Factorization of measurable matrix functions. Ed. by Georg Heinig. Transl. from the Russian by Bernd Luderer. With a foreword by Bernd Silbermann. (Licensed ed.)
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 account of the state of the theory up to 1984 (approximately). Many papers in the Russian literature from the seventies and early eighties (among others several due to the authors), which are of considerable interest but difficult to get in the West, are covered by the present book. As such the book is a most welcome addition to the \textit{K. Clancey}, \textit{I.Gohberg} book which concerns the same subject and appeared earlier in the same OT-series [Factorization of matrix functions and singular integral operators (1981; Zbl 0474.47023)]. The comments at the end of each chapter serve as an effective guide for the 31 pages of references. The book also discusses various open problems and it is source for further research. It would have been helpful if in the list of references all the Russian text would have been translated (and not just if a paper has appeared in translation). At a few places the translation is not optimal (e.g., numerical range is used in the Western literature and not numerical domain). But these are minor criticisms of an otherwise excellent book.
Wiener-Hopf factorization of matrix functions on a closed contour
boundary value problems of Hilbert and Riemann-Hilbert type
scientific article; zbMATH DE number 4061968