Finite-Dimensional Completion for the Matrix-Valued Truncated Complex Moment Problem

In this paper, we consider the matrix-valued truncated complex moment problem. We notice first that if a truncated complex matrix-valued sequence admits a representing measure, then it is the initial data of an infinite complex matrix-valued sequence verifying some suitable finite-dimensional property. We show that finite-dimensional completion of a truncated data provides a necessary and sufficient condition, and hence a solution, for the matrix-valued truncated complex moment problem. As a consequence, we obtain a matrix generalization of Curto–Fialkow’s result on flat positive extensions of moment matrices.