Factorization of measurable matrix functions. Ed. by Georg Heinig. Transl. from the Russian by Bernd Luderer. With a foreword by Bernd Silbermann. (Licensed ed.)
type: Item
instance of: scholarly article
title: Factorization of measurable matrix functions. Ed. by Georg Heinig. Transl. from the Russian by Bernd Luderer. With a foreword by Bernd Silbermann. (Licensed ed.)
zbMATH Open document ID: 0651.47010
author: G. S. Litvinchuk
author: Ilya M. Spitkovsky
published in: Operator Theory: Advances and Applications
publication date: 1987-01-01T00:00:00Z
review text: This book presents an up to date account of the theory of Wiener-Hopf factorization of matrix functions on a closed contour \(\Gamma\) in the complex plane. Let G be such a function, and assume that the countour separates the extended complex plane into an inner domain \(D^+(0\in D^+)\) and an outer domain \(D^-\). A (Wiener-Hopf) factorization of G relative to \(\Gamma\) is a representation of G in the form \[ G(t)=G_- (t)\Lambda (t)G_+(t),\quad t\in \Gamma, \] where \(G_{\pm}(t)\) are the boundary values of a matrix function that is analytic and non-singular on \(D^{\pm}\), and \(\Lambda (t)=diag(t^{\kappa_ 1},...,t^{\kappa_ n})\). Factorizations of this type have their roots deep into classical complex function theory (boundary value problems of Hilbert and Riemann- Hilbert type) and the theory of singular integral equations. They also play an important role in various branches of mathematical physics (including transport theory), in mathematical systems theory, in probability theory and other fields of applied mathematics. The present book surveys the theory. The contour \(\Gamma\) may be non- smooth and the boundary of a multi-connected domain. The functions G may be discontinuous or unbounded; in general, they are only assumed to be measurable. The first chapter contains background material from operator theory and complex function theory. Chapters 2-5 treat the general theory and the last three chapters deal with more specific topics (like stability of the indices \(\kappa_ 1,...,\kappa_ n\), factorization of Hermitian matrix-valued functions on the unit circle, the generalized Riemann boundary value problem). Roughly speaking, also in Chapters 2-5 the material is organized from the general to the specific. Factorization of functions of special classes (like \(H_{\infty}+C\) or piecewise continuous) appear in Chapter 5; the general properties of factorization are discussed in Chapter 2. The book is very informative. It provides an excellent and well-readable …
Mathematics Subject Classification ID: 47A68
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 30E25
Mathematics Subject Classification ID: 40E15
zbMATH DE Number: 4061968
zbMATH Keywords: Fredholm operator
zbMATH Keywords: Wiener-Hopf factorization of matrix functions on a closed contour
zbMATH Keywords: boundary value problems of Hilbert and Riemann-Hilbert type
zbMATH Keywords: singular integral equations
zbMATH Keywords: generalized Riemann boundary value
reviewed by: Marinus Kaashoek
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4061968