Solution of the Stieltjes truncated matrix moment problem

Abstract

The truncated Stieltjes matrix moment problem consisting in the description of all matrix distributions $\boldsymbol{\sigma}(t)$ on $[0,\infty)$ with given first $2n+1$ power moments $(\mathbf{C}j){n=0}^j$ is solved using known results on the corresponding Hamburger problem for which $\boldsymbol{\sigma}(t)$ are defined on $(-\infty,\infty)$. The criterion of solvability of the Stieltjes problem is given and all its solutions in the non-degenerate case are described by selection of the appropriate solutions among those of the Hamburger problem for the same set of moments. The results on extensions of non-negative operators are used and a purely algebraic algorithm for the solution of both Hamburger and Stieltjes problems is proposed.