Riesz's Theorem for Orthogonal Matrix Polynomials

Abstract

We describe the image through the Stieltjes transform of the set of solutions V of a matrix moment problem. We extend Riesz's theorem to the matrix setting, proving that those matrices of measures of V for which the matrix polynomials are dense in the corresponding $ {\cal L} $ 2 space are precisely those whose Stieltjes transform is an extremal point (in the sense of convexity) of the image set.