Holomorphic operator functions of one variable and applications. Methods from complex analysis in several variables
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 techniques of complex analysis of scalar and matrix functions of several variables. The applications concern: interpolation, holomorphic families of subspaces and frames, spectral theory of polynomials with operator coefficients, holomorphic equivalence and diagonalization, and Plemelj-Muschilishvili factorization. The book also contains a theory of Wiener-Hopf integral equations with operator-valued kernels and a theory of infinite Toeplitz matrices with operator entries. The book consists of an introduction and eleven chapters. It can be divided into four parts. In the first part, the straightforward generalizations of some results from the basic analysis of scalar functions of one complex variable is given. Also, some results in the scalar setting are presented, such as Pompeiju's integral formula, the Runge approximation theorem, the Mittag-Leffler theorem, and the Weierstrass product theorem. These results will be generalized to the case of operator functions by using Grauert's theory of cocycles. The second part, which is the main part of the theory, consists of some results that 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 of Banach space and Plemelj-Muschelishvili factorization. The fourth part presents a generalization of the theory of cocycles with restrictions. It is the main tool to improve the Weierstrass product theorem. This part contains also holomorphic equivalence and its applications to linearization and diagonalization. The titles of the chapters in this book are: elementary properties of holomorphic functions, solution of \(\bar{\partial}u=f\) and applications, splitting and factorization with respect to a contour, the Rouché theorem for operator functions, multiplicative cocycles, families of subspaces, Plemelj-Muschelishvili factorization, Wiener-Hopf operators, Toeplitz operators and factorization, multiplicative cocycles with restrictions, generalized interpolation problems, holomorphic equivalence, linearization and diagonalization.
scientific article; zbMATH DE number 5574489