Perturbation Theory for Linear Operators

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Springer Science & Business Media, 2012. dec. 6. - 623 oldal
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.
 

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Tartalomjegyzék

Linear forms and the adjoint space
10
Linear operators
16
4 Analysis with operators
25
The eigenvalue problem
34
Operators in unitary spaces
47
Chapter
62
erturbation series
74
3 4 5
77
Other stability theorems
236
Isolated eigenvalues
239
Another form of the stability theorem
241
Structure of the spectrum of a closed operator
242
1
267
11
281
Perturbation of selfadjoint operators
287
Analytic perturbation theory
364

Convergence radii and error estimates
88
4 imilarity transformations of the eigenspaces and eigenvectors
99
4
105
Diagonalization of a holomorphic matrix function
106
Perturbation of symmetric operators
120
Chapter Three
126
Linear operators in Banach spaces
142
Bounded operators
149
Compact operators
157
Closed operators
163
Resolvents and spectra
172
Isolated eigenvalues
180
The resolvent of the adjoint
183
The spectra of compact operators
185
Operators with compact resolvent
187
Chapter Four Stability theorems Stability of closedness and bounded invertibility 1 Stability of closedness under relatively bounded perturbation
189
Examples of relative boundedness
191
Relative compactness and a stability theorem
194
Stability of bounded invertibility
196
Generalized convergence of closed operators The gap between subspaces
197
The gap and the dimension
199
Duality
200
The gap between closed operators
201
Further results on the stability of bounded invertibility Generalized convergence
205
Perturbation of the spectrum Upper semicontinuity of the spectru
208
Lower semidiscontinuity of the spectrum
209
Continuity and analyticity of the resolvent
210
Semicontinuity of separated parts of the spectrum
212
Continuity of a finite system of eigenvalues
213
Change of the spectrum under relatively bounded perturbation
214
An application to Banach algebras Wieners theorem
216
Pairs of closed linear manifolds Definitions
218
Duality
221
Regular pairs of closed linear manifolds
223
The approximate nullity and deficiency
225
Stability theorems
226
Stability theorems for semiFredholm operators The nullity deficiency and index of an operator
229
The general stability theorem
232
Definition
375
olomorphic families of type
393
The relationship between holomorphic families of types A and
403
The problem The zeroth approximation for the resolvent
463
Asymptotic expansions for eigenvalues and eigenvectors
470
Asymptotic expansions
476
The Schrödinger and Dirac operators
483
Applications to the heat and Schrödinger equations
495
Perturbation of semigroups
497
2
498
Chapter
516
The trace class
525
Generalized wave operators
531
Existence and completeness of wave operators
537
Wave operators for functions of operators
545
Introduction
553
Solution of the integral equation for rank A 1
560
Supplementary Notes
568
Simultaneous consideration of an infinite number of eigenvalues 180
587
365
591
Supplementary Bibliography
596
185
597
371
598
187
600
189
601
379
604
516
607
201
608
427
610
254
612
206
613
Application to differential operators
614
208
616
6 Degenerate perturbations 1 The WeinsteinAronszajn determinants 2 The WA formulas 3 Proof of the WA formulas 4 Conditions excluding the si...
617
249
618
569
619
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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.

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