Table Of ContentSPECTRAL METHODS IN INFINITE-DIMENSIONAL ANALYSIS
Mathematical Physics and Applied Mathematics
Editor:
M. Flato, Universite de Bourgogne, Dijon, France
The titles published in this series are listed at the end of this volume.
Volume 12/1
SPECTRAL METHODS
IN
INFINITE-DIMENSIONAL
ANALYSIS
Volume 1
by
Y. M. Berezansky
and
Y. G. Kondratiev
Institute of Mathematics,
Ukrainian Academy of Science,
Kiev, Ukraine
W
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging-in-Publication Data
Berezanskll, fU. M. (fünf Makarovlch)
[Spektra 1 'nye metody v beskonechnomernom anallze. English]
Spectral methods in Infinite-dimensional analysis / by Yurlj M.
Berezansky and Yurlj G. Kondratlev.
p. cm. — (Mathematical physics and applied mathematics ; v.
12)
Includes bibliographical references and index.
ISBN 978-94-010-4227-7 ISBN 978-94-011-0509-5 (eBook)
DOI 10.1007/978-94-011-0509-5
1. Quantum field theory—Mathematics. 2. Degrees of freedom.
3. Statistical physics. 4. Spectral theory (Mathematics)
5. Infinite groups. 6. Mathematical physics. I. Kondrat'ev, fD.
G. (fDrll Gr Igor 'evich) II. Title. III. Series.
QC174.52.D43B47 1994
530. 1'55782—dc20 94-14885
ISBN 978-94-010-4227-7
This is the updated and revised translation of the original Russian work,
Spectral Methods in Infinite-Dimensional Analysis.
Translated by P.V. Malyshev and D. V. Malyshev.
Published by Naukova Dumka, Kiev, © 1988.
Printed on acid-free paper
All Rights Reserved
© 1995 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1995
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Volume 1
Preface to the English Edition ix
Preface xi
Introduction xiii
CHAPTER 1. RIGGED SPACES 1
§ 1. Rigged Hilbert Spaces 3
1.1. Hilbert riggings 3
1.2. Rigging of a Hilbert space by linear topologic spaces 13
§ 2. Tensor Products and the Kernel Theorem 21
2.1. Finite tensor products 21
2.2. Infinite tensor products 31
2.3. The kernel theorem 39
2.4. The case of bilinear forms 48
§ 3. Bilinear Forms 52
3.1. Some remarks on the completion of spaces 52
3.2. Two equivalent languages: pre-chains and positive forms 56
3.3. Semi-bounded forms and operators 59
3.4. Form-sums of operators 66
3.5. Nonclosable forms 71
CHAPTER 2. GENERALIZED FUNCTIONS OF INFINITELY MANY
VARIABLES. GAUSSIAN MEASURES 79
§ 1. Measures on Linear Spaces 81
l.l. Cylindrical sets and product measures 81
v
VI Contents
1.2. Functions summable with respect to infinite products of measures 86
l.3. Products of Gaussian measures 91
1.4. Measures in Hilbert spaces 96
1.5. Gaussian measures in finite-dimensional spaces 101
1.6. Gaussian measures in Hilbert spaces 105
1.7. Gaussian measures in Hilbert riggings 109
1.8. Gaussian measures in Hilbert riggings. The language of the
Fourier transform 115
1.9. Gaussian measures in nuclear riggings 119
1.10. Cylindrical functions 126
1.11. The case of abstract duality 128
§2. Some Problems ofA nalysis in the Case of Gaussian Measures 130
2.1. Measurable linear and multilinear functionals 130
2.2. The Fock space and the Wiener-Ito decomposition 138
2.3. The Segal isomorphism 145
2.4. Shifts of Gaussian measures and absolute continuity 151
2.5. The Fourier-Wiener transform 157
§3. Differentiable Functions on Infinite-Dimensional Spaces 161
3.1. Derivatives of functions on a Hilbert space 162
3.2. Derivatives of functions given in riggings 170
3.3. Smooth functions in nuclear riggings 173
§4. Spaces of Test and Generalized Functions of Infinitely Many
Variables as Infinite Tensor Products 183
4.1. Infinite tensor products of nuclear riggings 183
4.2. The space Jl(IR 1) 188
4.3. The spaces of functions of infinitely many variables 193
4.4. Some properties of test and generalized functions 200
4.5. The space q(IR~) 209
§s. Spaces of Test and Generalized Functions of Infinitely Many
Variables in Invariant Form 216
5.1. Riggings of the Fock space 217
5.2. Functional realizations of riggings of the Fock space 225
5.3. Relation between functional realizations of the Fock space 234
5.4. Internal description for functional realizations of the riggings
of the Fock space 239
5.5. Examples of generalized functions of infinitely many variables 251
Contents vii
CHAPTER 3. SPECTRAL THEOREM 259
§ 1. Joint Resolution of the Identity 264
1.1. General resolution of the identity 264
1.2. Products of finitely many resolutions of the identity 266
1.3. Construction of a joint resolution of the identity in the general case 269
104. Topologization 271
1.5. Regularity of a joint resolution of the identity 273
1.6. The concept of the support of a measure and its properties 275
1.7. Families of multiplication operators 279
1.8. Construction of a measure on a larger space from a measure
on a smaller space. Compactification 283
1.9. Construction of a measure on a smaller space from a measure
on a larger space. Modification of a measure 287
1.10. Properness of a joint resolution of the identity 290
1.11. Spectral representation of a family of commuting normal operators 295
§ 2. Spectral Projection Theorem 297
2.1. Differentiation of an operator-valued measure with respect to its trace 297
2.2. Differentiation of a resolution of the identity. The spectral measure 304
2.3. Differentiation of a joint resolution of the identity 306
204. The case of a nuclear rigging 309
2.5. The concepts of a generalized joint eigenvector and of a generalized
spectrum of a family of operators 310
2.6. Spectral theorem. The case of at most countably many operators 314
2.7. Spectral theorem. The general case 317
2.8. Spectral theorem. The case of at most countably many unbounded
operators 335
2.9. Continuity and smoothness of eigenvalues 336
2.10. Supplementary remarks. Three classical examples of expansion 337
§ 3. Supplementary Facts about Expansions 344
3.1. The Fourier transform. The direct integral of Hilbert spaces 344
3.2. The case of an imbedding that is not quasinuclear 351
3.3. The expansion in eigenfunctions of Carleman operators 355
304. Existence of a rigging 365
3.5. Connection with the theory of commutative normalized algebras and
with the nuclear spectral theorem 377
3.6. The example of a family of commuting selfadjoint operators:
Operators acting upon different variables 386
viii Contents
3.7. The Fourier-Wiener transformation. Spectral approach 395
3.8. The creation and annihilation operators. The Segal isomorphism
(spectral approach) 401
CHAPTER 4. REPRESENTATIONS BY COMMUTING OPERAT ORS 409
§ 1. Representations of Some Algebraic Structures by Commuting
Operators 411
1.1. Representations of a group 411
1.2. The generalizations of Stone and Sz.-Nagy-Hille theorems 413
1.3. Representation of semi-groups 428
1.4. Representations of linear spaces 432
1.5. Representations of an algebra 436
§ 2. Hypercomplex Systems with Locally Compact Basis. Representations of
These Systems by Commuting Operators 438
2.1. Concept of a hypercomplex system with locally compact basis 439
2.2. Some properties of hypercomplex systems. Operators of a
generalized shift 444
2.3. Nuclear space of functions on a basis of a hypercomplex system 455
2.4. Representation of hypercomplex systems 476
2.5. Examples 485
§ 3. Applications to Representations of Commutation Relations 492
3.1. Change of variables in spectral integrals 492
3.2. A commutation relation and its representation 497
3.3. Examples of commutative models 507
Bibliographical Notes 523
References 533
Subject Index of Volume 1 573
List of Notations of Volume 1 577
Contents of Volume 2 579
Contents
Volume 2
CHAPTER 5. APPLICATION OF THE THEORY OF EXPANSIONS
TO HARMONIC ANALYSIS 1
§ 1. Some Criteria of the Selfadjointness of Operators 4
1.1. Schrodinger and hyperbolic criteria of selfadjointness 4
1.2. Some generalizations of the hyperbolic criterion 13
1.3. Quasianalytical criteria of selfadjointness 21
104. Parabolic criteria of selfadjointness 28
1.5. Selfadjointness of perturbations of an operator by a potential 31
1.6. Another evolutionary criterion of selfadjointness 39
1.7. Approximation theorem 39
1.8. Verification of the commutativity of selfadjoint operators 44
§ 2. Generalized Power Moment Problem 51
2.1. Generalized symmetric power moment problem 52
2.2. Examples 73
2.3. Two more proofs of the theorem on representation 82
204. Generalized power moment problem 85
2.5. A family of shift operators and the corresponding Fourier transform 94
2.6. Some additional remarks 100
§ 3. Positive Definite Functions of Infinitely Many Variables 102
3.1. Some auxiliary facts 102
3.2. Positive definite functions and their spectral representation 106
3.3. Positive definite functions on a layer 121
§ 4. Positive Definite Functions on a Hilbert Space 125
4.1. Positive definite functions on a layer in the Hilbert space 125
v
vi Contents
4.2. Extension of a positive definite function from a layer in the
Hilbert space to the whole of the space. Connection with the
moment problem 137
4.3. Other formulations of the results concerning positive definite
functions on Hilbert spaces 146
4.4. Connection between unitary representations of a Hilbert space
and positive definite functions on this space 149
4.5. Some additional remarks 157
§s.
General Scheme oft he Representation of Positive Definite Kernels
in Terms of Elementary Ones 160
5.1. Hilbert space constructed for a given positive definite kernel
and its rigging 160
5.2. Operators acting in the space generated by a positive definite
kernel 165
5.3. Spectral representation of positive definite kernels in terms of
elementary ones 170
5.4. The case of smooth positive definite kernels 175
CHAPTER 6. INFINITE-DIMENSIONAL ELLIPTIC DIFFERENTIAL
OPERA TORS OF THE SECOND ORDER 181
§ 1. Second Quantization Operators in the Schrodinger
Representation 184
1.1. The second quantization operators 184
1.2. Mehler's formula in the infinite case 188
1.3. Definition of the second quantization operators by using
a differential expression 195
1.4. Properties of the second quantization operators 202
1.5. Functional integrals corresponding to the second quantization
operators 209
§ 2. Perturbations of the Second Quantization Operators by Potentials 224
2.1. Kato's inequality and its consequences 225
2.2. The Feynman - Kac formula 231
2.3. Properties of perturbed operators 238
2.4. More singular potential perturbations. Interactions in constructive
quantum field theory 245
