Perturbation theory for linear operators:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1976
|
| Edition: | 2. ed. |
| Series: | Grundlehren der mathematischen Wissenschaften
132 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 619 S. |
| ISBN: | 3540075585 0387075585 |
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| 245 | 1 | 0 | |a Perturbation theory for linear operators |c Tosio Kato |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1976 | |
| 300 | |a XXI, 619 S. | ||
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| 650 | 4 | |a Perturbation (Mathématiques) | |
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| adam_text | Contents
page
Introduction XVII
Chapter One
Operator theory in finite dimensional vector spaces
§ 1. Vector spaces and normed vector spaces 1
1. Basic notions 1
2. Bases 2
3. Linear manifolds 3
4. Convergence and norms 4
5. Topological notions in a normed space 6
6. Infinite series of vectors 7
7. Vector valued functions 8
§ 2. Linear forms and the adjoint space 10
1. Linear forms 10
2. The adjoint space 11
3. The adjoint basis 12
4. The adjoint space of a normed space 13
5. The convexity of balls 14
6. The second adjoint space 15
§3. Linear operators 16
1. Definitions. Matrix representations 16
2. Linear operations on operators 18
3. The algebra of linear operators 19
4. Projections. Nilpotents 20
5. Invariance. Decomposition 22
6. The adjoint operator 23
§ 4. Analysis with operators 25
1. Convergence and norms for operators 25
2. The norm of T» 27
3. Examples of norms 28
4. Infinite series of operators 29
5. Operator valued functions 31
6. Pairs of projections 32
§ 5. The eigenvalue problem 34
1. Definitions 34
2. The resolvent 36
3. Singularities of the resolvent 38
4. The canonical form of an operator 40
5. The adjoint problem 43
6. Functions of an operator 44
7. Similarity transformations 46
X Contents
§ 6. Operators in unitary spaces 47
1. Unitary spaces 47
2. The adjoint space 48
3. Orthonormal families 49
4. Linear operators 51
5. Symmetric forms and symmetric operators 52
6. Unitary, isometric and normal operators 54
7. Projections 55
8. Pairs of projections 56
9. The eigenvalue problem 58
10. The minimax principle 60
Chapter Two
Perturbation theory in a finite dimensional space 62
§ 1. Analytic perturbation of eigenvalues 63
1. The problem 63
2. Singularities of the eigenvalues 65
3. Perturbation of the resolvent 66
4. Perturbation of the eigenprojections 67
5. Singularities of the eigenprojections 69
6. Remarks and examples 70
7. The case of T(x) linear in x 72
8. Summary 73
§ 2. Perturbation series 74
1. The total projection for the A group 74
2. The weighted mean of eigenvalues 77
3. The reduction process 81
4. Formulas for higher approximations 83
5. A theorem of Motzkin Taussky 85
6. The ranks of the coefficients of the perturbation series 86
§ 3. Convergence radii and error estimates 88
1. Simple estimates 88
2. The method of majorizing series 89
3. Estimates on eigenvectors 91
4. Further error estimates 93
5. The special case of a normal unperturbed operator 94
6. The enumerative method 97
§ 4. Similarity transformations of the eigenspaces and eigenvectors .... 98
1. Eigenvectors 98
2. Transformation functions 99
3. Solution of the differential equation 102
4. The transformation function and the reduction process 104
5. Simultaneous transformation for several projections 104
6. Diagonalization of a holomorphic matrix function 106
§ 5. Non analytic perturbations 106
1. Continuity of the eigenvalues and the total projection 106
2. The numbering of the eigenvalues 108
3. Continuity of the eigenspaces and eigenvectors 110
4. Differentiability at a point Ill
Contents XI
5. Differentiability in an interval 113
6. Asymptotic expansion of the eigenvalues and eigenvectors .... 115
7. Operators depending on several parameters 116
8. The eigenvalues as functions of the operator 117
§ 6. Perturbation of symmetric operators 120
1. Analytic perturbation of symmetric operators 120
2. Orthonormal families of eigenvectors 121
3. Continuity and differentiability 122
4. The eigenvalues as functions of the symmetric operator 124
5. Applications. A theorem of Lidskii 124
Chapter Three
Introduction to the theory of operators in Banach spaces
§ 1. Banach spaces 127
1. Normed spaces 127
2. Banach spaces 129
3. Linear forms 132
4. The adjoint space 134
5. The principle of uniform boundedness 136
6. Weak convergence 137
7. Weak* convergence 140
8. The quotient space 140
§ 2. Linear operators in Banach spaces 142
1. Linear operators. The domain and range 142
2. Continuity and boundedness 145
3. Ordinary differential operators of second order 146
§ 3. Bounded operators 149
1. The space of bounded operators 149
2. The operator algebra S) (X) 153
3. The adjoint operator 154
4. Projections 155
§ 4. Compact operators 157
1. Definition 157
2. The space of compact operators 158
3. Degenerate operators. The trace and determinant 160
§ 5. Closed operators 163
1. Remarks on unbounded operators 163
2. Closed operators 164
3. Closable operators 165
4. The closed graph theorem 166
5. The adjoint operator 167
6. Commutativity and decomposition 171
§ 6. Resolvents and spectra 172
1. Definitions 172
2. The spectra of bounded operators 176
3. The point at infinity 176
4. Separation of the spectrum 178
XII Contents
5. Isolated eigenvalues 180
6. The resolvent of the adjoint 183
7. The spectra of compact operators 185
8. Operators with compact resolvent 187
Chapter Four
Stability theorems
§ 1. Stability of closedness and bounded invertibility 189
1. Stability of closedness under relatively bounded perturbation . .189
2. Examples of relative boundedness 191
3. Relative compactness and a stability theorem 194
4. Stability of bounded invertibility 196
§ 2. Generalized convergence of closed operators 197
1. The gap between subspaces 197
2. The gap and the dimension 199
3. Duality 200
4. The gap between closed operators 201
5. Further results on the stability of bounded invertibility 205
6. Generalized convergence 206
§ 3. Perturbation of the spectrum 208
1. Upper semicontinuity of the spectrum 208
2. Lower semi discontinuity of the spectrum 209
3. Continuity and analyticity of the resolvent 210
4. Semicontinuity of separated parts of the spectrum 212
5. Continuity of a finite system of eigenvalues 213
6. Change of the spectrum under relatively bounded perturbation . .214
7. Simultaneous consideration of an infinite number of eigenvalues . . 215
8. An application to Banach algebras. Wiener s theorem 216
§ 4. Pairs of closed linear manifolds 218
1. Definitions 218
2. Duality 221
3. Regular pairs of closed linear manifolds 223
4. The approximate nullity and deficiency 225
5. Stability theorems 227
§ 5. Stability theorems for semi Fredholm operators 229
1. The nullity, deficiency and index of an operator 229
2. The general stability theorem 232
3. Other stability theorems 236
4. Isolated eigenvalues 239
5. Another form of the stability theorem 241
6. Structure of the spectrum of a closed operator 242
§ 6. Degenerate perturbations 244
1. The Weinstein Aronszajn determinants 244
2. The W A formulas 246
3. Proof of the W A formulas 248
4. Conditions excluding the singular case 249
Contents XIII
Chapter Five
Operators in Hilbert spaces
§ 1. Hilbert space 251
1. Basic notions 251
2. Complete orthonormal families 254
§ 2. Bounded operators in Hilbert spaces 256
1. Bounded operators and their adjoints 256
2. Unitary and isometric operators 257
3. Compact operators 260
4. The Schmidt class 262
5. Perturbation of orthonormal families 264
§ 3. Unbounded operators in Hilbert spaces 267
1. General remarks 267
2. The numerical range 267
3. Symmetric operators 269
4. The spectra of symmetric operators 270
5. The resolvents and spectra of selfadjoint operators 272
6. Second order ordinary differential operators 274
7. The operators T*T 275
8. Normal operators 276
9. Reduction of symmetric operators 277
10. Semibounded and accretive operators 278
11. The square root of an m accretive operator 281
§ 4. Perturbation of selfadjoint operators 287
1. Stability of selfadjointness 287
2. The case of relative bound 1 289
3. Perturbation of the spectrum 290
4. Semibounded operators 291
5. Completeness of the eigenprojections of slightly non selfadjoint
operators 293
§ 5. The Schrodinger and Dirac operators 297
1. Partial differential operators 297
2. The Laplacian in the whole space 299
3. The Schrodinger operator with a static potential 302
4. The Dirac operator 305
Chapter Six
Sesquilinear forms in Hilbert spaces and associated operators
§ 1. Sesquilinear and quadratic forms 308
1. Definitions 308
2. Semiboundedness 310
3. Closed forms 313
4. Closable forms 315
5. Forms constructed from sectorial operators 318
6. Sums of forms 319
7. Relative boundedness for forms and operators 321
§ 2. The representation theorems 322
1. The first representation theorem 322
2. Proof of the first representation theorem 323
3. The Friedrichs extension 325
4. Other examples for the representation theorem 326
XIV Contents
5. Supplementary remarks 328
6. The second representation theorem 331
7. The polar decomposition of a closed operator 334
§ 3. Perturbation of sesquilinear forms and the associated operators . . . 336
1. The real part of an m sectorial operator 336
2. Perturbation of an m sectorial operator and its resolvent 338
3. Symmetric unperturbed operators 340
4. Pseudo Friedrichs extensions 341
§ 4. Quadratic forms and the Schrodinger operators 343
1. Ordinary differential operators 343
2. The Dirichlet form and the Laplace operator 346
3. The Schrodinger operators in R3 348
4. Bounded regions 352
§ 5. The spectral theorem and perturbation of spectral families 353
1. Spectral families 353
2. The selfadjoint operator associated with a spectral family .... 356
3. The spectral theorem 360
4. Stability theorems for the spectral family 361
Chapter Seven
Analytic perturbation theory
§ 1. Analytic families of operators 365
1. Analyticity of vector and operator valued functions 365
2. Analyticity of a family of unbounded operators 366
3. Separation of the spectrum and finite systems of eigenvalues . . . 368
4. Remarks on infinite systems of eigenvalues 371
5. Perturbation series 372
6. A holomorphic family related to a degenerate perturbation .... 373
§ 2. Holomorphic families of type (A) 375
1. Definition 375
2. A criterion for type (A) 377
3. Remarks on holomorphic families of type (A) 379
4. Convergence radii and error estimates 381
5. Normal unperturbed operators 383
§ 3. Selfadjoint holomorphic families 385
1. General remarks 385
2. Continuation of the eigenvalues 387
3. The Mathieu, Schrodinger, and Dirac equations 389
4. Growth rate of the eigenvalues 390
5. Total eigenvalues considered simultaneously 392
§ 4. Holomorphic families of type (B) 393
1. Bounded holomorphic families of sesquilinear forms 393
2. Holomorphic families of forms of type (a) and holomorphic families
of operators of type (B) 395
3. A criterion for type (B) 398
4. Holomorphic families of type (Bo) 401
5. The relationship between holomorphic families of types (A) and (B) 403
6. Perturbation series for eigenvalues and eigenprojections 404
7. Growth rate of eigenvalues and the total system of eigenvalues . . . 407
8. Application to differential operators 408
9. The two electron problem 410
Contents XV
§ 5. Further problems of analytic perturbation theory 413
1. Holomorphic families of type (C) 413
2. Analytic perturbation of the spectral family 414
3. Analyticity of H(x) and |H(x)|9 416
§ 6. Eigenvalue problems in the generalized form 416
1. General considerations 416
2. Perturbation theory 419
3. Holomorphic families of type (A) 421
4. Holomorphic families of type (B) 422
5. Boundary perturbation 423
Chapter Eight
Asymptotic perturbation theory
§ 1. Strong convergence in the generalized sense 427
1. Strong convergence of the resolvent 427
2. Generalized strong convergence and spectra 431
3. Perturbation of eigenvalues and eigenvectors 433
4. Stable eigenvalues 437
§ 2. Asymptotic expansions 441
1. Asymptotic expansion of the resolvent 441
2. Remarks on asymptotic expansions 444
3. Asymptotic expansions of isolated eigenvalues and eigenvectors . . 445
4. Further asymptotic expansions 448
§ 3. Generalized strong convergence of sectorial operators 453
1. Convergence of a sequence of bounded forms 453
2. Convergence of sectorial forms from above 455
3. Nonincreasing sequences of symmetric forms 459
4. Convergence from below 461
5. Spectra of converging operators 462
§ 4. Asymptotic expansions for sectorial operators 463
1. The problem. The zeroth approximation for the resolvent 463
2. The 1/2 order approximation for the resolvent 465
3. The first and higher order approximations for the resolvent .... 466
4. Asymptotic expansions for eigenvalues and eigenvectors 470
§ 5. Spectral concentration 473
1. Unstable eigenvalues 473
2. Spectral concentration 474
3. Pseudo eigenvectors and spectral concentration 475
4. Asymptotic expansions 476
Chapter Nine
Perturbation theory for semigroups of operators
§ 1. One parameter semigroups and groups of operators 479
1. The problem 479
2. Definition of the exponential function 480
3. Properties of the exponential function 482
4. Bounded and quasi bounded semigroups 486
5. Solution of the inhomogeneous differential equation 488
6. Holomorphic semigroups 489
7. The inhomogeneous differential equation for a holomorphic semi¬
group 493
8. Applications to the heat and Schrodinger equations 495
XVI Contents
§ 2. Perturbation of semigroups 497
1. Analytic perturbation of quasi bounded semigroups 497
2. Analytic perturbation of holomorphic semigroups 499
3. Perturbation of contraction semigroups 501
4. Convergence of quasi bounded semigroups in a restricted sense . . . 502
5. Strong convergence of quasi bounded semigroups 503
6. Asymptotic perturbation of semigroups 506
§ 3. Approximation by discrete semigroups 509
1. Discrete semigroups 509
2. Approximation of a continuous semigroup by discrete semigroups . 511
3. Approximation theorems 513
4. Variation of the space 514
Chapter Ten
Perturbation of continuous spectra and unitary equivalence
§ 1. The continuous spectrum of a self adjoint operator 516
1. The point and continuous spectra 516
2. The absolutely continuous and singular spectra 518
3. The trace class 521
4. The trace and determinant 523
§ 2. Perturbation of continuous spectra 525
1. A theorem of Wbyl von Neumann 525
2. A generalization 527
§ 3. Wave operators and the stability of absolutely continuous spectra . . . 529
1. Introduction 529
2. Generalized wave operators 531
3. A sufficient condition for the existence of the wave operator . . . 535
4. An application to potential scattering 536
§ 4. Existence and completeness of wave operators 537
1. Perturbations of rank one (special case) 537
2. Perturbations of rank one (general case) 540
3. Perturbations of the trace class 542
4. Wave operators for functions of operators 545
5. Strengthening of the existence theorems 549
6. Dependence of W± (Ht, H,) on Ht and Ht 553
§ 5. A stationary method 553
1. Introduction 553
2. The r operations 555
3. Equivalence with the time dependent theory 557
4. The F operations on degenerate operators 558
5. Solution of the integral equation for rank A = 1 560
6. Solution of the integral equation for a degenerate A 563
7. Application to differential operators 565
Supplementary Notes
Chapter I 568
Chapter II 568
Chapter III 569
Chapter IV 570
Chapter V 570
Contents XVII
Chapter VI 573
Chapter VII 574
Chapter VIII 574
Chapter IX 575
Chapter X 576
Bibliography 583.
Articles 583
Books and monographs 593
Supplementary Bibliography 596
Articles 596
Notation index 606
Author index 608
Subject index 112
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| id | DE-604.BV001968255 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:38:05Z |
| institution | BVB |
| isbn | 3540075585 0387075585 |
| language | English |
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| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Kato, Tosio 1917-1999 Verfasser (DE-588)17218083X aut Perturbation theory for linear operators Tosio Kato 2. ed. Berlin [u.a.] Springer 1976 XXI, 619 S. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 132 Opérateurs linéaires Perturbation (Mathématiques) Linear operators Perturbation (Mathematics) Störungstheorie (DE-588)4128420-3 gnd rswk-swf Linearer Operator (DE-588)4167721-3 gnd rswk-swf Störungstheorie (DE-588)4128420-3 s Linearer Operator (DE-588)4167721-3 s DE-604 Grundlehren der mathematischen Wissenschaften 132 (DE-604)BV000000395 132 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001283639&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kato, Tosio 1917-1999 Perturbation theory for linear operators Grundlehren der mathematischen Wissenschaften Opérateurs linéaires Perturbation (Mathématiques) Linear operators Perturbation (Mathematics) Störungstheorie (DE-588)4128420-3 gnd Linearer Operator (DE-588)4167721-3 gnd |
| subject_GND | (DE-588)4128420-3 (DE-588)4167721-3 |
| title | Perturbation theory for linear operators |
| title_auth | Perturbation theory for linear operators |
| title_exact_search | Perturbation theory for linear operators |
| title_full | Perturbation theory for linear operators Tosio Kato |
| title_fullStr | Perturbation theory for linear operators Tosio Kato |
| title_full_unstemmed | Perturbation theory for linear operators Tosio Kato |
| title_short | Perturbation theory for linear operators |
| title_sort | perturbation theory for linear operators |
| topic | Opérateurs linéaires Perturbation (Mathématiques) Linear operators Perturbation (Mathematics) Störungstheorie (DE-588)4128420-3 gnd Linearer Operator (DE-588)4167721-3 gnd |
| topic_facet | Opérateurs linéaires Perturbation (Mathématiques) Linear operators Perturbation (Mathematics) Störungstheorie Linearer Operator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001283639&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT katotosio perturbationtheoryforlinearoperators |