Basic classes of linear operators:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Basel [u.a.]
Birkhäuser
2003
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Frühere Ausg. u.d.T.: Gohberg, Yiśrāʿēl Z.: Basic operator theory |
| Beschreibung: | XVII, 423 S. |
| ISBN: | 3764369302 0817669302 |
Internformat
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| 245 | 1 | 0 | |a Basic classes of linear operators |c Israel Gohberg ; Seymour Goldberg ; Marinus A. Kaashoek |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2003 | |
| 300 | |a XVII, 423 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Frühere Ausg. u.d.T.: Gohberg, Yiśrāʿēl Z.: Basic operator theory | ||
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| 650 | 7 | |a Operadores lineares (teoria) |2 larpcal | |
| 650 | 4 | |a Opérateurs linéaires | |
| 650 | 4 | |a Linear operators | |
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| 650 | 0 | 7 | |a Linearer Operator |0 (DE-588)4167721-3 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-010588840 | ||
Datensatz im Suchindex
| _version_ | 1804130358752444416 |
|---|---|
| adam_text | Table of Contents
Preface
............................................................. xiii
Introduction
......................................................... xv
Chapter I Hubert Spaces
............................................ 1
1.1
Complex n-Space
................................................ 1
1.2
The Hubert Space £2
............................................. 3
1.3
Definition of Hubert Space and its Elementary Properties
............ 5
1.4
Distance from a Point to a Finite Dimensional Space
................ 8
1.5
The Gram Determinant
.......................................... 10
1.6
Incompatible Systems of Equations
............................... 13
1.7
Least Square Fit
................................................ 15
1.8
Distance to a Convex Set and Projections onto Subspaces
........... 16
1.9
Orthonormal
Systems
........................................... 18
1.10
Szegő
Polynomials
............................................. 19
1.11
Legendre Polynomials
.......................................... 24
1.12
Orthonormal
Bases
............................................. 26
1.13
Fourier Series
.................................................. 29
1.14
Completeness of the Legendre Polynomials
....................... 31
1.15
Bases for the Hubert Space of Functions on a Square
............... 32
1.16
Stability of
Orthonormal
Bases
................................... 34
1.17
Separable Spaces
............................................... 35
1.18
Isometry of Hubert Spaces
...................................... 36
1.19
Example of aNon Separable Space
............................... 38
Exercises
...................................................... 38
Chapter II Bounded Linear Operators on Hubert Spaces
............. 51
2.1
Properties of Bounded Linear Operators
.......................... 51
2.2
Examples of Bounded Linear Operators with Estimates of Norms
... 52
2.3
Continuity of a Linear Operator
.................................. 56
2.4
Matrix Representations of Bounded Linear Operators
.............. 57
2.5
Bounded Linear Functionals
..................................... 60
2.6
Operators of Finite Rank
........................................ 63
2.7
Invertible Operators
............................................. 64
2.8
Inversion of Operators by the Iterative Method
..................... 69
2.9
Infinite Systems of Linear Equations
.............................. 71
2.10
Integral Equations of the Second Kind
............................ 73
Table of
Contents
2.11
Adjoint
Operators .............................................. 76
2.12
Self Adjoint Operators
.......................................... 80
2.13
Orthogonal Projections
.......................................... 81
2.14
Two Fundamental Theorems
..................................... 82
2.15
Projections and One-Sided Invertibility of Operators
............... 84
2.16
Compact Operators
............................................. 91
2.17
The Projection Method for Inversion of Linear Operators
........... 96
2.18
The Modified Projection Method
................................ 105
2.19
Invariant Subspaces
............................................ 108
2.20
The Spectrum of an Operator
................................... 109
Exercises
..................................................... 118
Chapter III Laurent and Toeplitz Operators on Hubert Spaces
...... 135
3.1
Laurent Operators
.............................................. 135
3.2
Toeplitz Operators
............................................. 141
3.3
Band Toeplitz operators
......................................... 143
3.4
Toeplitz Operators with Continuous Symbols
..................... 152
3.5
Finite Section Method
.......................................... 159
3.6
The Finite Section Method for Laurent Operators
.................. 163
Exercises
..................................................... 166
Chapter IV Spectral Theory of Compact Self Adjoint Operators
..... 171
4.1
Example of an Infinite Dimensional Generalization
................ 171
4.2
The Problem of Existence of Eigenvalues and Eigenvectors
......... 172
4.3
Eigenvalues and Eigenvectors of Operators of Finite Rank
.......... 174
4.4
Existence of Eigenvalues
........................................ 175
4.5
Spectral Theorem
.............................................. 178
4.6
Basic Systems of Eigenvalues and Eigenvectors
................... 180
4.7
Second Form of the Spectral Theorem
............................ 182
4.8
Formula for the Inverse Operator
................................ 183
4.9
Minimum-Maximum Properties of Eigenvalues
................... 185
Exercises
..................................................... 188
Chapter V Spectral Theory of Integral Operators
................... 193
5.1
Hubert-Schmidt Theorem
....................................... 193
5.2
Preliminaries for Mercer s Theorem
.............................. 196
5.3
Mercer s Theorem
............................................. 197
5.4
Trace Formula for Integral Operators
............................. 200
Exercises
...................................................... 200
Chapter VI Unbounded Operators on Hilbert Space
................ 203
6.1
Closed Operators and First Examples
............................ 203
6.2
The Second Derivative as an Operator
............................ 204
Table of Contents
6.3
The Graph Norm
............................................... 206
6.4
Adjoint Operators
.............................................. 208
6.5
Sturm-Liouville Operators
...................................... 211
6.6
Self Adjoint Operators with Compact Inverse
..................... 214
Exercises
...................................................... 215
Chapter
VII
Oscillations of an Elastic String
........................ 219
7.1
The Displacement Function
..................................... 219
7.2
Basic Harmonic Oscillations
.................................... 220
7.3
Harmonic Oscillations with an External Force
..................... 222
Chapter
VIII
Operational Calculus with Applications
............... 225
8.1
Functions of a Compact Self Adjoint Operator
..................... 225
8.2
Differential Equations in Hilbert Space
........................... 230
8.3
Infinite Systems of Differential Equations
......................... 232
8.4
Integro-Differential Equations
................................... 233
Exercises
...................................................... 234
Chapter IX Solving Linear Equations by Iterative Methods
.......... 237
9.1
The Main Theorem
............................................. 237
9.2
Preliminaries for the Proof
...................................... 238
9.3
Proof of the Main Theorem
..................................... 240
9.4
Application to Integral Equations
................................. 242
Chapter X Further Developments of the Spectral Theorem
.......... 243
10.1
Simultaneous Diagonalization
.................................. 243
10.2
Compact Normal Operators
.................................... 244
10.3
Unitary Operators
............................................. 246
10.4
Singular Values
............................................... 248
10.5
Trace Class and Hilbert Schmidt Operators
...................... 253
Exercises
.................................................... 254
Chapter XI Banach Spaces
......................................... 259
11.1
Definitions and Examples
...................................... 259
11.2
Finite Dimensional Normed Linear Spaces
...................... 262
11.3
Separable Banach Spaces and
Schauder
Bases
................... 264
11.4
Conjugate Spaces
............................................. 265
11.5
Hahn-Banach Theorem
........................................ 267
Exercises
..................................................... 272
Table
of Contents
Chapter
XII
Linear Operators on a Banach Space
.................. 277
12.1
Description of Bounded Operators
.............................. 277
12.2
Closed Linear Operators
....................................... 279
12.3
Closed Graph Theorem
........................................ 281
12.4
Applications of the Closed Graph Theorem
...................... 283
12.5
Complemented Subspaces and Projections
....................... 286
12.6
One-Sided Invertibility Revisited
............................... 288
12.7
The Projection Method Revisited
............................... 289
12.8
The Spectrum of an Operator
................................... 290
12.9
Volterra Integral Operator
...................................... 293
12.10
Analytic Operator Valued Functions
............................ 295
Exercises
..................................................... 296
Chapter
XIII
Compact Operators on a Banach Space
............... 299
13.1
Examples of Compact Operators
............................... 299
13.2
Decomposition of Operators of Finite Rank
...................... 302
13.3
Approximation by Operators of Finite Rank
..................... 303
13.4
First Results in
Fredholm
Theory
............................... 305
13.5
Conjugate Operators on a Banach Space
......................... 306
13.6
Spectrum of a Compact Operator
............................... 310
13.7
Applications
.................................................. 313
Exercises
..................................................... 314
Chapter
XIV Poincaré
Operators: Determinant and Trace
.......... 317
14.1
Determinant and Trace
........................................ 317
14.2
Finite Rank Operators, Determinants and Traces
................. 321
14.3
Theorems about the
Poincaré
Determinant
....................... 327
14.4
Determinants and Inversion of Operators
........................ 330
14.5
Trace and Determinant Formulas for
Poincaré
Operators
.......... 336
Exercises
..................................................... 340
Chapter XV
Fredholm
Operators
.................................. 347
15.1
Definition and Examples
....................................... 347
15.2
First Properties
............................................... 347
15.3
Perturbations Small in
Nomi
................................... 352
15.4
Compact Perturbations
........................................ 355
15.5
Unbounded
Fredholm
Operators
................................ 356
Exercises
.................................................... 358
Table of Contents
xi
Chapter
XVI
Toeplitz and Singular Integral Operators
.............. 361
16.1
Laurent Operators on
ip(Ł)
.................................... 361
16.2
Toeplitz Operators onlp
....................................... 364
16.3
An Illustrative Example
....................................... 372
16.4
Applications to Pair Operators
.................................. 377
16.5
The Finite Section Method Revisited
............................ 384
16.6
Singular Integral Operators on the Unit Circle
................... 390
Exercises
..................................................... 395
Chapter
XVII
Non
Linear Operators
............................... 401
17.1
Fixed Point Theorems
......................................... 401
17.2
Applications of the Contraction Mapping Theorem
............... 402
17.3
Generalizations
............................................... 405
Appendix
1 :
Countable sets and Separable Hubert Spaces
............ 409
Appendix
2:
The Lebesgue integral and Lp Spaces
.................... 411
Suggested Reading
.................................................. 415
References
.......................................................... 417
List of Symbols
...................................................... 419
Index
............................................................... 421
|
| any_adam_object | 1 |
| author | Gohberg, Yiśrāʿēl Z. 1928-2009 Goldberg, Seymour 1928- Kaashoek, Marinus A. 1937- |
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| id | DE-604.BV017599551 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T19:19:46Z |
| institution | BVB |
| isbn | 3764369302 0817669302 |
| language | English |
| lccn | 2003063015 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010588840 |
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| physical | XVII, 423 S. |
| publishDate | 2003 |
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| publisher | Birkhäuser |
| record_format | marc |
| spelling | Gohberg, Yiśrāʿēl Z. 1928-2009 Verfasser (DE-588)118915878 aut Basic classes of linear operators Israel Gohberg ; Seymour Goldberg ; Marinus A. Kaashoek Basel [u.a.] Birkhäuser 2003 XVII, 423 S. txt rdacontent n rdamedia nc rdacarrier Frühere Ausg. u.d.T.: Gohberg, Yiśrāʿēl Z.: Basic operator theory Lineaire operatoren gtt Operadores lineares (teoria) larpcal Opérateurs linéaires Linear operators Operatortheorie (DE-588)4075665-8 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Linearer Operator (DE-588)4167721-3 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 s DE-604 Linearer Operator (DE-588)4167721-3 s Operatortheorie (DE-588)4075665-8 s Goldberg, Seymour 1928- Verfasser (DE-588)128510900 aut Kaashoek, Marinus A. 1937- Verfasser (DE-588)122738497 aut Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010588840&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gohberg, Yiśrāʿēl Z. 1928-2009 Goldberg, Seymour 1928- Kaashoek, Marinus A. 1937- Basic classes of linear operators Lineaire operatoren gtt Operadores lineares (teoria) larpcal Opérateurs linéaires Linear operators Operatortheorie (DE-588)4075665-8 gnd Funktionalanalysis (DE-588)4018916-8 gnd Linearer Operator (DE-588)4167721-3 gnd |
| subject_GND | (DE-588)4075665-8 (DE-588)4018916-8 (DE-588)4167721-3 |
| title | Basic classes of linear operators |
| title_auth | Basic classes of linear operators |
| title_exact_search | Basic classes of linear operators |
| title_full | Basic classes of linear operators Israel Gohberg ; Seymour Goldberg ; Marinus A. Kaashoek |
| title_fullStr | Basic classes of linear operators Israel Gohberg ; Seymour Goldberg ; Marinus A. Kaashoek |
| title_full_unstemmed | Basic classes of linear operators Israel Gohberg ; Seymour Goldberg ; Marinus A. Kaashoek |
| title_short | Basic classes of linear operators |
| title_sort | basic classes of linear operators |
| topic | Lineaire operatoren gtt Operadores lineares (teoria) larpcal Opérateurs linéaires Linear operators Operatortheorie (DE-588)4075665-8 gnd Funktionalanalysis (DE-588)4018916-8 gnd Linearer Operator (DE-588)4167721-3 gnd |
| topic_facet | Lineaire operatoren Operadores lineares (teoria) Opérateurs linéaires Linear operators Operatortheorie Funktionalanalysis Linearer Operator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010588840&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT gohbergyisraʿelz basicclassesoflinearoperators AT goldbergseymour basicclassesoflinearoperators AT kaashoekmarinusa basicclassesoflinearoperators |