Lerer / Olshevsky / Spitkovsky Convolution Equations and Singular Integral Operators

Selected Papers
2010
ISBN: 978-3-7643-8956-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 206, 240 Seiten

Reihe: Operator Theory: Advances and Applications

This book consists of translations into English of several pioneering papers in the areas of discrete and continuous convolution operators and on the theory of singular integral operators published originally in Russian. The papers were wr- ten more than thirty years ago, but time showed their importance and growing in?uence in pure and applied mathematics and engineering. The book is divided into two parts. The ?rst ?ve papers, written by I. Gohberg and G. Heinig, form the ?rst part. They are related to the inversion of ?nite block Toeplitz matrices and their continuous analogs (direct and inverse problems) and the theory of discrete and continuous resultants. The second part consists of eight papers by I. Gohberg and N. Krupnik. They are devoted to the theory of one dimensional singular integral operators with discontinuous co- cients on various spaces. Special attention is paid to localization theory, structure of the symbol, and equations with shifts. ThisbookgivesanEnglishspeakingreaderauniqueopportunitytogetfam- iarized with groundbreaking work on the theory of Toepliz matrices and singular integral operators which by now have become classical. In the process of the preparation of the book the translator and the editors took care of several misprints and unessential misstatements. The editors would like to thank the translator A. Karlovich for the thorough job he has done. Our work on this book was started when Israel Gohberg was still alive. We see this book as our tribute to a great mathematician.

Autoren/Hrsg.

Lerer, Leonid

Olshevsky, Vadim

Spitkovsky, Ilya M.

Rodman, Leiba

Weitere Mitwirkende

Karlovych, Oleksiy

Inhaltsverzeichnis

1;Title Page;4
2;Copyright Page;5
3;Table of Contents;6
4;Preface;7
5;Introduction;8
5.1;References;17
6;Inversion of Finite Toeplitz Matrices;22
6.1;1. Inversion of Toeplitz matrices;22
6.2;2. Inverse problem;24
6.3;3. Continual analogue;26
6.4;References;27
7;Inversion of Finite Toeplitz Matrices Consisting of Elements of a Noncommutative Algebra ;28
7.1;1. Theorems on inversion of Toeplitz matrices;28
7.2;2. Properties of solutions of equations;36
7.3;3. Inverse problem;44
7.4;4. Other theorems on inversion of Toeplitz matrices;49
7.5;5. Properties of solutions of equations;58
7.6;6. Inverse problem for equations;62
7.7;References;66
8;Matrix Integral Operators on a Finite Interval with Kernels Depending on the Difference of the Arguments;68
8.1;1. Two lemmas;69
8.2;2. The inversion formula;72
8.2.1;2.1. The main result of this section is the following.;72
8.2.2;2.2. Formula;76
8.3;3. Properties of the solutions of main equations;77
8.4;4. Inverse problem;81
8.5;References;84
9;The Resultant Matrix and its Generalizations. I. The Resultant Operator for Matrix Polynomials;85
9.1;1. Lemma on multiple extensions of systems of vectors;86
9.1.1;1.1.;86
9.1.2;1.2.;87
9.2;2. Auxiliary propositions;93
9.2.1;2.1.;93
9.2.2;2.2.;95
9.2.3;2.3.;97
9.3;3. Main theorem;98
9.3.1;3.1.;98
9.3.2;3.2.;99
9.3.3;3.3.;101
9.4;4. Applications;102
9.4.1;4.1.;102
9.4.2;4.2.;104
9.5;5. Kernel of Bezoutian;105
9.5.1;5.1.;105
9.5.2;5.2.;107
9.6;References;108
10;The Resultant Matrix and its Generalizations. II. The Continual Analogue of the Resultant Operator;109
10.1;1. Formulation of the main theorem;110
10.1.1;1.1.;110
10.1.2;1.2.;111
10.1.3;1.3.;113
10.1.4;1.4.;113
10.1.5;1.5.;114
10.2;2. A lemma;115
10.3;3. Proof of the main theorem;118
10.4;4. Scalar case;121
10.4.1;4.1.;121
10.4.2;4.2.;123
10.5;5. Applications;123
10.5.1;5.1.;123
10.5.2;5.2.;126
10.5.3;5.3.;126
10.6;6. Continual analogue of the Bezoutian;126
10.7;References;128
11;The Spectrum of Singular Integral Operators in Lp Spaces;130
11.1;1. The spectrum of discrete Wiener-Hopf equations in hp spaces;131
11.1.1;1.1.;131
11.1.2;1.2.;131
11.1.3;1.3.;132
11.2;2. The spectrum of singular integral operators in Lp spaces;133
11.3;3. Estimate for the norm of the singular integral operator;139
11.4;4. The spectrum of singular integral operators in symmetric spaces;141
11.5;References;143
12;On an Algebra Generated by the Toeplitz Matrices in the Spaces hp;145
12.1;1. Toeplitz operators in the spaces hp;146
12.1.1;1.1.;146
12.1.2;1.2.;147
12.2;2. Algebra generated by the Toeplitz operators;148
12.2.1;2.1.;148
12.2.2;2.2.;150
12.3;References;151
13;On Singular Integral Equations with Unbounded Coefficients;152
13.1;1. Auxiliary propositions;153
13.2;2. On the boundedness of the operator of singular integration;155
13.3;3. Operators of the form;157
13.4;4. Operators of the form;159
13.5;References;161
14;Singular Integral Equations with Continuous Coefficients on a Composed Contour;162
14.1;1. Auxiliary propositions and theorems on solvability;164
14.2;2. Quotient algebra;169
14.3;3. Normal solvability and index of operators in the algebra;171
14.4;References;173
15;On a Local Principle and Algebras Generated by Toeplitz Matrices;174
15.1;1. Localizing classes;175
15.1.1;1.1.;175
15.1.2;1.2.;175
15.1.3;1.3.;176
15.2;2. First example;177
15.2.1;2.1.;177
15.2.2;2.2.;178
15.3;3. Some properties of operators generated by Toeplitz matrices in lp spaces ;179
15.3.1;3.1.;179
15.3.2;3.2.;180
15.3.3;3.3.;181
15.3.4;3.4.;182
15.4;4. Second example and its applications;183
15.4.1;4.1.;183
15.4.2;4.2.;184
15.5;5. Inversion of Toeplitz matrices;186
15.5.1;5.1.;186
15.5.2;5.2.;188
15.5.3;5.3.;191
15.6;6. Algebra of operators with Fredholm symbol;192
15.7;7. Algebras generated by paired operators;195
15.7.1;7.1.;195
15.7.2;7.2.;198
15.8;References;199
16;The Symbol of Singular Integral Operators on a Composed Contour;202
16.1;1. Auxiliary propositions;203
16.1.1;1.1.;203
16.1.2;1.2.;204
16.1.3;1.3.;205
16.2;2. Singular operators with coefficients in .+ and .-;205
16.3;3. Algebra generated by singular operators with coefficients in .+m and .-m;207
16.4;4. Singular integral operators with coefficients in .m;209
16.5;5. Symbols of singular operators;211
16.5.1;5.1.;211
16.5.2;5.2.;213
16.6;6. Concluding remarks;214
16.6.1;6.1.;214
16.6.2;6.2.;215
16.7;References;215
17;One-dimensional Singular Integral Operators with Shift;217
17.1;Introduction;217
17.2;1. Auxiliary statements;219
17.3;2. Main statement;222
17.4;3. Example;225
17.5;References;226
18;Algebras of Singular Integral Operators with Shift;228
18.1;1.;228
18.2;2.;231
18.3;References;232