Perturbation Theory for Linear Operators

1 C L A S S I C SI N M A T H E M AT I C STosio KatoPerturbation Theoryfor Linear OperatorsSpringerTosio KatoPerturbation Theoryfor Linear OperatorsReprint of the 1980 EditionSpringerTosio KatoDepartment of Mathematics, University of CaliforniaBerkeley, CA 94720-3840 USAO riginally published as Vol. 132 of theGrundlehren der mathematischen WissenschaftenMathematics Subject Classification (1991): 46 BXX, 46 CXX, 47 AXX, 47 BXX,47D03, 47E05, 47F05, 81Q10, 81Q15, 81 UXXISBN 3-540-58661-X Springer-Verlag Berlin Heidelberg New YorkCIP data applied forThis work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustration, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks.

2 Duplication of this publicationor parts thereof is permitted only under the provision of the German Copyright Law of September 9, 1965,in its current version, and permission for use must always be obtained from Springer-Verlag. Violations areliable for prosecution under the German Copyright Springer-Verlag Berlin Heidelberg 1995 Printed in GermanyThe use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protective lawsand regulations and therefore free for general 1048526041/3140 - 54 3 210 - Printed on acid-free paperTosio KatoPerturbation Theoryfor Linear OperatorsCorrected Printing of the Second EditionSpringer-VerlagBerlin Heidelberg New York 1980Dr.

3 Tosio KatoProfessor of Mathematics, University of California, BerkeleyAMS Subject Classification (1970): 46 Bxx, 46 Cxx, 47 Axx, 47 Bxx, 47D05,47 Exx, 47 Fxx, 81A09, 81A10, 81A45 ISBN 3-540-07558-5 2. Auflage Springer-Verlag Berlin HeidelbergNew YorkISBN 0-387-07558-5 2nd edition Springer-Verlag New York HeidelbergBerlinISBN 3-540-03526-51. Auflage Berlin Heidelberg New YorkISBN 0-387-03526-51st edition New York Heidelberg BerlinLibrary of Congress Cataloging in Publication Data. Kato, Tosio, 1917-Perturbationtheory for Linear Operators . (Grundlehren der mathematischen Wissenschaften;132). Bibliography: p.

4 Includes indexes. 1. Linear Operators . 2. Perturbation (Mathematics). I. Title. II. Series: Die Grundlehren der mathematischen Wissen-schaften in Einzeldarstellungen; Bd. '. work is subject to copyright. All rights are reserved, whether thewhole or part ofthe material is concerned, specifically those of translation, reprinting, re-useofillustrations, broadcasting, reproduction by photocopying machine or similar means,and storage in data banks. Under 54 of the German Copyright Lawwherecopies are made for other than private use, a fee is payable to thepublisher,the amount of the fee to be determined by agreement with the publisher.

5 By Springer-Verlag Berlin Heidelberg 1966, 1976 Printed in , printing and binding: Briihlsche Universit5tsdruckerei GiellenTo the memoryof my patentsPreface to the Second EditionIn view of recent development in Perturbation Theory , supplementarynotes and a supplementary bibliography are added at the end of the newedition. Little change has been made in the text except that the para-graphs V- , VI- , and VIII- have been completely rewritten,and a number of minor errors, mostly typographical, have been author would like to thank many readers who brought the errors tohis to these changes, some theorems, lemmas, and formulas of thefirst edition are missing from the new edition while new ones are new ones have numbers different from those attached to the oldones which they may have considerable expansion, the bibliography is not intended tobe.

6 April 1976 Tosio KATOP reface to the First EditionThis book is intended to give a systematic presentation of perturba-tion Theory for Linear Operators . It is hoped that the book will be usefulto students as well as to mature scientists, both in mathematics and inthe physical Theory for Linear Operators is a collection of diversifiedresults in the spectral Theory of Linear Operators , unified more or lessloosely by their common concern with the behavior of spectral propertieswhen the Operators undergo a small change. Since its creation by RAY-LEIGH and SCHRODINGER, the Theory has occupied an important place inapplied mathematics; during the last decades, it has grown into amathematical discipline with its own interest.

7 The book aims at a mathe-matical treatment of the subject, with due consideration of mathematical foundations of the Theory belong to functionalanalysis. But since the book is partly intended for physical scientists,who might lack training in functional analysis, not even the elements ofthat subject are presupposed. The reader is assumed to have only a basicknowledge of Linear algebra and real and complex analysis. The necessarytools in functional analysis, which are restricted to the most elementarypart of the subject, are developed in the text as the need for them arises(Chapters I, III and parts of Chapters V, VI).

8 An introduction, containing a brief historical account of the Theory ,precedes the main exposition. There are ten chapters, each prefaced by aVIIIP refacesummary. Chapters are divided into sections, and sections into para-graphs. I- , for example, means paragraph three of section two ofchapter one; it is simply written when referred to within the samechapter and par. 3 when referred to within the same section. Theorems,Corollaries, Lemmas, Remarks, Problems, and Examples are numberedin one list within each section: Theorem , Corollary , Lemma ,etc. Lemma means Lemma of chapter one, and it is referredto simply as Lemma within the same chapter.

9 Formulas are numberedconsecutively within each section; I-( ) means the third formula ofsection two of chapter one, and it is referred to as ( ) within the samechapter. Some of the problems are disguised theorems, and are quotedin later parts of the in [ ] refer to the first part of the bibliography containingarticles, and those in QJI to the second part containing books and are a subject index, an author index and a notation index at theend of the book was begun when I was at the University of Tokyo andcompleted at the University of California. The preparation of the bookhas been facilitated by various financial aids which enabled me to pursueresearch at home and other institutions.

10 For these aids I am gratefulto the following agencies: the Ministry of Education, Japan; Com-missariat General du Plan, France; National Science Foundation,Atomic Energy Commission, Army Office of Ordnance Research, Officeof Naval Research, and Air Force Office of Scientific Research, am indebted to a great many friends for their suggestions duringthe long period of writing the book. In particular I express my heartythanks to Professors C. CLARK, K. O. FRIEDRICHS, H. FUJITA, S. GOLD-BERG, E. HILLE, T. IKEBE, S. KAKUTANI, S. T. KURODA, G. NEUBAUER,R. S. PHILLIPS, J. and O. TODD, F. WOLF, and K. YOSIDA.