Holomorphic Operator Functions of One Variable and Applications: Methods from Complex Analysis in Several Variables

Israel Gohberg, Jürgen Leiterer (auth.)3034601255, 9783034601252, 9783034601269

This is a book on holomorphic operator functions of a single variable and applications, which is focused 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 tools adapted 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 fourth 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.

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Table of contents :
Front Matter….Pages i-xx
Elementary properties of holomorphic functions….Pages 1-28
Solution of $$ bar partial u = f $$ and applications….Pages 29-58
Splitting and factorization with respect to a contour….Pages 59-95
The Rouché theorem for operator functions….Pages 97-112
Multiplicative cocycles ( $$ mathcal{O}^G $$ -cocycles)….Pages 113-158
Families of subspaces….Pages 159-217
Plemelj-Muschelishvili factorization….Pages 219-267
Wiener-Hopf operators, Toeplitz operators and factorization….Pages 269-341
Multiplicative cocycles with restrictions ( $$ mathcal{F} $$ -cocycles)….Pages 343-367
Generalized interpolation problems….Pages 369-378
Holomorphic equivalence, linearization and diagonalization….Pages 379-411
Back Matter….Pages 413-422