The main part of this paper concerns Toeplitz operators of which the symbol W is an m x m matrix function defined on a disconnected curve r. The curve r is assumed to be the union of s + 1 nonintersecting simple smooth closed contours rOo r -. . . - rs which form the positively l oriented boundary of a finitely connected bounded domain in t. Our main requirement on the symbol W is that on each contour rj the function W is the restriction of a rational matrix function Wj which does not have poles and zeros on rj and at infinity. Using the realization theorem from system theory (see. e. g . - [1]. Chapter 2) the rational matrix function Wj (which differs from contour to contour) may be written in the form 1 (0. 1) W . (A) = I + C. (A - A. f B. A E r· J J J J J where Aj is a square matrix of size nj x n- say. B and C are j j j matrices of sizes n. x m and m x n . - respectively. and the matrices A. J x J J and Aj = Aj - BjC have no eigenvalues on r . (In (0. 1) the functions j j Wj are normalized to I at infinity. I: Canonical and Minimal Factorization.- Editorial introduction.- Left Versus Right Canonical Factorization.- Wiener-Hopf Equations With Symbols Analytic In A Strip.- On Toeplitz and Wiener-Hopf Operators with Contour-Wise Rational Matrix and Operator Symbols.- Canonical Pseudo-Spectral Factorization and Wiener-Hopf Integral Equations.- Minimal Factorization of Integral operators and Cascade Decompositions of Systems.- II: Non-Canonical Wiener-Hopf Factorization.- Editorial introduction.- Explicit Wiener-Hopf Factorization and Realization.- Invariants for Wiener-Hopf Equivalence of Analytic Operator Functions.- Multiplication by Diagonals and Reduction to Canonical Factorization.- Symmetric Wiener-Hopf Factorization of Self-Adjoint Rational Matrix Functions and Realization.
