Perturbation Theory for Linear Operators

By: Tosio Kato

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Paperback | 15 February 1995 | Edition Number 2

At a Glance

Paperback
648 Pages

Edition Type
Revised

Dimensions(cm)
23.39 x 15.6 x 3.3

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In view of recent development in perturbation theory, supplementary notes and a supplementary bibliography are added at the end of the new edition. Little change has been made in the text except that the para- graphs V- 4.5, VI- 4.3, and VIII- 1.4 have been completely rewritten, and a number of minor errors, mostly typographical, have been corrected. The author would like to thank many readers who brought the errors to his attention. Due to these changes, some theorems, lemmas, and formulas of the first edition are missing from the new edition while new ones are added. The new ones have numbers different from those attached to the old ones which they may have replaced. Despite considerable expansion, the bibliography i" not intended to be complete. Berkeley, April 1976 TosIO RATO Preface to the First Edition This book is intended to give a systematic presentation of perturba- tion theory for linear operators. It is hoped that the book will be useful to students as well as to mature scientists, both in mathematics and in the physical sciences.

Industry Reviews

"The monograph by T. Kato is an excellent textbook in the theory of linear operators in Banach and Hilbert spaces. It is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
In chapters 1, 3, 5 operators in finite-dimensional vector spaces, Banach spaces and Hilbert spaces are introduced. Stability and perturbation theory are studied in finite-dimensional spaces (chapter 2) and in Banach spaces (chapter 4). Sesquilinear forms in Hilbert spaces are considered in detail (chapter 6), analytic and asymptotic perturbation theory is described (chapter 7 and 8). The fundamentals of semigroup theory are given in chapter 9. The supplementary notes appearing in the second edition of the book gave mainly additional information concerning scattering theory described in chapter 10.
The first edition is now 30 years old. The revised edition is 20 years old. Nevertheless it is a standard textbook for the theory of linear operators. It is user-friendly in the sense that any sought after definitions, theorems or proofs may be easily located. In the last two decades much progress has been made in understanding some of the topics dealt with in the book, for instance in semigroup and scattering theory. However the book has such a high didactical and scientific standard that I can recomment it for any mathematician or physicist interested in this field.
Zentralblatt MATH, 836

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Introduction
Operator theory in finite-dimensional vector spaces
Vector spaces and normed vector spaces	p. 1
Linear forms and the adjoint space	p. 10
Linear operators	p. 16
Analysis with operators	p. 25
The eigenvalue problem	p. 34
Operators in unitary spaces	p. 47
Perturbation theory in a finite-dimensional space	p. 62
Analytic perturbation of eigenvalues	p. 63
Perturbation series	p. 74
Convergence radii and error estimates	p. 88
Similarity transformations of the eigenspaces and eigenvectors	p. 98
Non-analytic perturbations	p. 106
Perturbation of symmetric operators	p. 120
Introduction to the theory of operators in Banach spaces
Banach spaces	p. 127
Linear operators in Banach spaces	p. 142
Bounded operators	p. 149
Compact operators	p. 157
Closed operators	p. 163
Resolvents and spectra	p. 172
Stability theorems
Stability of closedness and bounded invertibility	p. 189
Generalized convergence of closed operators	p. 197
Perturbation of the spectrum	p. 208
Pairs of closed linear manifolds	p. 218
Stability theorems for semi-Fredholm operators	p. 229
Degenerate perturbations	p. 244
Operators in Hilbert spaces
Hilbert space	p. 251
Bounded operators in Hilbert spaces	p. 256
Unbounded operators in Hilbert spaces	p. 267
Perturbation of selfadjoint operators	p. 287
The Schrodinger and Dirac operators	p. 297
Sesquilinear forms in Hilbert spaces and associated operators
Sesquilinear and quadratic forms	p. 308
The representation theorems	p. 322
Perturbation of sesquilinear forms and the associated operators	p. 336
Quadratic forms and the Schrodinger operators	p. 343
The spectral theorem and perturbation of spectral families	p. 353
Analytic perturbation theory
Analytic families of operators	p. 365
Holomorphic families of type (A)	p. 375
Selfadjoint holomorphic families	p. 385
Holomorphic families of type (B)	p. 393
Further problems of analytic perturbation theory	p. 413
Eigenvalue problems in the generalized form	p. 416
Asymptotic perturbation theory
Strong convergence in the generalized sense	p. 427
Asymptotic expansions	p. 441
Generalized strong convergence of sectorial operators	p. 453
Asymptotic expansions for sectorial operators	p. 463
Spectral concentration	p. 473
Perturbation theory for semigroups of operators
One-parameter semigroups and groups of operators	p. 479
Perturbation of semigroups	p. 497
Approximation by discrete semigroups	p. 509
Perturbation of continuous spectra and unitary equivalence
The continuous spectrum of a selfadjoint operator	p. 516
Perturbation of continuous spectra	p. 525
Wave operators and the stability of absolutely continuous spectra	p. 529
Existence and completeness of wave operators	p. 537
A stationary method	p. 553
Bibliography	p. 583
Supplementary Bibliography	p. 596
Notation index	p. 606
Author index	p. 608
Subject index	p. 612
Table of Contents provided by Blackwell. All Rights Reserved.

Perturbation Theory for Linear Operators

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