On a simultaneous approach to the even and odd truncated matricial Stieltjes moment problem I: An α-Schur-Stieltjes-type algorithm for sequences of complex matrices

The characterization of the solvability of matrix versions of truncated Stieltjes-type moment problems led to the class of α-Stieltjes non-negative definite sequences of complex q×q matrices. In [22], a parametrization of this class was introduced, the so-called α-Stieltjes parametrization. The main topic of this first part of the paper is the construction of a Schur-type algorithm which produces exactly the α-Stieltjes parametrization.