are two bosons p r e s e n t and s o on. The k e t s i n ( 8 ) a r e a l l o r t h o g o n a l . They e s t a b l i s h a complete set o f e i g e n k e t s f o r t h e m u t u a l l y commuting observables

N,, N, ,N,,

... .

I n C h a p t e r I1 w e have d e s c r i b e d t h e quantum m e c h a n i c a l harmonic o s c i l l a t o r . A s w e have n o t i c e d t h i s quantum s y s t e m i s c o m p l e t e l y d e s c r i b e d b y t h e ob-

servables

p

and

4 . I n s t e a d o f t h e r e a l dynamical v a r i a b l e s p and 4 w e c a n

370

THE FREE FIELD OPERATOR FORMALISM

take the complex dynamical variables q = recall that q and

1 (p+iq) and 0

fi

=

1 ( p -iq). we JT

have the following commutation relation

There is a connection between bosons and harmonic oscillators. To show this, we consider a system which is composed of a countable number of harmonic oscillators. We label these oscillators by the natural numbers. Thus we get complex dynamical variables q l , q 2 ,

... and -q1,q2, ...

with commutation re-

lations

Also

i

E

we introduce

Ni

ni!i.

=

Then

=

Ni + 1,

i

IN, mutually commute. The eigenvalues of each

IN. The observables Ni,

E

N.

are the nonnegative

integers. The i-th oscillator possesses a standard ket IO,i> with respect to the socalled Fock representation such that N.lO,i> = 0. Multiplying the standard kets IO,i> we get the standard ket

lo>

for the Fock representation of the

whole set of oscillators,

lo>

=

10,1> 10,2>

We observe that :.lo> 3

=

... .

0 for all j

E

IN. In addition, for given integers

nl,n2,... the ket

is a simultaneous eigenket of the commuting set of observables ?*lo eigenkets

exists j yields

E

(nyl ni2

.. . ) lo>

IN such that n

j

I

and (qy1

n;2

,

..

.)

lo>

{N, I i

E

IN).

are orthogonal if there

# n;. Comparison with the previous description

... 17id

10>

.

So the standard ket lo> represents the vacuum state.

371

INTRODUCTION

I f we r e p l a c e t h e d i s c r e t e s e t of b a s i c k e t s {la1> {lbk> Ik

E

Ij

IN} by another one,

F:

I N ) say, we can s e t up a s i m i l a r program. Then t h e b a s i c k e t s

a r e given by

I

> , Ibkl>,

Vlbkl bk2>

,.. . . -

Now we g e t complex dynamical v a r i a b l e s Bk with a d j o i n t O k ,

Vlbkl bk2

...> =

(0 0 k l k2

k

E

IN,

such t h a t

...) l o > .

In p a r t i c u l a r , it follows t h a t

J

J

and hence

So t h e dynamical v a r i a b l e s Ok can be obtained from t h e v a r i a b l e s q . accord7 k i n g t o t h e same law t o which t h e b a s i c k e t s Ib > can be obtained from t h e

b a s i c k e t la1>. Correspondingly, f o r t h e

0k

w e have

J

t I.' 0k

So the dynamical v a r i a b l e s

transform according to t h e same law a s

the basic bras.

Let T denote a dynamical v a r i a b l e with r e s p e c t t o t h e one p a r t i c l e k e t space. Then T i s expressed by t h e double s e r i e s

(12)

T =

1


la^>

laj>
.

i ,jeIN Here ( < a J l T l a i > ) . . i s t h e matrix of T with r e s p e c t t o t h e b a s i c k e t s 11 la'> , j E IN. Now we i n t r o d u c e t h e dynamical v a r i a b l e n ( T ) f o r t h e assemj i b l y by r e p l a c i n g la > by r l , and < a by {i i n t h e expansion (12) f o r T .

I

This y i e l d s

THE FREE FIELD OPERATOR FORMALISM

372

The dynamical v a r i a b l e Q ( T ) i s c a l l e d t h e second q u a n t i z a t i o n of

T.

If, for

i n s t a n c e , T i s t h e r e a l v a r i a b l e of t h e energy of one boson, then Q ( T ) i s t h e v a r i a b l e of t h e t o t a l energy of t h e assembly of bosons. I f T =

I , then

Q ( T ) equals t h e number o p e r a t o r N .

R denotes a dynam-

W e can extend i n d u c t i v e l y t h e p r o c e s s t h u s i n i t i a t e d . I f

i c a l v a r i a b l e with r e s p e c t t o t h e two-boson k e t space, then as

R

=

c


R Iakae>(Vlaiaj>

VRV la a

R can be w r i t t e n

ijkL (=

1

k e

>

VIaiaj>
kk? a

IV

)

.

ijkL The e q u a l i t y between b r a c k e t s i s v a l i d because

n , ~ and . 1 3

n(R)

k L

< a a I by :k{L

VRV

=

i

R . Replacing la a

j

> by

we get

i s t h e second q u a n t i z a t i o n of t h e dynamical v a r i a b l e R . I f R i s t h e

r e a l v a r i a b l e of t h e two-boson

i n t e r a c t i o n energy, then

Q(R) i s the real

v a r i a b l e of t h e sum of a l l two-boson i n t e r a c t i o n e n e r g i e s i n t h e assembly.

The second p a r t of t h i s i n t r o d u c t i o n i s devoted t o t h e d e s c r i p t i o n of an assembly of fermions. This case can be d e a l t with along s i m i l a r l i n e s a s t h e case of an assembly of bosons.

I n t h e l a t t e r , w e have introduced t h e

symmetrizing o p e r a t o r V . Now we introduce t h e anti-symmetrizing o p e r a t o r

summed over a l l permutations

P of o r d e r d. Hence E ( P )

E ( P ) = -1 if P i s odd. I f we apply

w

t o t h e k e t la:'

= 1 if

af2..

w.

P i s even and

. aid> d

( c f . (1)1 ,

we g e t t h e b a s i c k e t s f o r t h e assembly of fermions,

where we have l e f t o u t t h e l a b e l s 1 ,

...,d

because t h e r e i s no d i s t i n c t i o n

INTRODUCTION

313

between the particles. It can be readily checked that Wla'l

. .. aid> #

0 if

and only if all integers are mutually distinct. If we introduce the vari-

... as the numbers of fermions in the basic

ables n1,n2,

,... ,

states a1,a2,a3

takes only the value 1 or 0. So we can denote each basic ket

then each n

1 for the assembly of fermions by a sequence with components 1 or 0.

If the number of fermions is variable, then we can set up the complete set of kets

I > , tail>, Wla'l

.

. al*>, Wtail ai2 ai3>

,... .

Similar to the boson case we introduce a set of complex dynamical variables

vj,

ij corresponding to

each fermion state. Now we assume anti-commutation

relations, VjVk

(15)

+

- -

,

VkVj = 0

--

v.v t v v = Q , 3 k k j

-

vj . vk

t

-

vkv j

= 6.

j,k

jk'

E

IN

.

-2 If we put j = k we obtain v2 = 0, v . = Q and C.V. + v,;, = 1. Such a set of j 3 3 3 3 3 linear operators vj , v . can be constructed in the following way. Let u X'

3

u

Yl

u

z

denote the spinors

uX

=[:

;],

uy=[P

-11,

u z = [ 0 -1O ] .

They satisfy the commutation relations

0

0

X Y

- o a

Y X

=2iaz,

ozox - axaz = 2io Y' a o

Y Z

- o a

Z Y

=2iax.

Now to each state aJ we link such a triple u i

, uJ , u i , which satisfies the

Y above commutation relations. We assume that for j f k each operator k k u i , uJ , UJ commutes with each operator ux , by , uk We also introduce operY Z ators 5 . one for each state a', such that cjck + ck5 = 26, and such 3' 3k' that each 5 . commutes with all u-operators. Now put

.

3

THE FREE FIELD OPERATOR FORMALISM

374

Then the operators u

1

satisfy the Relation ( 1 5 ) .

It is clear that

(v.;.I2

= wjij

1 3

=

Nj

.

N. has eigenvalue 3 NkNj and

So the observable

lows that

N.N

=

1 k

-

vjNj

-

= 1

N. 3

.

Remark. Using the matrices

Nj

= Vjij

0 and 1. From the relations 15 it fol-

01

and u j we have

UJ

Y

x’

[ 1 ;]

=

and

Let lo> denote the simultaneous eigenket of the set of commuting observables

INj

Ij

E

IN} with j c IN.

N.lO>=o, 3

Then it follows that ;.lo>

= 0

3

So for each j

E

and further

IN, v.10> is an eigenket of 3

k # j NkwjlO>

= u.N

i k

lo>

= 0

N. with 3

eigenvalues 1.

Also

for

,

which shows that v.10> is an eigenket of Nk, k # j, with eigenvalue 0 . 3

In general the basic ket v the observables

N. 3

with

11

v.

12

... v 3.8 lo>

is a simultaneous eigenket of

375

INTRODUCTION

Thus we f i n d t h e following r e l a t i o n s

We observe t h a t both s i d e s a r e anti-symmetric i n t h e l a b e l s j we pass over t o another s e t of b a s i c k e t s ,

Ibk>

t h e boson c a s e w e g e t a new s e t of v a r i a b l e s I.! pk,

-

vk

Ik

-

E

# .

.. , I d . ’

If

IN} say, then a s i n

,pk. The dynamical v a r i a b l e s

s a t i s f y t h e following transformation r u l e s with r e s p e c t t o t h e dynam-

i c a l variables vj,

-

vj.

and

So t h e r e i s a procedure of second q u a n t i z a t i o n f o r fermions similar t o t h e

one f o r bosons. I f

T

i s a dynamical v a r i a b l e i n t h e one-fermion k e t space

generated by t h e b a s i c k e t s la]> t h e second q u a n t i z a t i o n O ( T ) of T i s given by

In p a r t i c u l a r ,

Q ( I )denotes t h e number o p e r a t o r .

i f R i s a dynamical v a r i a b l e i n t h e two-fermion k e t space which i s i j generated by t h e b a s i c k e t s Wla a >, then i t s second q u a n t i z a t i o n O ( R ) i s Also,

given by

Observe t h a t WRW = R. I t i s c l e a r t h a t t h e treatment of bosons and fermions i s s i m i l a r i n many

r e s p e c t s . But f o r assemblies of fermions t h e r e a r e o n l y two a l t e r n a t i v e s f o r t h e s t a t e s : a s t a t e i s occupied o r it i s not. These a l t e r n a t i v e s a r e symmetric, i . e . t h e r o l e o f t h e occupied and unoccupied s t a t e s can be i n t e r changed without a l t e r i n g t h e mathematical d e s c r i p t i o n . In f a c t , t h e v a r i a b l e

376

THE FREE FIELD OPERATOR FORMALISM

v . c r e a t e s a p a r t i c l e i n t h e unoccupied s t a t e a';

3 a t e s an occupied s t a t e . S i m i l a r l y , t h e v a r i a b l e v

p u t d i f f e r e n t l y , it c r e -

j

c r e a t e s an unoccupied

s t a t e a'.

I n quantum f i e l d t h e o r y one does n o t s t i c k t o d i s c r e t e s e t s of b a s i c k e t s . A l s o a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s based on continuum sets of b a s i c

k e t s occur. Then, boldly speaking, s e r i e s have t o be replaced by i n t e g r a l s , t h e Kronecker d e l t a by a Dirac d e l t a f u n c t i o n , e t c .

The m a t e r i a l of t h i s c h a p t e r i s presented i n t h e a b s t r a c t s e t t i n g of funct i o n a l a n a l y s i s . Thus we hope t o diminish t h e e v e r e x i s t i n g n o t a t i o n a l problems, b u t more importantly, t o c l a r i f y a l l a r i s i n g mathematical concepts i n t h e b e s t p o s s i b l e way. I t h a s s u r p r i s e d us t h a t t h e concept of Dirac b a s i s and t h e r e l a t e d expansions, p l a y s such an e s s e n t i a l r o l e i n t h e pres e n t mathematical i n t e r p r e t a t i o n of t h e f r e e f i e l d formalism.

111.1. The Fock space Let

Y

and

z

be separable H i l b e r t spaces with orthonormal bases ( e ) and n n6IN r e s p e c t i v e l y . Then general ( a l g e b r a i c ) v e c t o r space theory y i e l d s

(gm)me t h e e x i s t e n c e of a bilinear a l g e b r a i c t e n s o r product of denote by Y @a

z.

The b i l i n e a r mapping

@

Y and

z

which w e

sends t h e p a i r [en,g

1

t o en

M

N

m

@

gm

and so each p a i r of l i n e a r combinations C I: a e C Bmgm] i s send t o N M n=l n n ' m = l Z Z a B (en 8 g m ) . Here we have t h e usual i d e n t i f i c a t i o n n=l m = l n m

i f a;@;

= anBm, n = 1 , . . . , N ,

In Y Qa

z

Thus

Y

m

= 1,

...,M .

-

w e introduce t h e s e s q u i l i n e a r form ( - I ) y@z by

z

becomes a pre-Hilbert

(en a gm)n,mcm* and we c a l l Y 8

z

of a l l s e r i e s

space with orthonormal b a s i s

Y @a z with r e s p e c t t o ( ' , - ) y a z t h e H i l b e r t t e n s o r product of Y and z. So Y @ z c o n s i s t s

W e denote t h e completion of

311

THE FOCK SPACE

with

Now t h e c o n s t r u c t i o n of t h e Fock space i s a s follows. W e s t a r t w i t h a sepa r a b l e i n f i n i t e dimensional H i l b e r t space

In

x

x

with orthonormal b a s i s ( v . ) . 3 7Em' A by

w e define t h e p o s i t i v e s e l f - a d j o i n t operator

for

(

x

L2

(E.g. we may take = L (IR) and A = - - - + x2 + 1) . ) W e note t h a t the 2 whole set up of t h i s c h a p t e r does not depend on t h e c h o i c e of t h e eigenval u e s A . = j of

A. We make t h i s choice f o r combinatorial convenience.

7

x

Next we i n d u c t i v e l y d e f i n e t h e H i l b e r t spaces X(k) =

... x

O

@

(k-times).

The H i l b e r t space X(k) i s c a l l e d t h e k-fold H i l b e r t t e n s o r product of

...

x.

orthonormal b a s i s i n X(k) i s e s t a b l i s h e d by t h e v e c t o r s v . 8 0 v. k 31 3k ,j k ) E IN The space X ( k ) c o n s i s t s of a l l series

where j = ( j l , .

where

1eINk I w j

..

I

.

<

-. The inner

product i n X(k) i s given by

with

If f l ,

...,f k w

belong t o

j

= (f , v .

1

x,

defined by then t h e sequence ( w ) , j ~ E I N ~

1 (f2,vj2)

11

... ( f k , v j k )

I

j

k 6

IN

An

THE FREE FIELD OPERATOR FORMALISM

378

determines the element fl

B

... B

fk of X(k); we call this element the kfk).

...,

fold tensor product of (fl,

In X(k) we define the positive self-adjoint operator A(k) by

followed by linear self-adjoint extension. Here we put

Ij I

= j 1 + j2 +

. . . + j,.

We note that

111.1.1. Let

a.

Yl,Y 2,...

*,... be self-adjoint operators

be Hilbert spaces and let B1,g

... ,

in Yl,Y2,

respectively. We define the operator diag(Bk) in kern d) ‘k by

and

Then the operator diag(Bk) is self-adjoint in are nonnegative, then also diag(B

k

)

kFIN .,Y

If all operators 8k

is nonnegative.

0

Now we put m

F =

@

X(k)

k=O

and

where we put X(1)

=

X, X ( 0 ) = g, A(1) =

A

and A(0) = 0 = 01.

The operator H has a discrete spectrum. Each natural number and zero is an

..

eigenvalue of H. The multiplicity of 0 equals mo = 1 and of N = 1,2,. N-1 equals m = k=l (N-l k-l) = 2 . ( (N-1 k - l ) is the number of ways to write N as a

!

THE FOCK SPACE

s u m o f k n a t u r a l n u m b e r s , e.g.

379

5 = 1 + 1 + 3 = 1 + 3 + 1= 3 + 1 + 1 = 2 + 2 + 1 =

= 2 + 1 + 2 = 1 + 2 +2.)

I n t h i s s e c t i o n we c o n s t r u c t a r e p r e s e n t a t i o n of t h e i n f i n i t e dimensional

-

Lie algebra generated by t h e dynamical v a r i a b l e s qk , q k , k

E

IN,

as d i s -

cussed i n t h e i n t r o d u c t i o n . In a d d i t i o n we d e r i v e some t e c h n i c a l r e s u l t s f o r l a t e r use.

Let g

X. Then w e d e f i n e t h e o p e r a t o r a k ( g ) from X(k) i n t o X ( k - 1 ) by

6

where we employ t h e n o t a t i o n

Remark. I f f l ,

111.1.2. with

I''k

Proof.

E

then w e have

m. x

Let g E

...,f k x,

and l e t k ( g,

E

IN.

The o p e r a t o r a k ( g )

:

X(k)

-+

X(k

- 1)

x

I' (k)+x (k-1) = I1 gllx.

Let f(k)

E

X ( k ) . Then w e have t h e following e s t i m a t i o n

is bounded

380

THE FREE FIELD OPERATOR FORMALISM

Lemma.

111.1.3.

L e t t > 0 and l e t k E I N .

m.A

Then f o r a l l g

E

x

s i m p l e c a l c u l a t i o n shows t h a t f o r a l l n1

IN, n,m

E

E

INk-'

we d e n o t e t h e a d j o i n t of ak (g), g E x, k E IN, by C k ( 4 ) . Then Ck (9) i s a bounded l i n e a r o p e r a t o r from X ( k - 1 ) i n t o X ( k ) . Easy c o m p u t a t i o n shows t h a t C (g)f(k-1) = g 0 f(k-1)

k

Let k

f(k-1)

E

X(k-1)

Theorem.

111.1.4.

\

,

IN.

E

x

Then t h e mappings a : I+ B(X(k) k are c o n t i n u o u s .

,X ( k - 1 ) )

and

: X W B(X(k-1) , X ( k ) )

x.

L e t g1,g2 E

CoroZ lary

111.1.5. Let g

E

x

x.

Then w e have

.

and let k

E

IN.

W e have t h e f o l l o w i n g e x p a n s i o n s

.

381

THE FOCK SPACE

and

with convergence i n t h e sense of t h e Banach norm of B ( x ( k ) , X ( k - 1 ) and B(X(k - 1 )

Proof. ak

c.q.

,X ( k ) ) ,

respectively.

The a s s e r t i o n s follow from Theorem 111.1.4 and t h e a n t i - l i n e a r i t y of

0

t h e l i n e a r i t y of C

k’

In t h e Fock space F =

kzOX(k) w e m

introduce t h e dense l i n e a r subspace

D0 ‘

a > 0,

Also we introduce t h e a n n ih ila tio n o p e r a t o r U ( g ) , g

E

x,

by

and

For convenience we g i v e t h e o p e r a t o r matrix of A ( g ) :

0 Let C ( g ) denote t h e a d j o i n t a ( g ) * of a ( g ) . Then with s t a n d a r d arguments it can be checked t h a t c(g) i s a densely defined o p e r a t o r with D ( c ( g ) ) = m

For { f ( k ) } k = OE D, we have

D,.

382

THE FREE FIELD OPERATOR FORMALISM

where w e p u t C (9) = 0 and f ( - 1 ) = 0. 0

The o p e r a t o r matrix of C ( g ) i s given by

0 0

I t can be r e a d i l y checked t h a t U(g)

*

= C(g) and

c

9)

*

= a g) (as unbounded

l i n e a r o p e r a t o r s ) . So t h e o p e r a t o r s cp(g) = a ( g ) + c ( g ) and cp+(g) = a ( g ) - C ( g )

A

iA

are s e l f - a d j o i n t .

111.1.6.

e.

On D, w e have f o r a l l t > 0

- t A g) = et H U(g) e- t n

a(e

Proof.

.

Let { f ( k ) } i = OE D, and l e t t > 0. The following e q u a l i t i e s hold

111.1.7.

Corollary.

On e-tn(D1) we have f o r a l l t > 0

c ( e- t A g)

=

e

- t H c ( g ) etH

.

The following expansion r e s u l t s are v a l i d .

0

THE FOCK SPACE

111.1.8.

Let g

E

383

Theorem.

x.

m

Then for a l l I f ( k ) IksO

E

D, w e have

where t h e s e r i e s converges i n F. m

Proof.

Let {f ( k ) lkZOE D,.

Let N

E

IN and p u t

Since for a l l k E I N

I l a k ( 9 - 9,)

llX(k)*X(k-l)

=

Ilg- g N I1X

w e obtain t h e estimation

I t follows t h a t

S i m i l a r l y , w e can prove t h a t

III. 1.9. Theorem. Let g

E

x.

m

Then f o r a l l { f ( k ) j k E 0 E

where t h e s e r i e s converges i n F

D1

we have

n

THE FREE FIELD OPERATOR FORMALISM

384

The next s t e p i s t h e i n t r o d u c t i o n of t h e symmetric and anti-symmetric Fock space. Let 0 E

Ipk

denote t h e permutation group of o r d e r k . For each permutation

pk we introduce t h e l i n e a r o p e r a t o r

5

: X(k) -+

X ( k ) by

where

ifv.(k)=v. Bv. @ . . . @ v ,jEL?Jk. 3 31 32 lk We observe t h a t 5 i s a u n i t a r y o p e r a t o r . Now we d e f i n e

Then s t r a i g h t f o r w a r d c a l c u l a t i o n s y i e l d

( P ( + ) ) 2 = Pk(+)

=

k

(pL+))*. So Pk(+)

x(+) .

P(+) by ( k ) The k ( k ) i s c a l l e d t h e k-fold symmetric H i l b e r t t e n s o r product of

i s an orthogonal p r o j e c t i o n . We denote t h e range of

X(+)

subspace

x.

In p a r t i c u l a r , f o r a l l

3 f(k)

u

E

IPk and a l l f ( k )

E

x(+)( k )

w e have

= f(k).

I 11.1.1 0 .

Definition.

The orthogonal p r o j e c t i o n

Further, t h e range of

F, F(+)

P ( + ) on F is defined by

F(+). F(+) t h e

P ( + ) i n F i s denoted by

i s a H i l b e r t space i t s e l f . We c a l l

A s a closed subspace of

Boson Fock space. We

mention t h a t

I I I . 1.11.

Definition.

The s e l f - a d j o i n t o p e r a t o r

A ( + ) ( k ) i n X ( + ) ( k ) i s defined by

THE FOCK SPACE

385

Similarly, the self-adjoint operator ff (+) in F(+) is defined by

We note that P(+) (k) A(k) = A(k) P(+) (k) and hence that P(+) ff = ffP The eigenvalues of the self-adjoint operator ff

. . with

N = 0,1,2,.

multiplicity m r )

dure the multiplicity ) : m

(+)

(+)

.

are the numbers

. Because of the

symmetrization proce-

is strongly diminished in comparison with the

multiplicity mN. In fact the multiplicity m(+) is equal to the number of

decompositions of 5

=

1 +4

m:+)

= 2

= 7.

N

N

i n t o i n t e g e r s m a n d s without regard t o order. E.g.

+ 3 = 1+ I + 3 = 1+ 2 + 2

The asymptotics of

So for all t > 0 the series

Let

E

:

IPk

-t

{-l,l}

s(l+)

=

1 + 1 + 1 + 2 = 1 + 1 + 1 + 1 + 1 , whence

is given by

(+)

N=O mN

e-tN is convergent.

denote the function which has the value one for even

permutations and the value minus one for odd permutations. We introduce the bounded linear operator Pk-) ,

It is not hard to prove that Pk-) is an orthogonal projection on X(k). We denote its range by

x(-) (k) and call x(-) (k) the k-fold anti-symmetric x. Each f (k) E x ( - ) (k) satisfies

Hilbert tensor product of

s(f (k))

= E

1x1. I. 12.

.

(u) f (k) For completeness we put Ph-)

=

1.

Definition.

The orthogonal projection P ( - ) on

F

is defined by

The range of p(-) is denoted by F ( - ) and it is called the Fermion Fock space.

THE FREE FIELD OPERATOR FORMALISM

386

The eigenvalues of

H(-)

are the numbers N

... with multiplicity

= 0,1,2,

mA-). Now mi-) is equal to the number of decompositions of N i n t o d i s t i n c t

integer summands without regard to order. E.g. 5

=

1 + 4 = 2 +3, whence

m(-) = 3 . The asymptotics of m(-) for large N are given by

N

5

so for all t > 111.1.13.

Let g

E

o

the series N=O "N( - ) e-tN

Definition.

X. Then on D'') 1

u") (g)

=

c(+)(g)=

111.1.14.

=

P(')

(D,) we introduce the following operators

P (')

a ( g ) P(')

,

D(u'(g))

=

0:- + ) ,

P (*)

c ( g ) p(')

,

D(c'(g))

=

I):')

.

Coro~~arg.

The linear operators U " )

Also

is convergent.

( g ) , C")

( g ), g E

x,

are closed. They satisfy

the following relations are valid

and, correspondingly,

Proof.

The assertions are immediate consequences of the definition of

a'(g),

'C (9) and

.

a ( , ) , C (9)

0

THE FOCK SPACE

381

Theorem 111.1.8 yields the following expansions.

and

where the convergence is in

F (')

.

P(')

(D2 )

0

111.1.16. Theorem. Let g,h

E

x.

Then on

Da')

=

the following (anti-) commutation re-

lations are valid. (9) , U " ) (h)] T

CC

(')

0

(g) , c") ( h ) j = 0

+

(g) ,C(')

Proof.

=

(h)] T

=

(h,g)X1 " )

.

We only prove the third (anti-) commutation relation. The remaining

two can be dealt with in the same way. k Let k E IN and let j E IN Consider the following computations.

.

THE FREE FIELD OPERATOR FORMALISM

388

=

(h,g)X Pivj(k) f

Then from

and

(*)

(**)

e 1

(+l)'+'(v

j=l

Je

,g)?'k')(h@v.

we obtain

Recalling the definition of a") (g) and C")

where 1 " )

@...@v . 1 @...@v . @V. I1 11-1 lL+l Ik

(h) we get

denotes the identity on F " ) ,

Remark. For each 1

E

IN we put

Then the operators n1 and

:e,

IN, satisfy the following commutation re-

E

lations

1

-+ + r7'

rlk

-

+ -+

Qk

nL

=

-

6,e

So we have constructed a representation of the complex dynamical variables tll,

;L as introduced by Dirac, in the Hilbert space F(+).

Similarly, we put for each l?

E

IN

389

REQUIRED MATHEMATICAL TOOLS

-

n e n; -- --

n t nk -

-

rli

+

n; n e

= 0

I

Ile = 0

f

- -

+ ll;

-

n; + n; n;

=

.

6kR

So we have constructed a representation of the complex dynamical variables as introduced by Dirac, in the Hilbert space

vt,

..

Let c denote the vector c = (1,0,0,0,. j

= (jl,

...,jk), we have

It follows that any element of

F(')

)

E

F").

F(-).

Then for all j

E

k IN ,

can be written as

where

SO c

E

F(')

is a cyclic vector for the set of operators

+ -+ {rli , ni I j

E

IN].

111.2. Required mathematical tools In the previous section we have introduced the k-fold (linear) Hilbert tensor product X(k), k

E

IN, and in X(k) the positive self-adjoint operators

A(k) with orthonormal basis of eigenvectors established by v.(k) = v 3

jl

@

... -8 v

jk

,

j

E

k IN

such that A(k) v.(k) = 111 v.(k) I 3

.

It follows that the trajectory space TX(k) ,A(k) and the analyticity space

THE FREE FIELD OPERATOR FORMALISM

390

Sx(k) , A ( k ) a r e nuclear. The following r e s u l t i s a consequence of Chapter B . 1 ,

appendix.

Proposition.

111.2.1.

a . Let F1,F2,

...,Fk

E

T X , A ' Then t h e mapping

T

i s a member of

X(k) , A ( k ) '

This mapping i s denoted by F1

... 0

0

Fk. For

a l l t > 0,

The space T X(k) ,A(k) product of TX,A. b. Let w1

,...,wk

E

S X , A . Then i n p a r t i c u l a r w l , .

( l i n e a r ) topological tensor

. .,w k

E

e-TA(X)

f o r some

> 0. W e have

T

w1

Q

...

-TA (k) Q

sX ( k ) of s X,A'

product

111.2.2.

H

wk

E

e

.

( l i n e a r ) topological tensor

0

,

c f . 'Section 111.1. The space T

t r a j e c t o r y space and t h e space SF,H

F,H

i s c a l l e d t h e Fock-

t h e Fock-analyticity

space.

Proposition.

The spaces

Proof.

%(k),A(k)

Definition.

= d i a g ( A (k) )

111.2.3.

(X(k))

can be seen a s a k-fold

The space

Let

can be seen a s a k-fold

sF,H

and TFIH a r e Montel, b u t n o t n u c l e a r

...

The s e l f - a d j o i n t o p e r a t o r H has eigenvalues N = 0 , 1 , 2 , with multi2N- 1 , N E IN. So f o r each t > 0 t h e o p e r a t o r

p l i c i t i e s mo = 1 and mN =

exp(-tlf) i s compact. Since t h e s e r i e s

39 1

REQUIRED MATHEMATICAL TOOLS

is divergent for 0 < t < 'I log2, the operator exp(-tff) is not Hilbert-

0

Schmidt for these values of t. In the remaining part of this chapter we employ Dirac's bracket notation because of its expressiveness. To this end, we have to introduce some new notational conventions additional to those introduced in the previous chapter. represents the k-particles ket space. So The trajectory space T X(k) ,A(k) its elements are denoted by IF;k>. Therefore, the trajectory space TXt(k),A~(k) represents the k-particles bra space and correspondingly its elemtents are denoted by
TX' (k),A' (k) see

Section 1I.ii.) Instead of IF;1> and and
TF,H are

m

denoted by {IF;k>)k=O

. The k e t s

related to

and corresponding the bras related to

TF,,H,

are denoted by {
.. .

I

j E INk} is an orthonormal basis in As mentioned, the set {v. @I d v. 11 Jk is an eigenvector of A(k) with eigenvalue j X(k). Each v @ @ v jl jk We denote the ket corresponding to v . @ @ v by 71 jk

. ..

...

Let (vh)eEIN denote the orthonormal basis in i.e. v' (v 1 = 6 k 1 ke'

...

k,e c IN

I 1.

x' dual to the basis (ve ) e e l "

.

k Then the set (v' @ d v! Ij E IN } is an orthonormal basis in X'(k) 11 lk consisting of eigenvectors of A' (k) We denote the corresponding bras in

TX' (k), A '

(k) by

It follows that

and

.

THE FREE FIELD OPERATOR FORMALISM

392

with the usual interpretations. The relations

valid for all j

E

k

Dl

,

k

E

IN, lead to the following considerations:

...,IFk>) is the ket IF;k>,

Associated to the tuple (IF1>, \F;k>

:

t H IFl>(t)

0

...

0

IFk>(t)

.

Thus it makes sense to write

For the bra IF2>

.. .

IFk> we write

In particular, Proposition 111.3.1 yields that to each k-tuple of test kets

( 1 W1 >, .. . ,I Wk>) corresponds the k-particles test ket IW1>

... 'Wk>

and the k-particles test bra <Wll

...

<Wkl

-

The linear space of k-particles test kets (bras) can be identified with

(Sx,(k),A,(k)) under the identification mappings IW;k> H IW;k>(O) %(k) ,A (k) and <W;kl H <W;kl (0).Thus we can impose on the vector space of k-particles test kets (bras) the locally convex topology of sX(k) ,A(k) We arrive at the following definition.

csX'(k),A' (k)) '

III.2 . 4 . Definition.

Let <@I be a bra. Then the linear mapping Nk( < @ I )from

X'

(k-1), A (k-1)

is defined by

$ (k),A (k) into

393

R E Q U I R E D MATHEMATICAL TOOLS

+> ~ 0 and

where t

T

> 0 must b e t a k e n s u f f i c i e n t l y s m a l l .

The above d e f i n i t i o n makes sense and d o e s n o t depend on t h e c h o i c e of because f o r a l l

T

> 0

T,T'

and Uk(1@>(r))

: X(k)

-f

i s a bounded o p e r a t o r

X(k-1)

Theorem.

111.2.5.

<@I

For each b r a 'X(k-1) , A ( k - l )

M.L e t

t h e l i n e a r mapping Nk(

< @ I ) from

sx(k) , A ( k )

into

i s c o n t i n u o u s . W e have

<@I)e 11 I O > ( t ) IIx. Now t h e r e s u l t f o l l o w s from Theorem

t > 0 . Then t h e o p e r a t o r e tA(k-l)~k(

i s bounded w i t h norm

0

B. 1 1 . 1 . 2 .

tiernark. F o r ( l W l >

,..., lWk>)

a k - t u p l e of t e s t k e t s w e have

Nk(<@I~tlW1>lW2~

... IWk>'

=

<@I

W1>(0)

(IW2>

...

IWk>,

.

Theorem.

111.2.6.

The o p e r a t o r N ( < @ I ) e x t e n d s t o a ( c o n t i n u o u s ) l i n e a r mapping from k into T i f < @ Ii s a t e s t b r a . X(k-1) ,A(k-l) TX ( k ) , A ( k )

Proof. L e t < @ Ibe a t e s t b r a . Then t h e r e e x i s t s

(elt)

E

T

> 0 such t h a t

e-TA'(x')f o r a l l t > 0. I t follows t h a t f o r a l l t > 0 t h e opera-

tor e

-tA

N,

i s bounded f o r 0 <

(

< @ )I T'

eTIA

e-(t-T')AU k ( < @ l

(-TI))

< T . Now from Theorem B . I I . 1 . 6

t h e result f o l l o w s .

0

THE FREE FIELD OPERATOR FORMALISM

394

Remark.

<@I be a t e s t b r a . For (IF1>,

Let

...

have N k ( < @ l ) ( l F 1 >

IFk>)

Conjugate t o t h e o p e r a t o r TX(k-l) ,A(k-1)

into

...,IFk>) a k-tuple (IF2> ... IFk>).

is t h e o p e r a t o r M (I@>) from k defined by

Nk(<@l)

T

X ( k ) ,A(k)

Mk(l@>)IF; k - 1 > : t P e- ( t - T ) A ( k ) C k ( l @ > ( T ) )

111.2.7.

of k e t s we

<@IF1 > ( O )

=

where f o r each t > 0 w e t a k e 0 <

T

(IF; k - l > ( T ) )

< t . D e f i n i t i o n 111.2.7 does n o t depend

on t h e choice of T because f o r a l l t , t ' > 0 we have

111.2.8.

Theorem.

The l i n e a r mapping M (I@>). k * TX(k-l) ,A(k-i a l l t > 0 w e have

Proof.

X(k) .A(k)

is continuous. For

M, (10) e tA(k-l)c C k ( l @ > ( t ) ) i s bounded l@>(t) 11. Now t h e r e s u l t follows from Theorem B.II.1.6.

Let t > 0 . Then e-tA(k)

with norm 11

111.2.9.

Coro I Zary

0

.

Let I@> be a t e s t k e t . Then

'X

+ T

Mk(I@>)

maps S X ( k - l ) , A ( k - l )

continuously i n t o

( k ) , A (k) *

Proof.

There e x i s t s

T

> 0 such t h a t Vt>O

:

I@>(t) E e

-TA

(x).

I t follows t h a t

f o r a l l t > 0 and T ' with 0 < T' < m i n ( t , r ) T'A(k)

Mk(Ie>,

.-tA(k-l)

i s a bounded o p e r a t o r . Now t h e r e s u l t follows from Theorem B.II.1.2. Now l e t

be a Dirac b a s i s . For each k e t

t h e following h e u r i s t i c i n t e g r a l expressions

I@>we want t o i n t e r p r e t

0

REQUIRED MATHEMATICAL TOOLS

395

and

111.2.10.

s.

Let IW> be a t e s t k e t . Then

Nk ( < W I

)

can be expressed by t h e s e r i e s

m

where convergence of t h e s e r i e s i s i n t h e following sense

w.

L e t t > 0. The following e s t i m a t i o n i s v a l i d .

II e t A ( k - l ) N,( < L I

e - d ( k ) ~ ~=

~ tku ( l C > ( t ) ) II

=

II l C > ( t ) II

= e-lt

So f o r a l l t > 0 w e have

Let < W l

be a t e s t bra. Employing t h e expansion theorem f o r Dirac b a s e s we

THE FREE FIELD OPERATOR FORMALISM

396

have (cf. Corollary 111.1.5):

(*)

It is allowed to interchange summation and integration:

111.2.11.

Lemma.

Let 1W>be a test ket. Then there is T > 0 such that for all t > -T the function

I <W I

6 I+

c>(-T)

j IIe (t+T)A (k-1) Nk(<SI) e-(t+T)A(k)

is p-integrable and

=

J

<w I

<>(-T)e (t+T)A(k-l)Nk(<S1)

- (t+T) A ( k ) dp(S)

.

M where the integral converges strongly in

Proof. that

B (x (k) , x (k - 1)) .

The proof follows from previous considerations and the observation

REQUIRED MATHEMATICAL TOOLS

391

0 111.2.12. Let

I@>

Theorem.

be any k e t . Then

Here t h e

B (x ( k )

,x (k - 1)

-valued f u n c t i o n

5 w <@I & > ( t -

T)e

N~

(


)

e

-TA ( k )

i s strongly p-integrable.

m.We observe t h a t

f o r each T > 0 t h e mapping t

W

k e t . N o w t h e r e s u l t follows from Lemma 111.2.11.

S i m i l a r l y , by t a k i n g a d j o i n t s , we o b t a i n

111.2.13.

Let

I@>

CoroZZary.

be a k e t . Then

Mk(l@>)

=

M

i

<E I @ > M k ( l & > )

i n t h e sense t h a t f o r a l l t > 0 and a l l

dp(6)

T,

0 < T < t,

IO>(t

+T)

is a t e s t

0

THE FREE FIELD OPERATOR FORMALISM

3 98

, X ( k ) )-valued

Here t h e B ( X ( k - 1 )

GI-+

function

M,(I~>) e

< c I @ > ( t - ~ ) e

T A (k-1)

i s s t r o n g l y p-integrable.

Proof. -

FOK

each t > 0 we have -tA(k)

Let

L

:

TX , A

T X , A be a continuous l i n e a r mapping.

+.

Vt>0 Then f o r each

Mk(IO) e

6

%o E

M

:

-tA

LeTA :

x

+

x

i s bounded

So

L

satisfies

.

and each t > 0 t h e following e s t i m a t i o n i s v a l i d

II e - t A ( k ) ~ k ( ~ ~ c e> )

Because

k(<
The o p e r a t o r product

X'

(k), A (k)

into

s ( k - 1 ) ,A(k-1) C + T X ( k - l ) , A ( k - 1 ) and , we can compose t h e s e o p e r a t o r s . X (k) ,A (k) N k ( < 5 1 ) i s a continuous l i n e a r mapping from

maps ' X ( k ) , A ( k ) into into T

Mk(L15>) maps T X ( k - l ) J ( k - 1 )

Mk(LIS>)

T X ( k ) ,A(k) *

We have t h e following e s t i m a t i o n

So f o r t > 0 and a l l s u f f i c i e n t l y small T > 0

REQUIRED MATHEMATICAL TOOLS

< e-2t(k-1) II .-tA

,(

e~A

e-2kt)i

(

e-2k-c)4

399

.

k= 1

k=l Therefore the integral

M,(LI~>)

N,( <51

)

e -tA(k)dp ( 5 )

M exists in uniform operator sense and denotes a bounded linear operator in X(k).

111.2.14. D e f i n i t i o n . The integral expression

I

Mk(L15>) N k ( < 5 1 ) dv(5)

M

denotes the linear mapping from

sX ( k ) ,A (k) into TX ( k ) , A ( k ) ,

which acts on

the k-particles test ket space as follows

111.2.15.

-.

The linear mapping

Mk(L15>) N k ( < 5 1 ) dv(E.1 M is continuous. In addition, for all t > 0 and all

‘I,

0 < T <

t,

THE FREE FIELD OPERATOR FORMALISM

400

Proof.

The c o n t i n u i t y of M (LIS>) N ( < 5 1 ) d p ( c ) follows from t h e conM k k s i d e r a t i o n s leading t o D e f i n i t i o n 111.2.14.

The second a s s e r t i o n i s a consequence of t h e following s t r a i g h t f o r w a r d computation.

e-d(k) M ~ ( L I ~ >e)tA(k-1) -

0

Cf. Corollary 111.2.13.

111.3. Annihilation and c r e a t i o n o p e r a t o r s In t h e previous s e c t i o n we have introduced t h e Fock a n a l y t i c i t y space and t h e Fock t r a j e c t o r y space

TF,H. The members of T m F ,H

sF ,H

may be c a l l e d f i e l d

with t h e p r o p e r t y t h a t f o r a l l k e t s . So a f i e l d k e t i s a sequence {IF;k>} k=O t > 0, {IF;k>(t)}L=OE F. A f i e l d t e s t k e t i s a f i e l d k e t {lW;k>}E with t h e property t h a t t h e r e e x i s t s

T

> 0 such t h a t

TH

i n s t e a d of e ({lW;k>(t)}E=o). I t i s k=O c l e a r t h a t we can i d e n t i f y t h e vector space of f i e l d t e s t k e t s with t h e

A s usual we w r i t e {IW,k>(t - T ) j m

a n a l y t i c i t y space

111.3.1.

SF,H'

Definition. we d e f i n e t h e o p e r a t o r N (

For each k e t

111.3.2. Theorem. For each k e t

Proof.

I@>

t h e l i n e a r operator N ( < O I )

We have t o prove t h e following

maps

sF,H

continuously i n t o

ANNIHILATION AND CREATION OPERATORS

40 1

So l e t t > 0 and t a k e 0 < T < t. Then w e g e t

111 . 3 . 3 . Theorem. For each t e s t k e t IW>,

l i n e a r mapping from

Proof.

the l i n e a r operator N ( < W l )

TF,H

into

extends t o a continuous

TF,H.

We prove t h a t

3 T , ~

: eetH

N ( <wl)

eTH :

F + F i s bounded

So l e t t > 0 and t a k e 0 < T < t so s m a l l t h a t I W > ( O )

By Theorem 111.2.6 t h e wanted r e s u l t follows.

Cf. D e f i n i t i o n 111.1.5 and Lemma 111.1.6.

e e

-TA

. (x).

Then we have

402

THE FREE FIELD OPERATOR FORMALISM

Conjugate to the operators N ( < @ I ) we introduce the operators M ( I @ > ) . 111.3.4. Defiwi t i o n .

The linear mapping M(I@>) acts on the field ket space

T F,H

as follows.

111.3.5. Theorem. The linear mapping M(l@>)

:

TF,H *

TF,H

is continuous.

-

tH Let t > 0 and let 0 < T < t. Since e M ( l @ > ) eTH c e-(t-T)H CO@>(T)) C- (U(~@>(T))e-(t-T)H)*, the operator e-tH M(l!>)

Proof. C

-

is bounded. Now the result follows from Theorem B.II.1.7.

eTH

0

111.3.6. Theorem. Let IW> be a test ket. Then M(I@>) maps Proof.

sF,H into sF,H

Let t > 0 and let 0 < T < t so small that IW>(O)

continuously. E

e-‘*(X).

Then we

have TH

e

M (IW>)

(a(lw>(-r)) e-(t-T)”) *

e-tH

and the latter operator is bounded, whence by Corollary B.II.1.3 M ( l w > ) maps

SF,H into SF,H.

0

Remark. If I@>is a normalizable ket, then

111.3.7.

Lemma.

Let t > 0 and let 0 < T < t. Then we have I

TH

e

N ( < @ I ) e-tH

=

ANNIHILATION AND CREATION OPERATORS

where 0 0 <

7

C

C

7 <

T <

T

can be taken arbitrarily. For each t,

T

40 3

and 7 with

t the function

is strongly p-integrable as a B(F)-valued function.

Proof.

Consider the following estimation.

So the function

is p-integrable. Moreover, by Theorem 111.2.13 we have for all k r IN and O C ’ C ?

etA(k-l)N k ( < @ l ) e-tA(k)

Thus the result follows. 111.3.8. Theorem. The integral expansion

admits the following interpretation

=

THE FREE FIELD OPERATOR FORMALISM

404

where t h e i n t e g r a l s e x i s t i n uniform o p e r a t o r sense.

D

111.3.9. Theorem. The i n t e g r a l expansion

admits t h e following i n t e r p r e t a t i o n

where t h e i n t e g r a l s e x i s t i n uniform o p e r a t o r sense.

Proof.

D

Cf. Theorem 111.3.8.

F i n a l l y , w e p r e s e n t an i n t e r p r e t a t i o n of t h e i n t e g r a l expression

f o r each L

E

L(TX , A ) .

From t h e previous s e c t i o n we o b t a i n t h e following e s t i m a t i o n ,

where f o r each t > 0 we have t o take e

-t A

T >

0 so small t h a t t h e o p e r a t o r

LeTA i s bounded. Consequently, f o r all t > 0

A N N I H I L A T I O N AND CREATION OPERATORS

405

i s an i n t e g r a b l e function. So t h e i n t e g r a l expression

e-tHM(LIS>)

N ( < C I ) e-tHdu(S)

M converges i n uniform o p e r a t o r sense and thus d e f i n e s a bounded o p e r a t o r on

F. W e come t o t h e following d e f i n i t i o n . 111.3.10.

Definition.

By t h e i n t e g r a l expression

we mean t h e l i n e a r mapping from

sF ,H

into

T F,H

defined by

where T > 0 h a s t o be taken s u f f i c i e n t l y small.

111.3.1

I. Coro 1lary

.

The l i n e a r mapping

i s continuous. I n a d d i t i o n f o r a l l t > 0 and a l l T , 0 < T < t ,

406

Proof. The proof

THE FREE FIELD OPERATOR FORMALISM

can be immediately obtained from Lemma 111.2.15 and it

0

omitted.

W e a r r i v e a t t h e following i n t e r p r e t a t i o n .

111.4. The f r e e f i e l d formalism I n t h e second s e c t i o n of t h i s chapter w e have introduced t h e syunnetrizing p r o j e c t i o n P ( + ) and t h e anti-symmetrizing p r o j e c t i o n P(-) w e have introduced t h e Boson Fock space F( +) = space F(-) = P ( - ) (F)

. In

P(+)(F)

. Correspondingly,

and t h e Fermion Fock

F(+) we have defined t h e nonnegative s e l f - a d j o i n t

o p e r a t o r H ( + ) = P(+) H P ( + ) and i n F( - ) t h e nonnegative s e l f - a d j o i n t o p e r a t o r H(-) = p ( - ) HP(-).

and

In Section 111.1 w e have shown t h a t t h e o p e r a t o r e

-tH(+)

,

t > 0, i s a

THE FREE FIELD FORMALISM

Hilbert-Schmidt o p e r a t o r i n F ( + ) and t h a t e Schmidt o p e r a t o r i n F

(-)

.

So t h e spaces

a r e nuclear.

and TF(-),.,(-)

-t., ( - )

SF(+)

,.,(+),

,

407

t > 0, i s a Hilbert-

TF(+)

,.,(+),

SF(-) ,H(-)

(Cf. Remarks a t t h e end of t h i s s e c t i o n . )

The l i n e a r mappings N ( < o l ) and M ( I Y > )

do not s a t i s f y t h e wanted canonical

( a n t i - ) commutation r e l a t i o n s . These CAR and CCR can be obtained by modifying t h e o p e r a t o r s N ( < @ l ) and M(IY>)

by means of t h e p r o j e c t i o n s P ( + ) and

p(-) 111.4.2. Let

Definition.

I@>be a k e t . W e introduce t h e following o p e r a t o r s .

We g a t h e r t h e r e s u l t s of t h e previous s e c t i o n s . 111.4.3. Let

Theorem.

I@>be a k e t and l e t Iw> be a t e s t k e t . The following statements a r e

valid. a . N ( ' ) ( < ~ I ) i s a continuous l i n e a r mapping from s ~ ( ' ) , ~ ( 'i n) t o ~ ~ ( ,.,(*). 2 ) On SF(k),H(k) we have

.

N ( ' ) ( < @ I )= N ( < @ I )P(') b . N ( ' ) ( <wI

)

TF ( t ),H(k) C.

i s a l s o a continuous l i n e a r mapping from T F ( + ) ,.,(?I

into

*

M(') (I@>)i s a continuous l i n e a r mapping from TF(-+ ) ,H(?) into T F ( k ) , On TF(k),H(') we have

I@>)= P(')

M(lo>)

.

a l s o a continuous l i n e a r mapping from SF(?) ,.,(k)

into

~ (* i )

THE FREE FIELD OPERATOR FORMALISM

408

For a Dirac basis (15>)

E

ScM

E

M, establish a basic

the linear operators N ( ' )

set

(

<51) and M(') (15>),

of operators. The following expansions are

valid. 111.4.4. Theorem.

The informal expansion

admits the following interpretation:

where the integrals converge in the uniform operator sense.

Proof.

The proof follows immediately from L e m a 111.3.7 and Theorem 111.4.3

by applying the projection

P(')

on both sides of the integral expressions

for N ( < @ I ) . Correspondingly, we have III.4.5. Theorem.

The informal expansion

admits the following interpretation:

where the integrals converge in the uniform operator sense.

0

THE FREE FIELD FORMALISM

409

Next we present a distributional interpretation of the so-called canonical (anti-) commutation relations. By Theorem 111.4.3 for all pairs of kets (I@>, IY>)

t h e following (anti-)

commutators are well-defined,

+/('I and

"(2)

is well-defined. 111 .4.6.

Theorem

Let I@>and IY> be kets and let IW> be a test ket.

P r o o f . Let t > 0 and let 0 < T < t. Then by Theorem 111.1.16 we have

THE FREE FIELD OPERATOR FORMALISM

410

Thus we obtain

regarded as a linear mapping from T,-(k),H(k)

into TF(k),H(t)

.

0

The previous theorem implies that for arbitrary kets lo> and \Y> the following equality is valid,

So, naturally, we arrive at the following definition. 111.4.8. Definition.

The informal (anti-) commutator expression

denotes the operator valued function on (0,m) defined by

111.4.9. Theorem.

The informal (anti-) commutator expression

We mention the following consequence. 111.4.10. CoroZZarx.

Let

(1~')~~~

be a Dirac basis. Then for all x,y

E

M:

THE FREE FIELD FORMALISM

411

0

Cf. Section 11.3. Thus we have established a mathematical interpretation of the CAR and CCR

in the free field formalism with respect to an arbitrary Dirac basis. Definition 111.3.10 and Corollary 111.3.11 yield a mathematical interpretation of the informal integral expression

II

du(6) du(f')M(lC'>) <5'1 L I F > N ( < S I )

.

M M

Multiplying both sides by P ( ' )

we obtain the corresponding relation in the

Boson case and the Fermion case. 111.4.11. Definition. Let L c L(TX,*).

Let

M(LIC>) N ( < S I ) ah(€) be the operator from + , H into

TF,H as introduced in Definition 111.3.10.

n(') ( L ) of L

Then the second q u m t i a a t i o n

is defined by

is bounded for all t > 0. Indeed, the following equalities are valid

THE FREE FIELD OPERATOR FORMALISM

412

Theorem. L ( T X , * ) . Then we have

111.4.12. Let L

E

i n t h e sense t h a t f o r a l l t > 0

Here t h e i n t e g r a l s converge i n t h e uniform o p e r a t o r sense. In a d d i t i o n ,

i n t h e sense t h a t f o r a l l t > 0 and a l l

T,

0 <

T

< t,

with convergence i n t h e uniform o p e r a t o r sense.

Proof.

W e observe t h a t f o r a l l k e t s

N o w t h e proof follows from Theorem 111.3.12.

R d m r r k . I f L = I , then $2"' satisfies :

( I ) i s t h e so-called number o p e r a t o r , which

0

THE FREE FIELD FORMALISM

'keM

'jCMk

:

Q ( " ( 1 ) P(")IO,O

= k PLf){O,O

413

,...,O,lj;k>,O,O ,...1

=

,...,O,lj,k>,O,O ,...1

.

Without proof we mention that it is possible to define a second quantization procedure for any mapping k,e

E

(A) Lke

E

L ( T x ( ~ () k ) , A ( ? ) ( k ) * T x ( ? )

,A(L)

(e)) I

M, as follows

We note that the function

is u-integrable for all t > 0.

Some comments and bibliographical notes As

general references to this chapter we mention Dirac [Di], Friedrichs

[Fr], Cook [Coo],

Kristensen et al. [KMP],

and Robertson [Rob].

A

Bogolubov et al. [BBI, [BLT],

very good starting point for work as presented in

this chapter is the reference Shvarts [Sh]. Shvarts' book is really halfway between physics and mathematics: Not too much (mathematically irrelevant) physical detail on one hand and not completely rigorous on the other hand. In literature one often finds the suggestion that every operator can be expanded in annihilation and creation operators. Two types of expansions seem to be fashionable: One with a projection on the vacuum state in the middle of each term and one without such a projection (i.e. the normal. form). The first of these claims can be proved in the context of this chapter. We can even make an expansion involving an arbitrary Dirac basis in

x.

We pre-

414

THE FREE FIELD OPEMTOR FORMALISM

sent a sketch of the proof of this result here. Writing out all mathematical details would necessitate us to introduce still more book-keeping machinery. First we remark, that starting with an arbitrary, but fixed, Dirac basis {lx>

I x E M}

in TX,* we have a Dirac basis in T given by F,H

with

The underlying Federer measure space for this Dirac basis is the measure " k space M(-) = U M , i.e. the disjoint union of all k-fold Cartesian prok=O ducts of the measure space M. Next, let E")

:

SF(')

,H ( 2 )

be a continuous linear mapping.

+ T F ( k ) ,"(?)

Because of the nuclearity of these spaces and because of the relation

there exists

E

such that

TTFaF;HsI,ZaH

P(')

g P ( * ) represents E ( i )

as

described in kernel theorem K2. Now with the matrix, cf. Section II.vii,

we have the expansion result -tH = -sH e -e =

f

k=O e=O

/...I - /...Ik

e

ake(xl

,...,xk ,yl,...,y,

;T,u)

-

THE FREE FIELD FORMALISM

415

and hence, a l s o , with a bit different notation

where

-tH

-

.e

This is a mathematical interpretation of the familiar heuristic expression which can be obtained by taking t, T , s ,

U

all equal to zero.

I n literature, e.g. [Sh] 522, one also encounters this type of expansion

expressions without the projection on the vacuum state 1 O > < O I

in the

middle. Perhaps convergence of such expressions involving an arbitrary Dirac

4 16

THE FREE F I E L D OPERATOR FORMALISM

b a s i s can be d e a l t with i n a s i m i l a r way. A t t h i s moment we do n o t know whether each operator can be represented i n t h i s , s o - c a l l e d , normal form. But i n l i t e r a t u r e it is s t a t e d , without proof or r e f e r e n c e , t h a t each operator admits a normal form representation.

417

REFERENCES

Achiezer, N.I. and I.M. Glasman, Theorie der linearen Operatoren im Hilbertraum. Akademie Verlag, Berlin, 1965. Antoine, J.P., General Dirac formalism. J. Math. Phys. 10, 1969, pp. 53-69. Antoine, J.P. and A. Grossman, Partial inner product spaces I.

J. Funct. Anal. 23, 1976, pp. 369-378.

, Partial inner product

spaces 11. J. Funct. Anal. 23, 1976,

pp. 379-391. Bargman, V., On a Hilbert space of analytic functions and an associated integral transform. Commun. Pure Appl. Math., 14, 1961, pp. 187-214.

-,

On a Hilbert space of analytic functions and an associated integral transform 11. Commun. Pure Appl. Math., 20, 1967, pp. 1-101.

Berezanskii, Ju.M., Expansions in eigenfunctions of self-adjoint operators. Transl. Math. Monographs, Vol. 17, AMS,Providence, Rhode Island, 1968.

-, Proekcionnaja spektral' naja teorema. Uspekhi Mat.

Nauk.,

39, 1984, pp. 4-54.

Bogolubov, N.N. and N.N. Bogolubov, jr., Introduction to quantum statistical mechanics. World Scientific, Singapore, 1982. Bogolubov, N.N., L.A. Logunov and T.T. Todorov, Introduction to axiomatic quantum field theory. Benjamin/Cummings Publ. Comp., Reading, Massachusetts, 1975. Bdhm, A., The rigged Hilbert space and quantum mechanics. Lect. Notes in Phys., Springer-Verlag, Berlin etc., 1978. Brown, A., A version of multiplicity theory. Topics in operator theory. Math. Surveys, 13, AMS,Providence, Rhode-Island, 1974. Bruijn, N.G. de, A theory of generalized functions with applications to Wigner distribution and Weyl correspondence. Nieuw Archief voor Wiskunde (3), 21, 1973, pp. 205-280.

418

References

Carleman, T., Sur les equations integrales singulieres 2 noyeaux reel symmetrique. Uppsala, 1923. Conway, J.B., A course in functional analysis. G.T.M.,

96,

Springer-Verlag, Berlin etc., 1985. Cook, J.M., The mathematics of second quantization. Trans.

AMS,

74, 1953, pp. 222-245. Dirac, P.A.M., The principles of quantum mechanics. Fourth edition, Clarendon press, Oxford, 1958. Dixmier, J., Von Neumann algebras. North-Holland publ. comp., Amsterdam, 1984. Dunford, N. and J.T. Schwartz, Linear operators, Vol. I, General theory. Interscience, New-York etc., 1958/1964. Eijndhoven, S.J.L. van, Analyticity spaces, trajectory spaces and linear mappings between them. Thesis, Eindhoven University of Technology, Eindhoven, 1983.

-, A theory of generalized functions based on one-parameter groups of unbounded self-adjoint operators. TH-Report 81-WSK-03, Eindhoven University of Technology, Eindhoven, 1981.

CEG

11

Eijndhoven, S.J.L. van and J. de Graaf, Some results on Hankel invariant distribution spaces. Proc. Koninklijke Nederlandse Academie van Wetenschappen, A(8611, 1983, pp. 77-87.

LEG

21

-, Analyticity spaces of self-adjoint operators subjected to perturbations with applications to Hankel invariant distribution spaces, SIAM J. Math. Anal., 16(5), 1985.

CEG

31

-,

On distribution spaces based on Jacobi polynomials. EUTReport 84-WSK-01, Eindhoven University of Technology, Eindhoven, 1984.

CEG

41

-,

A mathematical interpretation of Dirac's formalism. Part a, Dirac bases in trajectory spaces. Rep. on Math. Phys., 22, 1985. no. 2 .

CEG

51

,A

mathematical interpretation of Dirac's formalism. Part b,

Generalized eigenfunctions in trajectory spaces. Rep. on Math. Phys., 22, 1985, no. 2.

References

LEG 61

419

-, A mathematical interpretation of Dirac's formalism. Part c, Quantum field operators. To appear in Rep. on Math. Phys..

CEG 71

-,

A measure theoretical Sobolev lemma. J . Funct. Anal., 60(1), 1985, pp. 95-101.

CEG

81

-,

A fundamental approach to the generalized eigenvalue problem for self-adjoint operators. J. Funct. Anal., 63(1), 1985, pp. 74-85.

CEG

91

-,

Trajectory spaces, generalized functions and unbounded operators. Lect. Notes in Math. 1162, Springer-Verlag, Berlin etc., 1985.

CEG lo1 -,

Dirac basis: A measure theoretical concept of basis based on Carleman operators. Preprint, 1985.

CEGK]

Eijndhoven, S.J.L.

van, J. de Graaf and P. Kruszyllski, On dual

systems of inductive-projective limits of Hilbert spaces. Proc. Koninklijke Nederlandse Academie van Wetenschappen, A(88)3, 1985, pp. 277-297. LEGPI

Eijndhoven, S . J . L .

van, J. de Graaf and R . S . Pathak, A character-

:::::

ization of the spaces S

by means of holomorphic semi-

groups. SIAM J. Math. Anal. (16), 14, 1983, pp. 1180-1187. [EK]

Eijndhoven, S . J . L . van and P. Kruszyllski, On Gelfand triples originating from algebras of unbounded operators. EUT-Report 84-WSK-02, Eindhoven University of Technology, Eindhoven,

1984. [Fel

Federer, H., Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 153, Springer-Verlag, Berlin etc., 1969.

[Foi 11

Foias, C., Decompositions intdgrales des familles spectrales et semi-spectrales en operateurs qui sortent de l'espace hilbertien. Acta Sci. Math. (Szeged), 20, (1959), pp. 117-155.

CFoi 21

-,

DBcompositions en operateurs et vecteurs propres. I, I1 Rev. Math. Pures Appl. (Roumanie), 7, (19621, pp. 241-282; ibid., 7, (19621, pp. 571-602; Add., ibid., 9, (1964), pp. 805-809.

420

References

Friedrichs, K.O., Mathematical aspects of the quantum theory of fields. Interscience Publ., New-York, 1953. Gelfand, I.M. and G.E. Shilov, Generalized functions. Vol. 2, Ac. Press, New-York, 1968. Gelfand, I.M. and N.Ya. Vilenkin, Generalized functions. Vol. 4, Ac. Press, New-York, 1964. Gong-Zhing Zhang, Theory of distributions of S-type and pansions. Chinese Math. (21, 4, 1963, pp. 211-221. Goodman, R., Analytic and entire vectors for representations of Lie groups. Trans. AMS 143, 1969, pp. 55-76.

-, Some regularity theorems for operators in an enveloping algebra. J. Diff. Eq. 10, 1971, pp. 448-470.

-,

Differential operators of infinite order on a Lie group I. J. Maths. Mech. 19(10), 1970, pp. 879-894.

Graaf, J. de, A theory of generalized functions based on holomorphic semigroups. TH-Report 79-WSK-02, Eindhoven University

of Technology, Eindhoven, 1979. [G 21

-,

A theory of generalized functions based on holomorphic semigroups. Part a, Introduction and survey. Proc. Koninklijke Nederlandse Academie van Wetenschappen, A(8614, pp. 407-420.

[G 31

-,

A theory of generalized functions based on holomorphic semigroups. Part b, Analyticity spaces, trajectory spaces and their pairing. Proc. K.N.A.W.,

[G 41

A(87)2, pp. 155-171.

-, A theory of generalized functions based on holomorphic semigroups. Part c, Linear mappings, tensor products and kernel theorems. Proc. K.N.A.W.

[G 51

-,

A(87)2, pp. 173-187.

Generalized functions and operators on the unit sphere. In Proceedings Bar-le-Duc, Lect. Notes in Math. 1171, SpringerVerlag, Berlin etc., 1985.

Guichardet, A., Symmetric Hilbert Spaces and Related Topics. Lect. Notes in Math. 261, Springer-Verlag, New-York, 1972. Takashi Ichinose, Spectral properties of tensor products of linear operators. Trans. AMS,237, 1978, pp. 223-254.

References

CJal

Jauch, J.M.,

42 1

On bras and kets. Aspects of quantum theory, edited

by A. Salam and E. Wigner, Cambridge Un. Press, 1972. Jones, D.S., Generalized functions. McGraw-Hill, New-York, 1966. Klamer, F., Group representations in Hilbert subspaces of a locally convex space. Thesis, Groningen, 1979. Korevaar, J., Pansions and the theory of Fourier transforms. Trans. AMS 91, 1959, p. 53. C m r1

Korotkov, V.B., Integral operators with Carleman kernels. Doklady &ad.

C W I

Nauk SSSR 165, 1965, pp. 1496-1499.

Kristensen, P., L. Mejlbo and P.T. Poulsen, Tempered distributions in infinitely many dimensions I. Commun. Math. Phys. 1, 1965, pp. 175-214. Kufner, A., D. John and S. Fucik, Function spaces. Noordhof, Leiden, 1977. Lighthill, M.J.,

Fourier analysis and generalized functions.

Cambridge Un. Press, London, 1958. Lions, J.L. and E. Magenes. Non-homogeneous boundary value problems and applications. Vol. I, Grundlehren 181, SpringerVerlag, Berlin etc., 1972. Maurin, K.,

Methods of Hilbert spaces. Polish Scientific Publ.,

Warszawa, 1967. Melsheimer, O., Rigged Hilbert space formalism. J. Math. Phys. 15, 1974, p. 902. Minemura, K., Harmonic functions on real hyperbolic spaces. Hiroshima Math. J. 3, 1973, pp. 121-151. Misra, B., D. Speiser and G. Targonski, Integral operators in the theory of scattering. Helvetia Phys. Acta 36, 1963, pp. 963-980. Nagel, B., Generalized eigenvectors. In Proceedings NATO AS1 on Math. Phys., Istanbul 1970, Reidel. Nelson, E., Analytic vectors. Ann. Math. 70, 1959, pp. 572-615.

-,

Topics in dynamics I: Flows. Math. notes, Princeton Un. Press, Princeton, 1969.

References

422

[Neu 11

Neumann, J. von, Mathematical foundations of quantum mechanics. Princeton Un. Press, Princeton, 1955.

[Neu 21

-, Charakterisierung des Spektrums eines Integraloperators. Actualitis Sci. et Ind. 299, 1935.

[Nie]

Nielsen, O.A., Direct integral theory. Lecture Notes in pure and appl. math., Vol. 61, M. Dekker, New-York-Bazel, 1980.

[Rob]

Robertson, B., Introduction to field operators in quantum mechanics. Amer. J. Phys. 41, 1973, pp. 678-690.

[Ro]

Rogers, J.E., The Dirac bra and ket formalism. J. Math. Phys. 7, 1966, pp. 1097-1104.

[Sch]

Schaefer, H.H., Topological vector spaces. McMillan, New-York, 1966.

[Schr]

Schreiber, M., Semi-Carleman operators. Acta Sci. Math. 24, 1963, pp. 82-87.

[Schw 11 Schwartz, L., Theorie des distributions. Hermann, Paris, 1973. [Schw 21

-, Sous-espaces hilbertiens d'espaces

vectoriels topologiques

et noyeaux associis. J. Anal. Math. 13, 1964, pp. 115-256. [Shl

Shvarts, A.S., Matematicheskie osnovy kvantovoj teorii polya. Moskva, Atomisdat, 1975.

[Sil

Singer, I., Bases in Banach spaces. Vol. I, Springer-Verlag, Berlin etc., 1970.

[Sml

Smirnov, W.I., Lehrgang der hdheren Mathematik. Teil 5 VEB, Deutscher Verlag der Wissenschaften, Berlin, 1982.

[Stol

Stone, M., Linear transformations in Hilbert space and their applications to analysis. Am. Math. SOC., New-York, 1932.

[Te]

Temple, G., The theory of generalized functions. Cambridge Un. Press, London, 1958.

[Tho]

Thomas, E.G.F.,

Theorem of Bochner-Schwartz-Godement. Functional

analysis: Surveys and recent results 111. Math. Studies, North-Holland, Amsterdam, 1984. [Wei 1 1

Weidmann, J., Linear operators in Hilbert spaces. GTM, 68, Springer-Verlag, Berlin etc., 1980.

References

423

Lwei 21

-, Carleman-operatoren. Manuscripta

[Weir]

Weir, A.J., General integration and measure. Cambridge Un. Press,

Math. 2 , 1970, pp. 1-38.

Cambridge, 1974.

[wZ]

Wheeden, R . L . and

A.

Zygmund, Measure and integral. Marcel Dekker

Inc. , New-York, 1974. [Yo]

Yosida, K., Functional analysis. 6th Edition, Springer-Verlag, Berlin etc., 1980.

[Ze]

Zemanian, A.H.,

Generalized integral transformations. Interscience

Publ., New-York, 1968.

This Page Intentionally Left Blank

425

INDEX

A

commutative multiplicity theory 82 ff.

almost periodic function 318 ff.

continuous linear mappings

analyticity domain 245

- from

analyticity space 150

-

analytic vector 151, 245 angular momenta 359

sX,A

s

into

from S

Y,B

into T X,A Y,B into S X,A Y,B from T into T X,A Y,E from T

annihilation operator 382, 406

counting measure 8

atom, p- 17

creation operator 382, 406

184 186 188 190

CAR 373, 387, 409

B

CCR 370, 387, 409

Banach-Steinhaus theorem 174 barreled 160

D

Besicovitch covering lemma 33

delta function 44, 257

Borel field of Hilbert spaces 109

Dirac basis 56, 258, 325

Borel function 3 Borel measure 3

canonical- 64, 267, 325 Dirac-Riesz basis 70

Borel space 3 Borel vector field 109 bornological 160

canonical- 74 Dirac's bracket notation 281 ff., 310 ff.

Boson Fock space 384

Dirac's expansion 334 ff.

bra 313

Dirac's formalism 281 ff., 309 ff.

bracket 316

Dirac's matrix calculus 338 ff.

bra space 313

Dirac's orthogonality relations 331 Direct integral 110

C

Direct resolution 113, canonical- 126

candidate generalized eigenvector 94 canonical direct resolution 126

disjoint union 62, 90

canonical evaluation functional 39

disjoint measure classes 83

canonical representative 48, 257

distribution space 149 ff.

Carleman kernel 8

dominated convergence 6

Carleman operator 8, d- 119 Carleman triple 70

E

Carleman type operator 13

eigentuple 88

m

C -domain

equivalent measures 83

column finite matrix

Euclidean group 248

commutative multiplicity theorem 89

extendible mapping

364 83,

192-193

Index

426

F

L

Federer measure space 30

Laguerre polynomials 240

Fermion Fock space 385

Lie group,

Fock space 378 ff.

unitary representation of 246 ff.

Fubini-Tonelli theorem 7

Lie algebra, infinitesimal representation of 246 ff.

G Gelfand-Shilov space 149 generalized eigenprojection 140

M

generalized eigenvalue problem 81, 271

genuine generalized eigenvector 94

matrix representation 208 ff. measure space a-finite- 3

generalized function 148 ff.

purely atomic- 17 measure theoretical Sobolev lemma 23, 42

H Hankel transform 241

momentum representation 358

harmonic oscillator 353

monotone convergence theorem 5

Heisenberg group 247

Monte1 space 160

Hermite polynomials 236

multiplicity theorem 89, 277

Hilbert-Schmidt operator 8

multiplier 220

hyperfunctions

- 2n-periodic 225 - on [-1,11 245

N normal form 413 normalizable

-

I inductive limit (topology) 152

ket 312 bra 316

integrable, p- 5

0 J

observables 3 5 1

Jacobi polynomials 242 joint spectral resolution 86

P

joint spectrum 81, 88

pairing

K kernel theorems 202 ff.

of R(X) and R - l ( X ) 2 8

- of SX,A

and T

X,A

171

Parseval's identity 333

ket 310

Plancherel (type) formula 59, 265

ket space 311

Poisson kernel 221 polish space 110

Index

position representation 357

42 7

standard splitting 89, 281

postulates of quantum mechanics 301-302

projection valued measure 132

strong Carleman operator 19 support of a measure 82

T test bra 316

R regular Bore1 measure 3 1

test function 148 ff.

relative differentiation theorem

test ket 313

36, 92

representatives

- of bras -

332 ff.

test space 148 ff. topological tensor product 193

-

of kets 332 ff. of linear operators 338 ff.

Riesz-Schauder basis 58 rotation group 249, 359

of S X , A and of T

X,A

s Y, 8

and T

of SX,* and of TX,A and

trajectory 164

Y,B

T Y,B S Y,B

195 196 197 199

trajectory space 164

S Schauder basis 57

U

Schrodinger group 248

uniform multiplicity

sequence space 178 smoothed function 149

V

smooth function 149

von Neumann spectral theorem

smoothing operator 149 Sobolev embedding theorem 47

d

Sobolev triple 28

rleak topology

This Page Intentionally Left Blank

429

L I S T OF SYMBOLS

SMSI.1

212

TMTI.1

214

A, 'A

218

Ra

219

M

219

Tb

219

2,

219

J,

D

219

Ao, Ae A

A sJ,

153

TX,A emb <w,F>

S{*)

163

164 171 181

222, 223

223

P

A

plo'

Sq-l

s: 2 Hn )''L

p,e

226, 227

227 229

231

236

L(')

n ' n

240

L i s t of symbols

430

y(") My

241 241

([GI ,M, u,A, X ) Y

262

272

322 ff.

I*><-I

TTXeX;Aa7, I@A

32 1

S S ~ e ~ i ~ 322 a ~ , ~ e ~ T S ~; e A ~~II,

~ A323

TSXeX;IaA,AxI Ilx> I x E <x

Iy>

MI

324 326

= 6 (x) Y

[.Ixy339

331