On Dyukarev's resolvent matrix for a truncated Stieltjes matrix moment problem under the view of orthogonal matrix polynomials

Abstract

The main result of this paper is a new representation of Yu.M. Dyukarev's resolvent matrix for the non-degenerate truncated matricial version of the classical Stieltjes moment (TMCSM) problem. This new representation is based on the use of a quadruple of q×q matrix polynomials which are characterized by certain orthogonality properties. In this way, useful new interrelations between the moment problems of Stieltjes and Hamburger which are associated with a Stieltjes positive definite sequence (sj)j=02n of complex q×q matrices are found. This includes an explicit formula connecting the resolvent matrices used by Yu.M. Dyukarev and I.V. Kovalishina, respectively. Furthermore, the coefficients in the recurrence formulas for the orthogonal polynomials with respect to a Stieltjes positive definite sequence are expressed in terms of the Dyukarev–Stieltjes parameters. Additionally, we obtain two different representations for each extremal solution of the TMCSM problem via matrix continued fraction expansions.