Perturbation Theory for Linear Operators / Edition 2

by Tosio Kato

Overview

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 perturbation 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.

Product Details

ISBN-13:	9783540586616
Publisher:	Springer Berlin Heidelberg
Publication date:	02/15/1995
Series:	Classics in Mathematics , #132
Edition description:	2nd ed. 1995
Pages:	623
Sales rank:	600,578
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Biography of Tosio Kato

Tosio Kato was born in 1917 in a village to the north of Tokyo. He studied theoretical physics at the Imperial University of Tokyo. After several years of inactivity during World War II due to poor health, he joined the Faculty of Science at the University of Tokyo in 1951. From 1962 he was Professor of Mathematics at the University of California, Berkeley, where he is now Professor Emeritus.

Kato was a pioneer in modern mathematical physics. He worked in te areas of operator theory, quantum mechanics, hydrodynamics, and partial differential equations, both linear and nonlinear.

Table of Contents

One Operator theory in finite-dimensional vector spaces.- § 1. Vector spaces and normed vector spaces.- § 2. Linear forms and the adjoint space.- § 3. Linear operators.- § 4. Analysis with operators.- § 5. The eigenvalue problem.- § 6. Operators in unitary spaces.- Two Perturbation theory in a finite-dimensional space.- § 1. Analytic perturbation of eigenvalues.- § 2. Perturbation series.- § 3. Convergence radii and error estimates.- § . Similarity transformations of the eigenspaces and eigenvectors.- § 5. Non-analytic perturbations.- § 6. Perturbation of symmetric operators.- Three Introduction to the theory of operators in Banach spaces.- § 1. Banach spaces.- § 2. Linear operators in Banach spaces.- § 3. Bounded operators.- § 4. Compact operators.- § 5. Closed operators.- § 6. Resolvents and spectra.- Four Stability theorems.- §1. Stability of closedness and bounded invertibility.- § 2. Generalized convergence of closed operators.- § 3. Perturbation of the spectrum.- § 4. Pairs of closed linear manifolds.- § 5. Stability theorems for semi-Fredholm operators.- § 6. Degenerate perturbations.- Five Operators in Hilbert spaces.- § 1. Hilbert space.- § 2. Bounded operators in Hilbert spaces.- § 3. Unbounded operators in Hilbert spaces.- § 4. Perturbation of self adjoint operators.- § 5. The Schrödinger and Dirac operators.- Six Sesquilinear forms in Hilbert spaces and associated operators.- § 1. Sesquilinear and quadratic forms.- § 2. The representation theorems.- § 3. Perturbation of sesquilinear forms and the associated operators.- § 4. Quadratic forms and the Schrödinger operators.- § 5. The spectral theorem and perturbation of spectral families.- Seven Analytic perturbation theory.- § 1. Analytic families of operators.- § 2.Holomorphic families of type (A).- § 3. Selfadjoint holomorphic families.- § 4. Holomorphic families of type (B).- § 5. Further problems of analytic perturbation theory.- § 6. Eigenvalue problems in the generalized form.- Eight Asymptotic perturbation theory.- § 1. Strong convergence in the generalized sense.- § 2. Asymptotic expansions.- § 3. Generalized strong convergence of sectorial operators.- § 4. Asymptotic expansions for sectorial operators.- § 5. Spectral concentration.- Nine Perturbation theory for semigroups of operators.- § 1. One-parameter semigroups and groups of operators.- § 2. Perturbation of semigroups.- § 3. Approximation by discrete semigroups.- Ten Perturbation of continuous spectra and unitary equivalence.- §1. The continuous spectrum of a selfadjoint operator.- § 2. Perturbation of continuous spectra.- § 3. Wave operators and the stability of absolutely continuous spectra.- § 4. Existence and completeness of wave operators.- § 5. A stationary method.- Supplementary Notes.- Supplementary Bibliography.- Notation index.- Author index.

Editorial 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

From the Publisher