A flat extension theorem for truncated matrix-valued multisequences

Abstract

Given a truncated multisequence of p × p Hermitian matrices S := (Sγ1,...,γd ) (γ1,...,γd)∈Nd 0 0≤γ1+···+γd≤m , the truncated matrix-valued moment problem on R d asks whether or not there exists a p×p positive semidefinite matrix-valued measure T, with convergent moments of all orders, such that Sγ1,...,γd = Z · · · Z Rd x γ1 1 · · · x γd d dT(x1, . . . , xd) for all (γ1, . . . , γd) ∈ N d 0 which satisfy 0 ≤ Pd j=1 γj ≤ m. When such a measure exists we say that T is a representing measure for S. We shall see that if m is even, then S has a minimal representing measure (that is, Pκ a=1 rank Qa is as small as possible) if and only if a block matrix determined entirely by S has a rank-preserving positive extension. In this case, the support of the representing measure has a connection with zeros (suitably interpreted) of a system of matrix-valued polynomials which describe the rank-preserving extension. The proof of this result relies on operator theory and certain results for ideals of multivariate matrix-valued polynomials. Our result subsumes the celebrated flat extension theorem of Curto and Fialkow. We shall pay particularly close attention to the bivariate quadratic matrix-valued moment problem (that is, where d = 2 and m = 2).