On Toeplitz and Wiener-Hopf Operators with Contourwise Rational Matrix and Operator Symbols

Abstract

Explicit formulas for the (generalized) inverse and criteria of invertibility are given for block Toeplitz and Wiener-Hopf type operators. We consider operators with symbols defined on a curve composed of several non-intersecting simple closed contours. Also criteria and explicit formulas for canonical factorization of matrix functions relative to a compound contour are presented. The matrix functions we work with are rational on each of the compounding contours but the rational expressions may vary from contour to contour. We use realizations for each of the rational expressions and the final results are stated in terms of invertibility properties of a certain finite matrix called indicator, which is built from the realizations. The analysis does not depend on finite dimensionality and is carried out for operator valued symbols.