Perturbation theory for linear operators

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Springer Science & Business Media, 2013. jún. 29. - 592 oldal
 

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Tartalomjegyzék

Linear forms and the adjoint space
10
Linear operators
16
4 Analysis with operators
25
5 The eigenvalue problem
34
1
38
6 Operators in unitary spaces
47
6
49
3
50
Degenerate perturbations
244
The WA formulas
246
Proof of the WA formulas
248
Conditions excluding the singular case
249
Sesquilinear forms in Hilbert spaces and associated operators
308
The second representation theorem
331
The polar decomposition of a closed operator
334
Perturbation of sesquilinear forms and the associated operators 1 The real part of an msectorial operator
336

Chapter
62
Perturbation series
74
3
77
1
83
Convergence radii and error estimates
88
5
94
2
95
4 Similarity transformations of the eigenspaces and eigenvectors
98
4
115
Chapter Three
126
Linear forms
132
Weak convergence
140
Ordinary differential operators of second order
146
The operator algebra 4 X
153
Degenerate operators The trace and determinant
160
The closed graph theorem
166
Resolvents and spectra
172
5
178
6
179
Isolated eigenvalues
180
The resolvent of the adjoint
183
The spectra of compact operators
185
Operators with compact resolvent
187
Chapter Four Stability theorems Stability of closedness and bounded invertibility
189
Examples of relative boundedness
191
Relative compactness and a stability theorem
194
Stability of bounded invertibility
196
Generalized convergence of closed operators
197
The gap and the dimension
199
Duality
200
The gap between closed operators
201
Further results on the stability of bounded invertibility
205
Generalized convergence
206
Perturbation of the spectrum
208
Lower semidiscontinuity of the spectrum
209
Continuity and analyticity of the resolvent
210
Semicontinuity of separated parts of the spectrum
212
Continuity of a finite system of eigenvalues
213
Change of the spectrum under relatively bounded perturbation
214
Simultaneous consideration of an infinite number of eigenvalues
215
An application to Banach algebras Wieners theorem
216
Pairs of closed linear manifolds
218
Duality
221
Regular pairs of closed linear manifolds
223
The approximate nullity and deficiency
225
Stability theorems
227
Stability theorems for semiFredholm operators
229
The general stability theorem
232
Other stability theorems
236
Isolated eigenvalues
239
Another form of the stability theorem
241
Structure of the spectrum of a closed operator
242
Perturbation of an msectorial operator and its resolvent
338
Symmetric unperturbed operators
340
PseudoFriedrichs extensions
341
Quadratic forms and the Schrödinger operators 1 Ordinary differential operators
343
The Dirichlet form and the Laplace operator
346
The Schrödinger operators in
350
Bounded regions
352
The spectral theorem and perturbation of spectral families 1 Spectral families
353
The selfadjoint operator associated with a spectral family
356
The spectral theorem
359
Stability theorems for the spectral family
361
Chapter Seven Analytic perturbation theory
364
Analytic families of operators Analyticity of vector and operatorvalued functions
365
Separation of the spectrum and finite systems of eigenvalues
368
Perturbation series
372
Definition
375
A criterion for type
377
Remarks on holomorphic families of type
379
Total eigenvalues considered simultaneously
392
Holomorphic families of type B 1 Boundedholomorphic families of sesquilinear forms
393
Holomorphic families of forms of type a and holomorphic families of operators of type B
395
A criterion for type
398
Holomorphic families of type Bo
401
The relationship between holomorphic families of types A and
403
Perturbation series for eigenvalues and eigenprojections
404
Growth rate of eigenvalues and the total system of eigenvalues
407
Application to differential operators
408
The twoelectron problem
410
Convergence radii and error estimates Normal unperturbed operators elfadjoint holomorphic families
411
Holomorphic families of type
413
Asymptotic perturbation theory
426
Unstable eigenvalues
471
Perturbation theory for semigroups of operators
477
Perturbation of semigroups
495
Approximation by discrete semigroups
507
Chapter
514
Perturbation of continuous spectra
523
4 Existence and completeness of wave operators
535
Bibliography
566
General remarks Continuation of the eigenvalues The Mathieu Schrödinger and Dirac equations Growth rate of the eigenvalues 331 334 336 336
568
341
571
343
572
The inhomogeneous differential equation for a holomorphic semi
578
377
581
393
584
401
585
403
586
408
587
410
588
Errata
591
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