Operator functions and operator equations:
"This book is devoted to norm estimates for operator-valued functions of one and two operator arguments, as well as to their applications to spectrum perturbations of operators and to linear operator equations, i.e. to equations whose solutions are linear operators. Linear operator equations ar...
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo
World Scientific
[2018]
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | "This book is devoted to norm estimates for operator-valued functions of one and two operator arguments, as well as to their applications to spectrum perturbations of operators and to linear operator equations, i.e. to equations whose solutions are linear operators. Linear operator equations arise in both mathematical theory and engineering practice. The norm estimates suggested in the book have applications to the theories of ordinary differential, difference, functional-differential and integro-differential equations, as well as to the theories of integral operators and analytic functions. This book provides new tools for specialists in matrix theory and functional analysis. A significant part of the book covers the theory of triangular representations of operators that was developed by L de Branges, M S Brodskii, I C Gohberg, M G Krein, M S Livsic and other mathematicians."--Publisher's website |
| Physical Description: | xiii, 245 Seiten |
| ISBN: | 9789813221260 |
Staff View
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Record in the Search Index
| _version_ | 1804178127612542976 |
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| adam_text | Contents Preface v 1. Preliminaries 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1 2 2. Banach and Hilbert spaces................................................... Linear operators ................................................................. Functions of a bounded operator.................................. 6 Functions of an unbounded operator............................ 10 The operator logarithm .......................... Perturbations of operators in uniform topology ............... Perturbations of operators in strong topology ....... Spectral variations.................... Rotations of simple eigenvectors..................................... 18 Comments to Chapter 1 ............................................ Representations of Solutionsto Operator Equations 12 13 14 17 20 21 2.1 Generalized polynomial operator equations................... 22 2.2 The quasi-Sylvester equation.......................... 24 2.3 . The Sylvester equation........................................................ 26 2.4 Additional representations for solutions of the Sylvester equation.................................................................... 30 2.5 Polynomial operator equations . ........................... 32 2.6 Proof of Theorem 2.6....................................................... 33 2.7, Additional representations of solutions to equation (5.1) . 35 2.8 Additional representations of solutions to equation (5.2) . 35 2.9 Equations with scalar type spectral operators ....... 37 2.10 Perturbations of two-sided Sylvester equations............. 39 ix
x Operator Functions and Operator Equations 2.11 Differentiating of solutions to two-sided Sylvester equations 40 2.12 Perturbed Sylvester equations and spectral variations ... 42 2.13 Comments to Chapter 2...................................................... 42 3. 4. Functions of Finite Matrices 43 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 43 47 50 53 54 58 60 62 65 69 71 Solution Estimates for Polynomial Matrix Equations 73 4.1 4.2 4.3 73 75 77 77 78 78 79 84 85 4.4 4.5 4.6 4.7 5. Departure from normality................................................... A norm estimate for resolvents.......................................... Inequality between resolvents and determinants............... Functions regular on the convex hull of the spectrum ... Proof of Theorem 3.5....................... The function j(A)............................................................... Norm estimates for the matrix logarithm........................... Spectral representations for resolvents.............................. An additional inequality for powers of nilpotent matrices . Perturbations of matrices . . ................................. Comments to Chapter 3...................................................... The quasi-Sylvester matrix equation ............ The Sylvester equation......................................... Polynomial matrix equations ........................... 4.3.1 Equation (3.1).............. 4.3.2 Equation (3.2)......................................... Functions of two non-commuting matrices ......... Proof of Theorem 4.3............................. Generalized
polynomial matrix equations........................... Comments to Chapter 4.................... Two-sided Matrix Sylvester Equations 5.1 5.2 5.3 5.4 5.5 5.6 5.7 87 Auxiliary results ............................................... 87 The resolvent of operator Z ................. 90 5.2.1 Statement of the result..................... .................... ,90 5.2.2 Proof of Theorem 5.1........................ . ............... 92 Simultaneously triangularizable matrices........................... 93 Proof of Theorem 5.2 ............. ..................... 94 Particular cases of conditions (3.1) . . . ........................... 97 Solution estimates for two-sided matrix Sylvester equations 101 Perturbations of invariant subspaces of finite matrices . . 102
Contents 6. 7. xi 5.8 Comments to Chapter 5................... 104 Bounds for Condition Numbers of DiagonalizableMatrices 105 6.1 6.2 6.3 6.4 6.5 6.6 105 107 109 111 112 113 A bound for condition numbers of matrices . . . ... . . Auxiliary results ..................... Proof of Theorem 6.1 . . . ................................. The quasi-Sylvester equation with diagonalizable matrices The Sylvester equation with diagonalizable matrices ... Comments to Chapter 6........................................... .......... Functions of a Compact Operator in a Hilbert Space . 115 7.1 , Schatten-von Neumann operators ... . . . ..... . . . 115 7.2 The resolvent of a Hilbert-Schmidt operator . . . . . . . . 117 7.3 Resolvents of Schatten-von Neumann operators . . . . . . 119 7.4 . Proofs of Theorems 7.2 and 7.3.................... 120 7.5 Functions of a Hilbert-Schmidt operator...................... 123 7.6 Equations with compact operators . . . . . . ... . . . . 124 7.7 Perturbations of compact operators . ..................... .. . . . 125 7.8 Comments to Chapter 7................................................. 128 8. 9. Triangular Representations of Non-selfadjoint Operators 129 8.1 8.2 8.3 8.4 8.5 8.6 129 135 137 141 142 145 P-triangular operators . . ... . .... . ... . . . . . . ; Existence of invariant maximal chains............. .... . . . . Operators with real spectra..................................... . Operators with non-real spectra . . . . . . . . ...... v . Compactly perturbed unitary operators . ........ Comments to Chapter 8 ......... ... ........ Resolvents of
Bounded Non-selfadjoint Operators 147 9.1 9.2 147 9.3 9.4 9.5 9.6 Resolvents of ^-triangular operators . . . . . . . . . . . . Resolvents of operators with Hilbert-Schmidt Hermitian components . . . . ... . . . ..... . . ............ .. . . . Auxiliary results . ............................................................... Some properties of quasi-nilpotent Schatten-von Neumann operators ................................... . . Resolvent of operators with Schatten-von Neumann Hermitian components.................................................. Resolvents of operators close to unitary ones . . . . . . . . 148 151 154 156 157
Operator Functions and Operator Equations xii 9.7 Eigenvalues of compactly perturbed unitary operators . . 9.7.1 Eigenvalues outside the unit circle........................ 9.7.2 Eigenvalues inside the unit circle .......... 9.7.3 . The general case.......................... 9.8 Additional estimates for ő(A).............. 9.9 Multiplicative representations of resolvents . . . ... . . . 9.10 Comments to Chapter 9 .......................................... 10. Regular Functions of a Bounded Non-selfadjoint Operator 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 161 161 162 163 164 165 167 169 Preliminary results.......................................................... 169 Functions of an operator with a Hilbert-Schmidt component 174 Integral models of quasi-nilpotent operators ..................... 175 Operators with Hilbert-Schmidt nilpotent parts...... 179 Functions of an operator close to a unitary one.......... 181 Examples....................................................................... 184 The Sylvester equation with non-selfadjoint operators . . 185 Perturbations of invariant subspaces.............. 186 Comments to Chapter 10 . . ........................ 187 11. Functions of an Unbounded Operator 189 11.1 Boundedly perturbed selfadjoint operators................... 189 11.2 Unbounded operators with compact components..... 191 11.3 Resolvents of operators inverse to Schatten-von Neumann ones.................................................................................. 193 11.4 Proof of Theorem 11.2.................................................... 194 11.5
Hirsch type operator functions....................... 198 11.6 Comments to Chapter 11 ........................... .. . ............... 200 12. Similarity Condition Numbers of Unbounded Diagonalizable Operators 201 12.1 Condition numbers of operators with Schatten-von Neumann Hermitian components........................ 201 12.2 Operators with finite invariant chains .....՝.................. 204 12.3 The finite dimensional case....................... 206 12.4 Proof of Theorem 12.1................................... 209 12.5 Condition numbers of boundedly perturbed normal operators ........................................................... 210 12.6 Proof of Theorem 12.2............................................... 212
Contents xiii 12.7 Applications of condition numbers .................................... 215 12.8 Comments to Chapter 12 .................................................. 217 13. Commutators and Perturbations of Operator Functions 13.1 13.2 13.3 13.4 13.5 13.6 Representations of commutators.......................................... 219 The finite dimensional case ................................................ 221 Operators with. Hilbert-Schmidt components.....................224 Perturbations of entire Banach valued functions...............226 Conservation of stability...................................................... 227 Comments to Chapter 13 ................................................... 229 14. Functions of Two Non-commuting Operators in Hilbert Spaces 14.1 14.2 14.3 14.4 219 Statement of the result.................................................. Proof of Theorem 14.1 . . ................................................... Generalized polynomial equations in Hilbert spaces .... Comments to Chapter 14 ................................................... 231 231 232 233 235 Bibliography 237 List of Symbols 243 Index 245
|
| any_adam_object | 1 |
| author | Gil', Michail I. 1941- |
| author_GND | (DE-588)128577916 |
| author_facet | Gil', Michail I. 1941- |
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| author_sort | Gil', Michail I. 1941- |
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| dewey-full | 515.73 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.73 |
| dewey-search | 515.73 |
| dewey-sort | 3515.73 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV044676221 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:59:02Z |
| institution | BVB |
| isbn | 9789813221260 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-030073483 |
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| owner_facet | DE-824 DE-19 DE-BY-UBM DE-739 |
| physical | xiii, 245 Seiten |
| publishDate | 2018 |
| publishDateSearch | 2018 |
| publishDateSort | 2018 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Gil', Michail I. 1941- Verfasser (DE-588)128577916 aut Operator functions and operator equations Michael I Gil', Ben Gurion University of the Negev, Israel New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific [2018] © 2018 xiii, 245 Seiten txt rdacontent n rdamedia nc rdacarrier "This book is devoted to norm estimates for operator-valued functions of one and two operator arguments, as well as to their applications to spectrum perturbations of operators and to linear operator equations, i.e. to equations whose solutions are linear operators. Linear operator equations arise in both mathematical theory and engineering practice. The norm estimates suggested in the book have applications to the theories of ordinary differential, difference, functional-differential and integro-differential equations, as well as to the theories of integral operators and analytic functions. This book provides new tools for specialists in matrix theory and functional analysis. A significant part of the book covers the theory of triangular representations of operators that was developed by L de Branges, M S Brodskii, I C Gohberg, M G Krein, M S Livsic and other mathematicians."--Publisher's website Operator-valued functions Operatory theory Functional analysis Linear operators Perturbation (Mathematics) Spectral theory (Mathematics) Electronic books Operatorgleichung (DE-588)4043601-9 gnd rswk-swf Operatorfunktion (DE-588)4202830-9 gnd rswk-swf Operatorgleichung (DE-588)4043601-9 s Operatorfunktion (DE-588)4202830-9 s DE-604 Erscheint auch als Online-Ausgabe 978-981-3221-27-7 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=030073483&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gil', Michail I. 1941- Operator functions and operator equations Operator-valued functions Operatory theory Functional analysis Linear operators Perturbation (Mathematics) Spectral theory (Mathematics) Electronic books Operatorgleichung (DE-588)4043601-9 gnd Operatorfunktion (DE-588)4202830-9 gnd |
| subject_GND | (DE-588)4043601-9 (DE-588)4202830-9 |
| title | Operator functions and operator equations |
| title_auth | Operator functions and operator equations |
| title_exact_search | Operator functions and operator equations |
| title_full | Operator functions and operator equations Michael I Gil', Ben Gurion University of the Negev, Israel |
| title_fullStr | Operator functions and operator equations Michael I Gil', Ben Gurion University of the Negev, Israel |
| title_full_unstemmed | Operator functions and operator equations Michael I Gil', Ben Gurion University of the Negev, Israel |
| title_short | Operator functions and operator equations |
| title_sort | operator functions and operator equations |
| topic | Operator-valued functions Operatory theory Functional analysis Linear operators Perturbation (Mathematics) Spectral theory (Mathematics) Electronic books Operatorgleichung (DE-588)4043601-9 gnd Operatorfunktion (DE-588)4202830-9 gnd |
| topic_facet | Operator-valued functions Operatory theory Functional analysis Linear operators Perturbation (Mathematics) Spectral theory (Mathematics) Electronic books Operatorgleichung Operatorfunktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030073483&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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