Perturbation theory for linear operators:
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| Language: | English |
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Berlin <<[u.a.]>>
Springer
1995
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| Edition: | Reprint of the corr. print. of the 2. ed. 1980 |
| Series: | Classics in mathematics
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| Online Access: | Inhaltsverzeichnis |
| Item Description: | Auch als: Grundlehren der mathematischen Wissenschaften ; 132 |
| Physical Description: | XXI, 619 S. |
| ISBN: | 354058661X 9783540586616 |
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| 100 | 1 | |a Kato, Tosio |d 1917-1999 |e Verfasser |0 (DE-588)17218083X |4 aut | |
| 245 | 1 | 0 | |a Perturbation theory for linear operators |c Tosio Kato |
| 250 | |a Reprint of the corr. print. of the 2. ed. 1980 | ||
| 264 | 1 | |a Berlin <<[u.a.]>> |b Springer |c 1995 | |
| 300 | |a XXI, 619 S. | ||
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| 490 | 0 | |a Classics in mathematics | |
| 500 | |a Auch als: Grundlehren der mathematischen Wissenschaften ; 132 | ||
| 650 | 4 | |a Linear operators | |
| 650 | 4 | |a Perturbation (Mathematics) | |
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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. Basic notions...................................................................................... 2. Bases................................................................................................. 3. Linear manifolds............................................................................... 4. Convergence and norms.................................................................... 5. Topological notions in a normed space............................................. 6. Infinite series of vectors.................................................................... 7. Vector-valued functions........................................................ 1 1 2 3 4 6 7 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 Λ-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 5. 6. 7. 8. XI Differentiability in an interval........................................................... 113 Asymptotic expansion of the eigenvalues and eigenvectors .... 115 Operators depending on several parameters.............................. . 116 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 âi (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 traceand 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
Contents XII 5. 6. 7. 8. Isolated eigenvalues............................................................................... 180 The resolvent of the adjoint............................................................... 183 The spectra of compact operators........................................................ 185 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 thedimension................................................................... 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 oí 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 closedUnearmanifolds.............................................. 223 4. The approximate nullity anddeficiency............................................. 225 5. StabUity theorems.................................................................................. 227 § 5. StabiUty theorems for semi-Fredholm operators..................................... 229 1. The nuUity, deficiency and index of an operator............................... 229 2. The general stabUity
theorem............................................................. 232 3. Other stabUity 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 § 1. § 2. § 3. § 4. § 5. XIII Chapter Five Operators in Hilbert spaces Hilbertspace............................................................................................. 251 1. Basic notions....................................................................................... 251 2. Complete orthonormal families......................................................... 254 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 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 selfadjointoperators......................... 272 6. Second-order ordinary difierential
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 Perturbation of selfadjoint operators....................................................... 287 1. Stability of selfadjointness................................................................. 287 2. The case of relative bound1............................................................... 289 3. Perturbation of the spectrum............................................................. 290 4. Semibounded operators ..................................................................... 291 5. Completeness of the eigenprojections of slightly non-selfadjoint operators............................................................................................ 293 The Schrödinger and Dirac operators....................................................... 297 1. Partial differential operators.............................................................. 297 2. The Laplacian in the whole space....................................................... 299 3. The Schrödinger operator witha 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 sectorialoperators......................................... 318 6. Sums of forms ..................................................................................... 319 7. Relative boundedness for forms andoperators...................................... 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 Schrödinger operators..................................... 343 1. Ordinary differential operators.......................................................... 343 2. The Dirichlet form and the Laplace operator....................................346 3. The Schrödinger operators in R։.......................................................... 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 § 1. § 2. § 3. § 4. Chapter Seven Analytic perturbation theory 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 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 Selfadjoint holomorphic families............................................................... 385 1. General remarks...................................................................................... 385 2. Continuation of the
eigenvalues........................................................... 387 3. The Mathieu, Schrödinger, and Dirac equations................................. 389 4. Growth rate of the eigenvalues........................................................... 390 5. Total eigenvalues considered simultaneously..................................... 392 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 tire spectral family .................................... 414 3. Analyticity of pZ(x)| and IÆ(x)(®....................................................... 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 § 1. § 2. § 3. § 4. § 5. Chapter Eight Asymptotic perturbation theory 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 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 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 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 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 Schrôdinger 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 § 1. § 2. § 3. § 4. § 5. Perturbation of continuous spectra and unitary equivalence The continuous spectrum of a selfadj oint 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 Perturbation oí continuous spectra........................................................... 525 1. A theorem of Weyl-von Neumann................................................... 525 2. A generalization........................................................ 527 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 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 (H։, Ях) on Ях and Я։............................................ 553 A stationary method................................................................................. 553 1.
Introduction......................................................................................... 553 2. The Г operations................................................................................. 555 3. Equivalence with the time-dependent theory.................................... 557 4. The Г 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................................................................................................................. 612
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Contents page Introduction. XVII Chapter One Operator theory in finite-dimensional vector spaces § 1. Vector spaces and normed vector spaces. 1. Basic notions. 2. Bases. 3. Linear manifolds. 4. Convergence and norms. 5. Topological notions in a normed space. 6. Infinite series of vectors. 7. Vector-valued functions. 1 1 2 3 4 6 7 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 Λ-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 5. 6. 7. 8. XI Differentiability in an interval. 113 Asymptotic expansion of the eigenvalues and eigenvectors . 115 Operators depending on several parameters. . 116 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 âi (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 traceand 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
Contents XII 5. 6. 7. 8. Isolated eigenvalues. 180 The resolvent of the adjoint. 183 The spectra of compact operators. 185 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 thedimension. 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 oí 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 closedUnearmanifolds. 223 4. The approximate nullity anddeficiency. 225 5. StabUity theorems. 227 § 5. StabiUty theorems for semi-Fredholm operators. 229 1. The nuUity, deficiency and index of an operator. 229 2. The general stabUity
theorem. 232 3. Other stabUity 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 § 1. § 2. § 3. § 4. § 5. XIII Chapter Five Operators in Hilbert spaces Hilbertspace. 251 1. Basic notions. 251 2. Complete orthonormal families. 254 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 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 selfadjointoperators. 272 6. Second-order ordinary difierential
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 Perturbation of selfadjoint operators. 287 1. Stability of selfadjointness. 287 2. The case of relative bound1. 289 3. Perturbation of the spectrum. 290 4. Semibounded operators . 291 5. Completeness of the eigenprojections of slightly non-selfadjoint operators. 293 The Schrödinger and Dirac operators. 297 1. Partial differential operators. 297 2. The Laplacian in the whole space. 299 3. The Schrödinger operator witha 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 sectorialoperators. 318 6. Sums of forms . 319 7. Relative boundedness for forms andoperators. 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 Schrödinger operators. 343 1. Ordinary differential operators. 343 2. The Dirichlet form and the Laplace operator.346 3. The Schrödinger operators in R։. 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 § 1. § 2. § 3. § 4. Chapter Seven Analytic perturbation theory 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 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 Selfadjoint holomorphic families. 385 1. General remarks. 385 2. Continuation of the
eigenvalues. 387 3. The Mathieu, Schrödinger, and Dirac equations. 389 4. Growth rate of the eigenvalues. 390 5. Total eigenvalues considered simultaneously. 392 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 tire spectral family . 414 3. Analyticity of pZ(x)| and IÆ(x)(®. 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 § 1. § 2. § 3. § 4. § 5. Chapter Eight Asymptotic perturbation theory 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 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 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 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 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 Schrôdinger 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 § 1. § 2. § 3. § 4. § 5. Perturbation of continuous spectra and unitary equivalence The continuous spectrum of a selfadj oint 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 Perturbation oí continuous spectra. 525 1. A theorem of Weyl-von Neumann. 525 2. A generalization. 527 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 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 (H։, Ях) on Ях and Я։. 553 A stationary method. 553 1.
Introduction. 553 2. The Г operations. 555 3. Equivalence with the time-dependent theory. 557 4. The Г 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. 612 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T22:30:23Z |
| indexdate | 2024-07-09T21:23:29Z |
| institution | BVB |
| isbn | 354058661X 9783540586616 |
| language | English |
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| spelling | Kato, Tosio 1917-1999 Verfasser (DE-588)17218083X aut Perturbation theory for linear operators Tosio Kato Reprint of the corr. print. of the 2. ed. 1980 Berlin <<[u.a.]>> Springer 1995 XXI, 619 S. txt rdacontent n rdamedia nc rdacarrier Classics in mathematics Auch als: Grundlehren der mathematischen Wissenschaften ; 132 Linear operators Perturbation (Mathematics) Linearer Operator (DE-588)4167721-3 gnd rswk-swf Störungstheorie (DE-588)4128420-3 gnd rswk-swf Linearer Operator (DE-588)4167721-3 s Störungstheorie (DE-588)4128420-3 s DE-604 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016828772&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kato, Tosio 1917-1999 Perturbation theory for linear operators Linear operators Perturbation (Mathematics) Linearer Operator (DE-588)4167721-3 gnd Störungstheorie (DE-588)4128420-3 gnd |
| subject_GND | (DE-588)4167721-3 (DE-588)4128420-3 |
| title | Perturbation theory for linear operators |
| title_auth | Perturbation theory for linear operators |
| title_exact_search | Perturbation theory for linear operators |
| title_exact_search_txtP | 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 | Linear operators Perturbation (Mathematics) Linearer Operator (DE-588)4167721-3 gnd Störungstheorie (DE-588)4128420-3 gnd |
| topic_facet | Linear operators Perturbation (Mathematics) Linearer Operator Störungstheorie |
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