On a general moment problem and certain matrix equations

Abstract

Let K be a nonnegative matrix satisfying a matrix equation of the form A K A ∗ − B K B ∗ = M ∗JM (where A, B, and M are of a special form). We look for representations of K of the form K = ʃ Γ 0 X(γ) dI(γ) X(γ) ∗ where Γ 0 is a simple closed contour, dI is a positive matrix valued measure, and the matrix valued function X (depending on A, B, and M) defines a space of functions invariant with respect to a shift associated to Γ 0. The method used is that of the fundamental matrix inequality of Potapov, suitably adapted to the present framework.