Traces and determinants of linear operators:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2000
|
| Series: | Operator theory
116 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VIII, 258 S. |
| ISBN: | 3764361778 |
Staff View
MARC
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| 245 | 1 | 0 | |a Traces and determinants of linear operators |c Israel Gohberg ; Seymour Goldberg ; Nahum Krupnik |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface ix
Introduction 1
I Finite Rank Operators
1 Trace and determinant for finite rank operators 5
2 Properties of the trace and determinant 7
3 Representations of the trace and determinant 8
4 Uniqueness of the trace and determinant 10
5 Von Koch form of the determinant 12
6 Fredholm form of the determinant 14
7 Plemelj Smithies formulas 15
8 Polynomial operator pencils 17
9 Inversion formulas 20
10 Comments 23
II Continuous Extension of Trace and Determinant
1 Extension problems and embedded algebras 25
2 Main theorems 26
3 Analyticity of the determinant and the
Plemelj Smithies formulas 29
4 Lipschitz conditions 31
5 Several remarks 33
6 Connections between the zeros of the determinant and
the eigenvalues of an operator 34
7 Determinants of infinite matrices in Von Koch form 35
8 Comments 38
III First Examples
1 The Poincare determinant 39
2 Hill s method 40
3 The Von Koch Riesz algebra 42
4 The Mennicken Wagenfiihrer algebra 43
5 The algebra T {£i) 45
6 Comments 45
vi Contents
IV Trace Class and Hilbert Schmidt Operators in Hilbert Space
1 Preliminaries 47
2 Singular numbers 49
3 Inequalities for eigenvalues, diagonal elements
and singular numbers 52
4 Additional inequalities for singular numbers 56
5 Ideal of trace class operators 59
6 Lidskii trace theorem 63
7 Hilbert Schmidt operators 66
8 Tests of nuclearity for integral operators with continuous
and Hilbert Schmidt kernels 70
9 Integral operators with smooth kernels 78
10 Polynomial operator pencils 83
11 Classes Sp 87
12 Comments 90
V Nuclear Operators in Banach Spaces
1 The Ruston Grothendieck algebra of nuclear operators 91
2 Examples of nuclear operators in Banach spaces 97
3 Grothendieck trace theorem 102
4 Asymptotic behavior of eigenvalues of nuclear operators 106
5 Comments 109
VI The Predholm Determinant
1 The Fredholm determinant for integral operators with
continuous and piecewise continuous kernels Ill
2 The Algebra Vn{H). Hill s Method (revisited) 115
3 Diagonally modified Fredholm determinant 121
4 A modification of the Plemelj Smithies formula 124
5 Integral Operators in Li(T, 5^,/i) 126
6 Systems of integral equations 127
7 Comments 132
VII Possible Values of Traces and Determinants. Perelson Algebras
1 Perelson algebras 133
2 Possible values of traces and determinants
in Perelson algebras 137
3 Possible values in Dn(H) 139
4 Comments 141
VIII Inversion Formulas
1 General inversion formulas 143
2 Explicit formulas for resolvents of integral operators 145
3 Homogeneous integral equations 152
4 Comments 157
Contents vii
IX Regularized Determinants
1 Extension problems 159
2 The main (extension) theorems for regularized determinants . . . 162
3 Analyticity, Plemelj Smithies formulas 166
4 Comments 168
X Hilbert Carleman Determinants
1 Integral operators with degenerate kernels 169
2 Integral operators on a class of Banach spaces 172
3 Hilbert Schmidt integral operators 176
4 Mikhlin Itskovich algebra 177
5 Algebra S1 179
6 Diagonally modified Hilbert Carleman determinant 180
7 Hilbert Carleman determinant for infinite matrices 183
8 Comments 186
XI Regularized Determinants of Higher Order
1 Main extension theorems 187
2 Analyticity and Plemelj Smithies formulas 192
3 Preparation for the proof of Theorem IV.10.3 195
4 Proof of Theorem IV. 10.3 199
5 Comments 200
XII Inversion Formulas via Generalized Determinants
1 General case 201
2 Integral equations 205
3 Systems of Hill s equations 209
4 Comments 211
XIII Determinants of Integral Operators with Semi separable Kernels
1 Statement of the main theorem 213
2 Input output representations 216
3 Cascade connection of systems 219
4 Inverse systems 222
5 Inversion of integral operators with semi separable kernels .... 223
6 Indicator of integral operators 224
7 Computation of the Hilbert Carleman and the
Fredholm determinants 227
8 Spectra of integral operators with semi separable kernels 230
9 Time invariant systems 232
10 Counting negative eigenvalues of a Hilbert Schmidt operator
via sign changes of a determinant 235
11 Comments 242
viii Contents
XIV Algebras without the Approximation Property
1 A general class of algebras 243
2 Integral operators with a jump discontinuity
on the diagonal 244
3 Applications to integral operators with a
jump discontinuity on the diagonal 246
4 Applications to integral operators with
semi separable kernel 247
5 Comments 248
Bibliography 249
Index 253
List of Symbols 257
|
| any_adam_object | 1 |
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| author_sort | Gohberg, Yiśrāʿēl Z. 1928-2009 |
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| building | Verbundindex |
| bvnumber | BV013022357 |
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| discipline | Mathematik |
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| id | DE-604.BV013022357 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:37:50Z |
| institution | BVB |
| isbn | 3764361778 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008872556 |
| oclc_num | 633106887 |
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| physical | VIII, 258 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Operator theory |
| series2 | Operator theory |
| spelling | Gohberg, Yiśrāʿēl Z. 1928-2009 Verfasser (DE-588)118915878 aut Traces and determinants of linear operators Israel Gohberg ; Seymour Goldberg ; Nahum Krupnik Basel [u.a.] Birkhäuser 2000 VIII, 258 S. txt rdacontent n rdamedia nc rdacarrier Operator theory 116 Spur Mathematik (DE-588)4202272-1 gnd rswk-swf Linearer Operator (DE-588)4167721-3 gnd rswk-swf Banach-Raum (DE-588)4004402-6 gnd rswk-swf Determinante (DE-588)4138983-9 gnd rswk-swf Linearer Operator (DE-588)4167721-3 s Banach-Raum (DE-588)4004402-6 s Determinante (DE-588)4138983-9 s Spur Mathematik (DE-588)4202272-1 s DE-604 Goldberg, Seymour 1928- Verfasser (DE-588)128510900 aut Krupnik, Nahum 1932- Verfasser (DE-588)121758400 aut Operator theory 116 (DE-604)BV000000970 116 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008872556&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gohberg, Yiśrāʿēl Z. 1928-2009 Goldberg, Seymour 1928- Krupnik, Nahum 1932- Traces and determinants of linear operators Operator theory Spur Mathematik (DE-588)4202272-1 gnd Linearer Operator (DE-588)4167721-3 gnd Banach-Raum (DE-588)4004402-6 gnd Determinante (DE-588)4138983-9 gnd |
| subject_GND | (DE-588)4202272-1 (DE-588)4167721-3 (DE-588)4004402-6 (DE-588)4138983-9 |
| title | Traces and determinants of linear operators |
| title_auth | Traces and determinants of linear operators |
| title_exact_search | Traces and determinants of linear operators |
| title_full | Traces and determinants of linear operators Israel Gohberg ; Seymour Goldberg ; Nahum Krupnik |
| title_fullStr | Traces and determinants of linear operators Israel Gohberg ; Seymour Goldberg ; Nahum Krupnik |
| title_full_unstemmed | Traces and determinants of linear operators Israel Gohberg ; Seymour Goldberg ; Nahum Krupnik |
| title_short | Traces and determinants of linear operators |
| title_sort | traces and determinants of linear operators |
| topic | Spur Mathematik (DE-588)4202272-1 gnd Linearer Operator (DE-588)4167721-3 gnd Banach-Raum (DE-588)4004402-6 gnd Determinante (DE-588)4138983-9 gnd |
| topic_facet | Spur Mathematik Linearer Operator Banach-Raum Determinante |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008872556&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000970 |
| work_keys_str_mv | AT gohbergyisraʿelz tracesanddeterminantsoflinearoperators AT goldbergseymour tracesanddeterminantsoflinearoperators AT krupniknahum tracesanddeterminantsoflinearoperators |