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.

The first author acknowledges support from Professor P. Lancaster on a grant from the Natural Sciences and Engineering Research Council of Canada