Gohberg / Leiterer Holomorphic Operator Functions of One Variable and Applications

Methods from Complex Analysis in Several Variables
2009
ISBN: 978-3-0346-0126-9
Verlag: Birkhäuser Basel
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 192, 424 Seiten

Reihe: Operator Theory: Advances and Applications

This book presents holomorphic operator functions of a single variable and applications, which are focused on the relations between local and global theories. It is based on methods and technics of complex analysis of several variables.

Autoren/Hrsg.

Gohberg, Israel

Leiterer, Jürgen

Inhaltsverzeichnis

1;Contents;7
2;Preface;11
3;Introduction;13
4;Notation;18
5;1 Elementary properties of holomorphic functions;20
5.1;1.1 Definition and first properties;20
5.2;1.2 The maximum principle;22
5.3;1.3 Contour integrals;27
5.4;1.4 The Cauchy integral theorem;30
5.5;1.5 The Cauchy formula;31
5.6;1.6 The Hahn-Banach criterion;34
5.7;1.7 A criterion for the holomorphy of operator functions;37
5.8;1.8 Power series;38
5.9;1.9 Laurent series;42
5.10;1.10 Isolated singularities;44
5.11;1.11 Comments;47
6;2 Solution of .u = f and applications;48
6.1;2.1 The Pompeiju formula for solutions of on compact sets;48
6.2;2.2 Runge approximation;56
6.3;2.3 Solution of of .u = f on open sets;60
6.4;2.4 OE-cocycles and the Mittag-Leffler theorem;62
6.5;2.5 Runge approximation for invertible scalar functions and the Weierstrass product theorem;63
6.6;2.6 OE-cocycles with prescribed zeros and a stronger version of the Mittag-Leffler theorem;71
6.7;2.7 Generalization of the Weierstrass product theorem;73
6.8;2.8 Comments;77
7;3 Splitting and factorization with respect to a contour;78
7.1;3.1 Splitting with respect to a contour;78
7.2;3.2 Splitting and the Cauchy Integral;80
7.3;3.3 H¨ older continuous functions split;84
7.4;3.4 The splitting behavior of differentiable functions;91
7.5;3.5 Approximation of H¨ older continuous functions;94
7.6;3.6 Example: A non-splitting continuous function;96
7.7;3.7 The additive local principle;101
7.8;3.8 Factorization of scalar functions with respect to a contour. First remarks;103
7.9;3.9 Factorization of H¨ older functions;108
7.10;3.10 Factorization of Wiener functions;110
7.11;3.11 The multiplicative local principle;112
7.12;3.12 Comments;114
8;4 The Rouché theorem for operator functions;115
8.1;4.1 Finite meromorphic Fredholm functions;115
8.2;4.2 Invertible finite meromorphic Fredholm functions;119
8.3;4.3 Smith factorization;124
8.4;4.4 The Rouché theorem;129
8.5;4.5 Comments;130
9;5 Multiplicative cocycles (OG-cocycles);131
9.1;5.1 Topological properties of GL(E);132
9.2;5.2 Two factorization lemmas;138
9.3;5.3 OE-cocycles;141
9.4;5.4 Runge approximation ofG-valued functions - First steps;143
9.5;5.5 The Cartan lemma;147
9.6;5.6 OG-cocycles. Definitions and statement of the main result;149
9.7;5.7 Refinement of the covering;151
9.8;5.8 Exhausting by compact sets;155
9.9;5.9 Proof of Theorem 5.6.3 for simply connected open sets;159
9.10;5.10 Runge approximation of G-valued functions General case;161
9.11;5.11 Proof of Theorem 5.6.3 in the general case;165
9.12;5.12 OG8(E)-cocycles;169
9.13;5.13 Weierstrass theorems;171
9.14;5.14 Weierstrass theorems for G8(E) and G.(E)-valued functions;174
9.15;5.15 Comments;175
10;6 Families of subspaces;177
10.1;6.1 The gap metric;177
10.2;6.2 Kernel and image of operator functions;190
10.3;6.3 Holomorphic sections of continuous families of subspaces;202
10.4;6.4 Holomorphic families of subspaces;203
10.5;6.5 Example: A holomorphic family of subspaces with jumping isomorphism type;219
10.6;6.6 Injective families;221
10.7;6.7 Shubin families;222
10.8;6.8 Complemented families;224
10.9;6.9 Finite dimensional and finite codimensional families;227
10.10;6.10 One-sided and generalized invertible holomorphic operator functions;229
10.11;6.11 Example: A globally non-trivial complemented holomorphic family of subspaces;232
10.12;6.12 Comments;234
11;7 Plemelj-Muschelishvili factorization;236
11.1;7.1 Definitions and first remarks about factorization;237
11.2;7.2 The algebra of Wiener functions and other splitting R-algebras;239
11.3;7.3 Hölder continuous and differentiable functions;247
11.4;7.4 Reduction of the factorization problem to functions, holomorphic and invertible on C*;254
11.5;7.5 Factorization of holomorphic functions close to the unit;257
11.6;7.6 Reduction of the factorization problem to polynomials in z and 1/z;257
11.7;7.7 The finite dimensional case;259
11.8;7.8 Factorization of G8(E)-valued functions;262
11.9;7.9 The filtration of an operator function with respect to a contour;268
11.10;7.10 A general criterion for the existence of factorizations;276
11.11;7.11 Comments;284
12;8 Wiener-Hopf operators, Toeplitz operators and factorization;285
12.1;8.1 Holomorphic operator functions;285
12.2;8.2 Factorization of G.(E)-valued functions;289
12.3;8.3 The space L2(G,H);292
12.4;8.4 Operator functions with values acting in a Hilbert space;303
12.5;8.5 Functions close to the unit operator or with positive real part;307
12.6;8.6 Block Töplitz operators;313
12.7;8.7 The Fourier transform of;321
12.8;8.8 The Fourier isometry U of L2(R,H);329
12.9;8.9 The isometry V from L2(T,H) onto L2(R,H);333
12.10;8.10 The algebra of operator functions L(H).L1(R, L(H));336
12.11;8.11 Factorization with respect to the real line;340
12.12;8.12 Wiener-Hopf integral operators in L2([0,8[,H);341
12.13;8.13 An example;354
12.14;8.14 Comments;356
13;9 Multiplicative cocycles with restrictions (F-cocycles);358
13.1;9.1 F-cocycles;358
13.2;9.2 The main results on cocycles with restrictions. Formulation and reduction to OD,Z,m;361
13.3;9.3 The Cartan lemma with restrictions;362
13.4;9.4 Splitting over simply connected open sets after shrinking;367
13.5;9.5 Runge approximation on simply connected open sets;369
13.6;9.6 Splitting over simply connected open sets without shrinking;373
13.7;9.7 Runge approximation. The general case;377
13.8;9.8 The Oka-Grauert principle;380
13.9;9.9 Comments;382
14;10 Generalized interpolation problems;383
14.1;10.1 Weierstrass theorems;383
14.2;10.2 Right- and two-sided Weierstrass theorems;385
14.3;10.3 Weierstrass theorems for G8(E)- and G.(E)-valued functions;388
14.4;10.4 Holomorphic G8(E)-valued functions with given principal parts of the inverse;391
14.5;10.5 Comments;392
15;11 Holomorphic equivalence, linearization and diagonalization;393
15.1;11.1 Introductory remarks;393
15.2;11.2 Linearization by extension and equivalence;394
15.3;11.3 Local equivalence;400
15.4;11.4 A theorem on local and global equivalence;406
15.5;11.5 The finite dimensional case;408
15.6;11.6 Local and global equivalence for finite meromorphic Fredholm functions;413
15.7;11.7 Global diagonalization of finite meromorphic Fredholm functions;420
15.8;11.8 Comments;425
16;Bibliography;426
17;Index;432