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About the work These two volumes provide a modem treatment of Hilbert space theory and linear operators for use by mathematicians, mathematical physicists and research students. They are entirely new translations of the third Russian edition with much new material. Volume I represents a standard text, with new material in Chapter 5, treating the theory of Hilbert space, linear functionals and bounded linear operators, projection and unitary operators, general concepts and theorems in the theory of linear operators and the spectral analysis of completely continuous operators and unitary and self- adjoint operators. Volume n contains many new results, including a new chapter on the spectrum and perturbations of self-adjoint operators. This is followed by treatment of the theory of extension of symmetric operators, generalised exten sions and spectral functions of symmetric operators, an appendix on differential operators and an entirely new appendix on integral operators containing material hitherto unavailable in book form. In its comprehensive treatment of the subject, with its wide range of applications to integral and differential equations, the work gives an unparalleled and definitive modem treatment for essential reference by specialists and students. Volume 11980,339pages ISBN 0273 08495 X. Full contents shown inside this volume. Theory of Linear Operators in Hilbert Space Volume II Main Editors A. Jeffrey, University of Newcastle-upon-Tyne R. G. Douglas, State University of New York at Stony Brook Editorial Board F. F. Bonsall, University of Edinburgh H. Brezis, Université de Paris G. Fichera, Université di Roma R. P. Gilbert, University of Delaware K. Kirchgässner, Universität Stuttgart R. E. Meyer, University of Wisconsin-Madison J. Nitsche, Universität Freiburg L. E. Payne, Cornell University I. N. Stewart, University of Warwick S. J. Taylor, University of Liverpool Theory of Linear Operators in Hilbert Space Volume II N. I. Akhiezer University of Kharkov The late I. M. Glazman Polytechnic Institute, Kharkov Translated by E. R. Dawson University of Dundee English translation edited by W. N. Everitt University of Dundee rM ^f Published in association with w Scottish Academic Press, Edinburgh by Pitman Advanced Publishing Program Boston • London • Melbourne PITMAN PUBLISHING LIMITED 39 Parker Street, London WC2B 5PB PITMAN PUBLISHING INC 1020 Plain Street, Marshfield, Massachusetts Associated Companies Pitman Publishing Pty Ltd, Melbourne Pitman Publishing New Zealand Ltd, Wellington Copp Clark Pitman, Toronto AMS Subject Classifications: (main) 46C, 46E, 47A, 47B, 47E05, 47G0S (subsidiary) 34A30, 34B20, 34B25 Library of Congress Cataloging in Publication Data Akhiezer, Naum Il’ich, 1901-1980 Theory of linear operators in Hilbert space. (Monographs and Studies in Mathematics) Revised translation of Teoriya lineinykh operatorov. Bibliography: p. Includes index. 1. Hilbert space. 2. Linear operators. QA322.4.A3813 1980 515.ГЗЗ 80-18258 ISBN 0-273-08495-8-(v.2) Translated from TEORIYA LINEINYKH OPERATOROV V GILBERTOVOM PROSTRANSTVE (TOM II) by E. R. Dawson © Vishcha Shkola 1978 Third Russian edition published by Vishcha Shkola, Kharkov English edition published by Pitman in association with Scottish Academic Press © E. R. Dawson 1981 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 and/or otherwise without the prior written permission of the publishers. ISBN 0 273 08496 8 Printed and bound in Great Britain at The Pitman Press, Bath Contents Volumes 1 and II Author’s preface to the English edition X Preface to the second edition xi Notation xii References to volumes I and II xiii Index to volumes I and II xxiii Volume I !• Hilbert space 1. Linear systems 1 2. Linear manifolds 2 3. The scalar product (inner product) 4 4. Some general concepts 7 5. Hilbert space 8 6. Distance of a point from a convex set in H 12 7. Projection of a vector on to a sub-space 14 8. Orthogonalization of a sequence of vectors 19 9. Bessel’s inequality and Parseval’s equation 21 10. Complete orthonormal systems of vectors in H 27 11. The space L^ 32 12 Complete orthonormal sequences in L^ 36 13 Bi-orthogonal systems of vectors in H 40 14. The space L^ 43 15. The space of almost-periodic functions 49 16. The concept of a basis of a space 50 2. Linear functionals and bounded linear operators 56 17. Point-functions 56 18. Linear functionals 58 19. A theorem of F. Riesz 60 20. A criterion for a given system of vectors to be closed in H 63 21. A lemma on convex functionals 64 CONTENTS 22. Bounded linear operators 67 23. Bilinear functionals 69 24. The general form of a bilinear functional 71 25. Adjoint operators 73 26. Weak convergence in H 77 27. Compactness 79 28. A criterion for an operator to be bounded 82 29. Linear operators in a separable space 82 30. Completely continuous operators (compact operators) 88 31. Absolute norms 91 32. Hilbert-Schmidt operators 95 33. Convergent sequences of bounded linear operators 97 34. Sets of bounded linear operators in a separable Hilbert space 99 3. Projection operators and unitary operators 103 35. Definition of a projection operator 103 36. Properties of projection operators 104 37. Operations on projection operators 105 38. Sequences of projection operators 108 39. The aperture of two linear manifolds 109 40. Unitary operators 112 41. Isometric operators 114 42. The Fourier-Plancherel operator 116 4. General concepts and theorems in the theory of linear operators 120 43. Closed operators 120 44. General definition of an adjoint operator 121 45. Eigenvectors, invariant sub-spaces, and the reducibility of linear operators 124 46. Symmetric operators 128 47. More about isometric and unitary operators 131 48. The concept of the spectrum 132 49. The resolvent 136 50. Conjugation operators 139 51. The graph method 141 52. A generalization of the concept of a projection operator 145 53. Matrix representation of unbounded symmetric operators 147 54. The operator of multiplication by the independent variable 152 55. The operator of differentiation 156 CONTENTS 5. Spectral analysis of completely continuous operators 164 56. Two lemmas 164 57. On the eigenvalues of completely continuous operators in R 166 58. Fredholm theorems for completely continuous operators 170 59. F. Riesz’s method in the theory of linear functional equa tions 173 60. Completely continuous self-adjoint operators in R 178 61. Completely continuous, normal operators 183 62. Application to the theory of almost-periodic functions 186 63. Expansion of an arbitrary completely continuous operator as a series of one-dimensional operators 193 64. Nuclear operators 196 65. Schauder’s fixed-point theorem 202 66. A theorem on the existence of an invariant sub-space for any completely continuous operator, and a generalization 208 6. Spectral analysis of unitary and self-adjoint operators 212 67. The resolution of the identity 212 68. The trigonometrical problem of moments 215 69. Analytic functions with values lying in a half-plane 218 70. The Bochner-Khinchin theorem 224 71. The spectral resolution of a unitary operator 228 72. Stieltjes-integral operators 233 73. Integral representation of a group of unitary operators 239 74. Integral representation of the resolvent of a self-adjoint operator 241 75. Spectral resolution of self-adjoint operators 247 76. On sets of operator-measure zero in a separable space 253 77. Functions of unitary operators 256 78. Direct derivation of the spectral resolution of a unitary operator 261 79. The Cayley transform 264 80. Commutative operators 269 81. Spectral resolution of bounded normal operators 270 82. The spectrum of a self-adjoint operator and of a unitary operator 272 83. The simple spectrum 277 84. Spectral types 283 85. The multiple spectrum 286 86. Canonical form of a self-adjoint operator with a spectrum of finite multiplicity 288 CONTENTS 87. Unitary invariants of self-adjoint operators 292 88. General definition of functions of self-adjoint operators 294 89. Examples 296 90. Rings of bounded, self-adjoint operators 303 91. A characteristic property of functions of a self-adjoint operator 308 92. A theorem on the generating operator 312 References to volumes I and II xiii Index to volumes I and II xxiii Volume II 7. The spectnim and perturbations of self-adjoint operators 313 93. The continuous spectrum of a self-adjoint operator 313 94. H. Weyl’s theorem and von Neumann’s theorem on com pletely continuous perturbations 317 95. The absolutely continuous part and the singular part of the spectrum 325 96. Invariance of the absolutely continuous part of the spec trum relative to finite-dimensional perturbations 327 97. Definition and formal properties of wave operators 332 98. Existence of wave operators in the case of finite dimensional perturbations 337 99. Transition to the general case of nuclear perturbations 341 8. Theory of extensions of symmetric operators 347 100. Deficiency indices 347 101. More about the Cayley transform 351 102. The von Neumann formulae 354 103. Simple symmetric operators 357 104. The structure of maximal operators 360 105. The spectra of self-adjoint extensions of a given symmetric operator 364 106. M. G. Krein’s formula for the resolvents of self-adjoint extensions of a given symmetric operator 367 107. Self-adjoint extensions of semi-bounded operators 372 108. Norm-preserving, self-adjoint extensions of a bounded symmetric operator whose domain is not dense in H 376 109. Self-adjoint extensions of a semi-bounded symmetric operator which preserve its lower bound 382