An edition of Perturbation theory for linear operators (1966)

Perturbation theory for linear operators

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An edition of Perturbation theory for linear operators (1966)

Perturbation theory for linear operators

Corr. printing of the 2nd ed.

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Springer-Verlag
Language
English
Pages
619

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Subjects

Linear operators, Perturbation (Mathematics), Partial Differential equations, Mathematics, Differential equations, partial, Mathematical optimization, Calculus of Variations and Optimal Control; Optimization

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Cover of: Perturbation theory for linear operators
Perturbation theory for linear operators
1995, Springer
in English
Libraries near you: WorldCat
Cover of: Perturbation theory for linear operators
Perturbation theory for linear operators
1984, Springer-Verlag
in English - Corr. printing of the 2nd ed.
Libraries near you: WorldCat
Cover of: Perturbation theory for linear operators
Perturbation theory for linear operators
1976, Springer-Verlag
in English - 2d ed.
Libraries near you: WorldCat
Cover of: Perturbation theory for linear operators.
Perturbation theory for linear operators.
1966, Springer-Verlag
in English
Libraries near you: WorldCat
Cover of: Perturbation theory for linear operators
Perturbation theory for linear operators
Publish date unknown, Springer-Verlag
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Book Details

Table of Contents

Introduction
Page XVII
Chapter One. Operator Theory In Finite-Dimensional Vector Spaces
§ 1. Vector Spaces And Normed Vector Spaces
Page 1
1. Basic Notions
Page 1
2. Bases
Page 2
3. Linear Manifolds
Page 3
4. Convergence And Norms
Page 4
5. Topological Notions In A Normed Space
Page 6
6. Infinite Series Of Vectors
Page 7
7. Vector-Valued Functions
Page 8
§ 2. Linear Forms And The Adjoint Space
Page 10
1. Linear Forms
Page 10
2. The Adjoint Space
Page 11
3. The Adjoint Basis
Page 12
4. The Adjoint Space Of A Normed Space
Page 13
5. The Convexity Of Balls
Page 14
6. The Second Adjoint Space
Page 15
§ 3. Linear Operators
Page 16
1. Definitions. Matrix Representations
Page 16
2. Linear Operations On Operators
Page 18
3. The Algebra Of Linear Operators
Page 19
4. Projections. Nilpotents
Page 20
5. Invariance. Decomposition
Page 22
6. The Adjoint Operator
Page 23
§ 4. Analysis With Operators
Page 25
1. Convergence And Norms For Operators
Page 25
2. The Norm Of T^n
Page 27
3. Examples Of Norms
Page 28
4. Infinite Series Of Operators
Page 29
5. Operator-Valued Functions
Page 31
6. Pairs Of Projections
Page 32
§ 5. The Eigenvalue Problem
Page 34
1. Definitions
Page 34
2. The Resolvent
Page 36
3. Singularities Of The Resolvent
Page 38
4. The Canonical Form Of An Operator
Page 40
5. The Adjoint Problem
Page 43
6. Functions Of An Operator
Page 44
7. Similarity Transformations
Page 46
§ 6. Operators In Unitary Spaces
Page 47
1. Unitary Spaces
Page 47
2. The Adjoint Space
Page 48
3. Orthonormal Families
Page 49
4. Linear Operators
Page 51
5. Symmetric Forms And Symmetric Operators
Page 52
6. Unitary, Isometric And Normal Operators
Page 54
7. Projections
Page 55
8. Pairs Of Projections
Page 56
9. The Eigenvalue Problem
Page 58
10. The Minimax Principle
Page 60
Chapter Two. Perturbation Theory In A Finite-Dimensional Space
Page 62
§ 1. Analytic Perturbation Of Eigenvalues
Page 63
1. The Problem
Page 63
2. Singularities Of The Eigenvalues
Page 65
3. Perturbation Of The Resolvent
Page 66
4. Perturbation Of The Eigenprojections
Page 67
5. Singularities Of The Eigenprojections
Page 69
6. Remarks And Examples
Page 70
7. The Case Of T(x) Linear In x
Page 72
8. Summary
Page 73
§ 2. Perturbation Series
Page 74
1. The Total Projection For The λ-Group
Page 74
2. The Weighted Mean Of Eigenvalues
Page 77
3. The Reduction Process
Page 81
4. Formulas For Higher Approximations
Page 83
5. A Theorem Of Motzkin-Tausky
Page 85
6. The Ranks Of The Coefficients Of The Perturbation Series
Page 86
§ 3. Convergence Radii And Error Estimates
Page 88
1. Simple Estimates
Page 88
2. The Method Of Majorizing Series
Page 89
3. Estimates On Eigenvectors
Page 91
4. Further Error Estimates
Page 93
5. The Special Case Of A Normal Unperturbed Operator
Page 94
6. The Enumerative Method
Page 97
§ 4. Similarity Transformations Of The Eigenspaces And Eigenvectors
Page 98
1. Eigenvectors
Page 98
2. Transformation Functions
Page 99
3. Solution Of The Differential Equation
Page 102
4. The Transformation Function And The Reduction Process
Page 104
5. Simultaneous Transformation For Several Projections
Page 104
6. Diagonalization Of A Holomorphic Matrix Function
Page 106
§ 5. Non-Analytic Perturbations
Page 106
1. Continuity Of The Eigenvalues And The Total Projection
Page 106
2. The Numbering Of The Eigenvalues
Page 108
3. Continuity Of The Eigenspaces And Eigenvectors
Page 110
4. Differentiability At A Point
Page 111
5. Differentiability In An Interval
Page 113
6. Asymptotic Expansion Of The Eigenvalues And Eigenvectors
Page 115
7. Operators Depending On Several Parameters
Page 116
8. The Eigenvalues As Functions Of The Operator
Page 117
§ 6. Perturbation Of Symmetric Operators
Page 120
1. Analytic Perturbation Of Symmetric Operators
Page 120
2. Orthonormal Families Of Eigenvectors
Page 121
3. Continuity And Differentiability
Page 122
4. The Eigenvalues As Functions Of The Symmetric Operator
Page 124
5. Applications: A Theorem Of Lidskii
Page 124
Chapter Three. Introduction To The Theory Of Operators In Banach Spaces
§ 1. Banach Spaces
Page 127
1. Normed Spaces
Page 127
2. Banach Spaces
Page 129
3. Linear Forms
Page 132
4. The Adjoint Space
Page 134
5. The Principle Of Uniform Boundedness
Page 136
6. Weak Convergence
Page 137
7. Weak* Convergence
Page 140
8. The Quotient Space
Page 140
§ 2. Linear Operators In Banach Spaces
Page 142
1. Linear Operators: The Domain And Range
Page 142
2. Continuity And Boundedness
Page 145
3. Ordinary Differential Operators Of Second Order
Page 146
§ 3. Bounded Operators
Page 149
1. The Space Of Bounded Operators
Page 149
2. The Operator Algebra B(X)
Page 153
3. The Adjoint Operator
Page 154
4. Projections
Page 155
§ 4. Compact Operators
Page 157
1. Definition
Page 157
2. The Space Of Compact Operators
Page 158
3. Degenerate Operators: The Trace And Determinant
Page 160
§ 5. Closed Operators
Page 163
1. Remarks On Unbounded Operators
Page 163
2. Closed Operators
Page 164
3. Closable Operators
Page 165
4. The Closed Graph Theorem
Page 166
5. The Adjoint Operator
Page 167
6. Commutativity And Decomposition
Page 171
§ 6. Resolvents And Spectra
Page 172
1. Definitions
Page 172
2. The Spectra Of Bounded Operators
Page 176
3. The Point At Infinity
Page 176
4. Separation Of The Spectrum
Page 178
5. Isolated Eigenvalues
Page 180
6. The Resolvent of the Adjoint
Page 183
7. The Spectra of Compact Operators
Page 185
8. Operators with Compact Resolvent
Page 187
Chapter Four. Stability Theorems
§ 1. Stability of Closedness and Bounded Invertibility
Page 189
1. Stability of Closedness Under Relatively Bounded Perturbation
Page 189
2. Examples of Relative Boundedness
Page 191
3. Relative Compactness and a Stability Theorem
Page 194
4. Stability of Bounded Invertibility
Page 196
§ 2. Generalized Convergence of Closed Operators
Page 197
1. The Gap Between Subspaces
Page 197
2. The Gap and the Dimension
Page 199
3. Duality
Page 200
4. The Gap Between Closed Operators
Page 201
5. Further Results on the Stability of Bounded Invertibility
Page 205
6. Generalized Convergence
Page 206
§ 3. Perturbation of the Spectrum
Page 208
1. Upper Semicontinuity of the Spectrum
Page 208
2. Lower Semi-Discontinuity of the Spectrum
Page 209
3. Continuity and Analyticity of the Resolvent
Page 210
4. Semicontinuity of Separated Parts of the Spectrum
Page 212
5. Continuity of a Finite System of Eigenvalues
Page 213
6. Change of the Spectrum Under Relatively Bounded Perturbation
Page 214
7. Simultaneous Consideration of an Infinite Number of Eigenvalues
Page 215
8. An Application to Banach Algebras: Wiener's Theorem
Page 216
§ 4. Pairs of Closed Linear Manifolds
Page 218
1. Definitions
Page 218
2. Duality
Page 221
3. Regular Pairs of Closed Linear Manifolds
Page 223
4. The Approximate Nullity and Deficiency
Page 225
5. Stability Theorems
Page 227
§ 5. Stability Theorems for Semi-Fredholm Operators
Page 229
1. The Nullity, Deficiency and Index of an Operator
Page 229
2. The General Stability Theorem
Page 232
3. Other Stability Theorems
Page 236
4. Isolated Eigenvalues
Page 239
5. Another Form of the Stability Theorem
Page 241
6. Structure of the Spectrum of a Closed Operator
Page 242
§ 6. Degenerate Perturbations
Page 244
1. The Weinstein-Aronszajn Determinants
Page 244
2. The W-A Formulas
Page 246
3. Proof of the W-A Formulas
Page 248
4. Conditions Excluding the Singular Case
Page 249
Chapter Five. Operators in Hilbert Spaces
§ 1. Hilbert Space
Page 251
1. Basic Notions
Page 251
2. Complete Orthonormal Families
Page 254
§ 2. Bounded Operators in Hilbert Spaces
Page 256
1. Bounded Operators and Their Adjoints
Page 256
2. Unitary and Isometric Operators
Page 257
3. Compact Operators
Page 260
4. The Schmidt Class
Page 262
5. Perturbation of Orthonormal Families
Page 264
§ 3. Unbounded Operators in Hilbert Spaces
Page 267
1. General Remarks
Page 267
2. The Numerical Range
Page 267
3. Symmetric Operators
Page 269
4. The Spectra of Symmetric Operators
Page 270
5. The Resolvents and Spectra of Selfadjoint Operators
Page 272
6. Second-Order Ordinary Differential Operators
Page 274
7. The Operators T*T
Page 275
8. Normal Operators
Page 276
9. Reduction of Symmetric Operators
Page 277
10. Semibounded and Accretive Operators
Page 278
11. The Square Root of an m-Accretive Operator
Page 281
§ 4. Perturbation of Selfadjoint Operators
Page 287
1. Stability of Selfadjointness
Page 287
2. The Case of Relative Bound 1
Page 289
3. Perturbation of the Spectrum
Page 290
4. Semibounded Operators
Page 291
5. Completeness of the Eigenprojections of Slightly Non-Selfadjoint Operators
Page 293
§ 5. The Schrödinger and Dirac Operators
Page 297
1. Partial Differential Operators
Page 297
2. The Laplacian in the Whole Space
Page 299
3. The Schrödinger Operator with a Static Potential
Page 302
4. The Dirac Operator
Page 305
Chapter Six. Sesquilinear Forms in Hilbert Spaces and Associated Operators
§ 1. Sesquilinear and Quadratic Forms
Page 308
1. Definitions
Page 308
2. Semiboundedness
Page 310
3. Closed Forms
Page 313
4. Closable Forms
Page 315
5. Forms Constructed from Sectorial Operators
Page 318
6. Sums of Forms
Page 319
7. Relative Boundedness for Forms and Operators
Page 321
§ 2. The Representation Theorems
Page 322
1. The First Representation Theorem
Page 322
2. Proof of the First Representation Theorem
Page 323
3. The Friedrichs Extension
Page 325
4. Other Examples for the Representation Theorem
Page 326
5. Supplementary Remarks
Page 328
6. The Second Representation Theorem
Page 331
7. The Polar Decomposition of a Closed Operator
Page 334
§ 3. Perturbation of Sesquilinear Forms and the Associated Operators
Page 336
1. The Real Part of an M-Sectorial Operator
Page 336
2. Perturbation of an M-Sectorial Operator and Its Resolvent
Page 338
3. Symmetric Unperturbed Operators
Page 340
4. Pseudo-Friedrichs Extensions
Page 341
§ 4. Quadratic Forms and the Schrödinger Operators
Page 343
1. Ordinary Differential Operators
Page 343
2. The Dirichlet Form and the Laplace Operator
Page 346
3. The Schrödinger Operators in R³
Page 348
4. Bounded Regions
Page 352
§ 5. The Spectral Theorem and Perturbation of Spectral Families
Page 353
1. Spectral Families
Page 353
2. The Selfadjoint Operator Associated with a Spectral Family
Page 356
3. The Spectral Theorem
Page 360
4. Stability Theorems for the Spectral Family
Page 361
Chapter Seven. Analytic Perturbation Theory
§ 1. Analytic Families of Operators
Page 365
1. Analyticity of Vector- and Operator-Valued Functions
Page 365
2. Analyticity of a Family of Unbounded Operators
Page 366
3. Separation of the Spectrum and Finite Systems of Eigenvalues
Page 368
4. Remarks on Infinite Systems of Eigenvalues
Page 371
5. Perturbation Series
Page 372
6. A Holomorphic Family Related to a Degenerate Perturbation
Page 373
§ 2. Holomorphic Families of Type (A)
Page 375
1. Definition
Page 375
2. A Criterion for Type (A)
Page 377
3. Remarks on Holomorphic Families of Type (A)
Page 379
4. Convergence Radii and Error Estimates
Page 381
5. Normal Unperturbed Operators
Page 383
§ 3. Selfadjoint Holomorphic Families
Page 385
1. General Remarks
Page 385
2. Continuation of the Eigenvalues
Page 387
3. The Mathieu, Schrödinger, and Dirac Equations
Page 389
4. Growth Rate of the Eigenvalues
Page 390
5. Total Eigenvalues Considered Simultaneously
Page 392
§ 4. Holomorphic Families of Type (B)
Page 393
1. Bounded-Holomorphic Families of Sesquilinear Forms
Page 393
2. Holomorphic Families of Forms of Type (A) and Holomorphic Families of Operators of Type (B)
Page 395
3. A Criterion for Type (B)
Page 398
4. Holomorphic Families of Type (B₀)
Page 401
5. The Relationship Between Holomorphic Families of Types (A) and (B)
Page 403
6. Perturbation Series for Eigenvalues and Eigenprojections
Page 404
7. Growth Rate of Eigenvalues and the Total System of Eigenvalues
Page 407
8. Application to Differential Operators
Page 408
9. The Two-Electron Problem
Page 410
§ 5. Further Problems of Analytic Perturbation Theory
Page 413
1. Holomorphic Families of Type (C)
Page 413
2. Analytic Perturbation of the Spectral Family
Page 414
3. Analyticity of
Page H(x)
§ 6. Eigenvalue Problems in the Generalized Form
Page 416
1. General Considerations
Page 416
2. Perturbation Theory
Page 419
3. Holomorphic Families of Type (A)
Page 421
4. Holomorphic Families of Type (B)
Page 422
5. Boundary Perturbation
Page 423
Chapter Eight. Asymptotic Perturbation Theory
§ 1. Strong Convergence in the Generalized Sense
Page 427
1. Strong Convergence of the Resolvent
Page 427
2. Generalized Strong Convergence and Spectra
Page 431
3. Perturbation of Eigenvalues and Eigenvectors
Page 433
4. Stable Eigenvalues
Page 437
§ 2. Asymptotic Expansions
Page 441
1. Asymptotic Expansion of the Resolvent
Page 441
2. Remarks on Asymptotic Expansions
Page 444
3. Asymptotic Expansions of Isolated Eigenvalues and Eigenvectors
Page 445
4. Further Asymptotic Expansions
Page 448
§ 3. Generalized Strong Convergence of Sectorial Operators
Page 453
1. Convergence of a Sequence of Bounded Forms
Page 453
2. Convergence of Sectorial Forms "From Above"
Page 455
3. Nonincreasing Sequences of Symmetric Forms
Page 459
4. Convergence from Below
Page 461
5. Spectra of Converging Operators
Page 462
§ 4. Asymptotic Expansions for Sectorial Operators
Page 463
1. The Problem. The Zeroth Approximation for the Resolvent
Page 465
2. The 1/2-Order Approximation for the Resolvent
3. The First and Higher Order Approximations for the Resolvent
Page 466
4. Asymptotic Expansions for Eigenvalues and Eigenvectors
Page 470
§ 5. Spectral Concentration
Page 473
1. Unstable Eigenvalues
Page 473
2. Spectral Concentration
Page 474
3. Pseudo-Eigenvectors and Spectral Concentration
Page 475
4. Asymptotic Expansions
Page 476
Chapter Nine. Perturbation Theory for Semigroups of Operators
§ 1. One-Parameter Semigroups and Groups of Operators
Page 479
1. The Problem
Page 479
2. Definition of the Exponential Function
Page 480
3. Properties of the Exponential Function
Page 482
4. Bounded and Quasi-Bounded Semigroups
Page 486
5. Solution of the Inhomogeneous Differential Equation
Page 488
6. Holomorphic Semigroups
Page 489
7. The Inhomogeneous Differential Equation for a Holomorphic Semigroup
Page 493
§ 2. Perturbation of Semigroups
Page 497
1. Analytic Perturbation of Quasi-Bounded Semigroups
Page 499
2. Analytic Perturbation of Holomorphic Semigroups
Page 501
3. Perturbation of Contraction Semigroups
Page 502
4. Convergence of Quasi-Bounded Semigroups in a Restricted Sense
Page 503
5. Strong Convergence of Quasi-Bounded Semigroups
Page 506
6. Asymptotic Perturbation of Semigroups
Page 509
§ 3. Approximation by Discrete Semigroups
Page 509
1. Discrete Semigroups
Page 511
2. Approximation of a Continuous Semigroup by Discrete Semigroups
Page 513
3. Approximation Theorems
Page 514
4. Variation of the Space
Page 514
Chapter Ten. Perturbation of Continuous Spectra and Unitary Equivalence
§ 1. The Continuous Spectrum of a Selfadjoint Operator
Page 516
1. The Point and Continuous Spectra
Page 516
2. The Absolutely Continuous and Singular Spectra
Page 518
3. The Trace Class
Page 521
4. The Trace and Determinant
Page 523
§ 2. Perturbation of Continuous Spectra
Page 525
1. A Theorem of Weyl-Von Neumann
Page 525
2. A Generalization
Page 527
§ 3. Wave Operators and the Stability of Absolutely Continuous Spectra
Page 529
1. Introduction
Page 529
2. Generalized Wave Operators
Page 531
3. A Sufficient Condition for the Existence of the Wave Operator
Page 535
4. An Application to Potential Scattering
Page 536
§ 4. Existence and Completeness of Wave Operators
Page 537
1. Perturbations of Rank One (Special Case)
Page 537
2. Perturbations of Rank One (General Case)
Page 540
3. Perturbations of the Trace Class
Page 542
4. Wave Operators for Functions of Operators
Page 545
5. Strengthening of the Existence Theorems
Page 549
6. Dependence of W+ (H₂, H₁) on H₁ and H₂
Page 553
§ 5. A Stationary Method
Page 553
1. Introduction
Page 553
2. The I Operations
Page 555
3. Equivalence with the Time-Dependent Theory
Page 557
4. The I Operations on Degenerate Operators
Page 558
5. Solution of the Integral Equation for Rank A = 1
Page 560
6. Solution of the Integral Equation for a Degenerate A
Page 563
7. Application to Differential Operators
Page 565
Supplementary Notes
Chapter I.
Page 568
Chapter II.
Page 568
Chapter III.
Page 569
Chapter IV.
Page 570
Chapter V.
Page 570
Chapter VI.
Page 573
Chapter VII.
Page 574
Chapter VIII.
Page 574
Chapter IX.
Page 575
Chapter X.
Page 576
Bibliography
Page 583
Articles
Page 583
Books and Monographs
Page 593
Supplementary Bibliography
Page 596
Articles
Page 596
Notation Index
Page 606
Author Index
Page 608
Subject Index
Page 612

Edition Notes

Bibliography: p.
Cataloging based on CIP information.
Includes indexes.

Published in
Berlin, New York
Series
Grundlehren der mathematischen Wissenschaften ;, 132, Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete ;, Bd. 132.

Classifications

Dewey Decimal Class
515.7/246
Library of Congress
QA329.2 .K37 1984, QA329.2 .K37 1976

The Physical Object

Pagination
p. cm.
Number of pages
619

ID Numbers

Open Library
OL2839551M
Internet Archive
perturbationtheo0132kato
ISBN 10
0387075585
LCCN
84001362, 76004553
OCLC/WorldCat
13237239, 2072789, 6829171

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