The matricial subnormal completion problem

Given a truncated sequence of positive numbers a=(a_j)_j=0^m, the subnormal completion problem (originally posed and geometrically solved by J. Stampfli) asks whether or not there exists a subnormal weighted shift operator on ℓ₂ whose initial weights are given by a. Subsequently a concrete solution based on a solution of the truncated Stieltjes moment problem was discovered by Curto and Fialkow. In this paper we will consider a matricial analogue of the subnormal completion problem, where the truncated sequence of positive numbers (a_j)_j=0^m is replaced by a truncated sequence of positive definite matrices. We will provide concrete conditions for a solution based on the parity of m and make a connection with a matricial truncated Stieltjes moment problem. We will also put forward a certain canonical subnormal completion which is minimal in the sense of the norm of the corresponding weighted shift operator and describe the support of the corresponding matricial Berger measure in terms of the zeros of a matrix polynomial which describes the initial positive rank preserving extension.