On Canonical Factorization of Dissipative and Positive Matrix Functions Relative to Non-Simple Contours

Abstract

Let G + be an open set on the complex plane bounded by a non-simple curve Γ and z 0 ∈ G +. It is proved that any dissipative continuous matrix function of the form A(t) = (t−z 0)−1 A 0+ B+(t)(t ∈ Γ), where Ao is a constant matrix and B +(z)is analytic in G +,admits a canonical factorization. Also it is shown that for any non-simple contour Γ there exist 2 × 2 rational dissipative matrix functions and 2 × 2 Hölder continuous positive matrix functions which admit non-canonical factorization.