Holomorphic Operator Functions of One Variable and Applications

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ISBN: 9783034601252 | 3034601255
Cover: Hardcover
Copyright: 7/1/2009

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This is a book on holomorphic operator functions of a single variable and applications, which is focussed on the relations between local and global theories. It is based on methods and technics of complex analysis of several variables. The first part of the theory starts with a straightforward generalization of some results from the basics of analysis of scalar functions of one complex variable. In the second part, which is the main part of the theory, results are obtained by methods and toolsadapted from complex analysis of functions of several variables. We have in mind the theory of holomorphic cocycles (fiber bundles) with values in infinite-dimensional non-commutative groups. As a rule, these results do not appear in traditional complex analysis of one variable, not even for matrix valued cocycles. The third part consists of applications to operator theory. Here applications are presented for holomorphic families of subspaces and Plemelj-Muschelishvili factorization. The forth part presents a generalization of the theory of cocycles to cocycles with restrictions. This part contains also applications to interpolation problems, to the problem of holomorphic equivalence and diagonalization.

Preface	p. xi
Introduction	p. xiii
Notation	p. xix
Elementary properties of holomorphic functions	p. 1
Definition and first properties	p. 1
The maximum principle	p. 3
Contour integrals	p. 8
The Cauchy integral theorem	p. 11
The Cauchy formula	p. 12
The Hahn-Banach criterion	p. 15
A criterion for the holomorphy of operator functions	p. 18
Power series	p. 19
Laurent series	p. 23
Isolated singularities	p. 25
Comments	p. 28
Solution of $$ = f and applications	p. 29
The Pompeiju formula for solutions of $$ = f on compact sets	p. 29
Runge approximation	p. 37
Solution of $$ = f on open sets	p. 41
OE-cocycles and the Mittag-Leffler theorem	p. 43
Runge approximation for invertible scalar functions and the Weierstrass product theorem	p. 44
OE-cocycles with prescribed zeros and a stronger version of the Mittag-Leffler theorem	p. 52
Generalization of the Weierstrass product theorem	p. 54
Comments	p. 58
Splitting and factorization with respect to a contour	p. 59
Splitting with respect to a contour	p. 59
Splitting and the Cauchy Integral	p. 61
Höet;lder continuous functions split	p. 65
The splitting behavior of differentiable functions	p. 72
Approximation of Höet;lder continuous functions	p. 75
Example: A non-splitting continuous functions	p. 77
The additive local principle	p. 82
Factorization of scalar functions with respect to a contour. First remarks	p. 84
Factorization of Höet;lder functions	p. 89
Factorization of Wiener functions	p. 91
The multiplicative local principle	p. 93
Comments	p. 95
The Rouché theorem for operator functions	p. 97
Finite meromorphic Fredholm functions	p. 97
Invertible finite meromorphic Fredholm functions	p. 101
Smith factorization	p. 106
The Rouché theorem	p. 111
Comments	p. 112
Multiplicative cocycles ($$G-cocycles)	p. 113
Topological properties of GL(E)	p. 114
Two factorization lemmas	p. 120
$$E -cocycles	p. 123
Runge approximation of G-valued functions. First steps	p. 125
The Cartan lemma	p. 129
$$G -cocycles. Definitions and statement of the main result	p. 131
Refinement of the covering	p. 133
Exhausting by compact sets	p. 137
Proof of the main theorem in the case of simply connected open sets	p. 141
Runge approximation of G-valued functions. General case	p. 143
Proof of the main theorem in the general case	p. 147
$$(E) -cocycles	p. 151
Weierstrass theorems	p. 153
Weierstrass theorems for $$(E) and Gw(E)-valued functions	p. 156
Comments	p. 157
Families of subspaces	p. 159
The gap metric	p. 159
Kernel and image of operator functions	p. 172
Holomorphic sections of continuous families of subspaces	p. 184
Holomorphic families of subspaces	p. 185
Example: A holomorphic family of subspaces with jumping isomorphism type	p. 201
Injective families	p. 203
Shubin families	p. 204
Complemented families	p. 206
Finite dimensional and finite codimensional families	p. 209
One-sided and generalized invertible holomorphic operator functions	p. 211
Example: A globally non-trivial complemented holomorphic family of subspaces	p. 214
Comments	p. 216
Plemelj-Muschelishvili factorization	p. 219
Definitions and first remarks about factorization	p. 220
The algebra of Wiener functions and other splitting R-algebras	p. 222
Höet;lder continuous and differentiable functions	p. 230
Reduction of the factorization problem to functions, holomorphic and invertible on C	p. *237
Factorization of holomorphic functions close to the unit	p. 240
Reduction of the factorization problem to polynomials in z and 1/z	p. 240
The finite dimensional case	p. 242
Factorization of g$$(E)-valued functions	p. 245
The filtration of an operator function with respect to a contour	p. 251
A general criterion for the existence of factorizations	p. 259
Comments	p. 267
Wiener-Hopf operators, Toeplitz operators and factorization	p. 269
Holomorphic operator functions	p. 269
Factorization of Gw(E)-valued functions	p. 273
The space L2 (¿ H)	p. 276
Operator functions with values acting in a Hilbert space	p. 287
Functions close to the unit operator or with positive real part	p. 291
Block Töet;plitz operators	p. 297
The Fourier transform of L1 (R, E)	p. 305
The Fourier isometry U of L2 (R, H)	p. 313
The isometry V from L2 (T, H) onto L2 (R, H)	p. 317
The algebra of operator functions L(H) $$ L1 (R, L(H))	p. 320
Factorization with respect to the real line	p. 324
Wiener-Hopf integral operators in L2 ([o, $$[,H)	p. 325
An example	p. 338
Comments	p. 340
Multiplicative cocycles with restrictions (F-cocycles)	p. 343
F-cocycles	p. 343
The main results on cocycles with restrictions. Formulation and reduction to $$D,Z,m	p. 346
The Cartan lemma with restrictions	p. 347
Splitting over simply connected open sets after shrinking	p. 352
Runge approximation on simply connected open sets	p. 354
Splitting over simply connected open sets without shrinking	p. 358
Runge approximation. The general case	p. 362
The Oka-Grauert principle	p. 365
Comments	p. 367
Generalized interpolation problems	p. 369
Weierstrass theorems	p. 369
Right- and two-sided Weierstrass theorems	p. 371
Weierstrass theorems for G$$ (E)- and gw (E)-valued functions	p. 374
Holomorphic g$$(E) -valued functions with given principal parts of the inverse	p. 377
Comments	p. 378
Holomorphic equivalence, linearization and diagonalization	p. 379
Introductory remarks	p. 379
Linearization by extension and equivalence	p. 380
Local equivalence	p. 386
A theorem on local and global equivalence	p. 392
The finite dimensional case	p. 394
Local and global equivalence for finite meromorphic Fredholm functions	p. 399
Global diagonalization of finite meromorphic Fredholm functions	p. 406
Comments	p. 411
Bibliography	p. 413
Index	p. 419
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Holomorphic Operator Functions of One Variable and Applications

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