Factorization of piecewise valued matrix functions and its applications (Q1094622)

Let \(\Gamma\equiv \cup^{s-1}_{j=0}\Gamma_ j\) be the positively oriented boundary of an s-connected compact domain \(D^+_{\Gamma}\), \(\Gamma_ j\) (0\(\leq j\leq s-1)\) be closed Lyapunov contours with \(\Gamma_ j\cap \Gamma_ k=\emptyset\) (j\(\neq k)\), and assume that \(\Gamma_ 0\) contains the contours \(\Gamma_ j\) \((j=1,2,...,s-1)\) in its interior. Let T(z) be an \(n\times n\) piecewise matrix valued function defined on each of components \(\Gamma_ j\) of \(\Gamma\) as follows: For \(z\in \Gamma_ j\), \(j=0,1,...,s-1\), (1) \(T(z)=T_ j(z)\), where \(T_ j(z)\) is a holomorphic matrix function in some neighborhood of \(\Gamma_ j.\) The main purpose of this paper is to present explicit expressions for the factors and partial indices of factorization of some classes of piecewise matrix functions given by (1). These formulas are given in terms of common zeros in \(D^+_{\Gamma}\) of some polynomials associated with the matrix functions \(T_ j(z)\).