The Matrix Moment Problem

We expose the recent extensions to the matrix case of classical results in the theory of the moment problem: the theorem of Riesz, the parametrization of Nevanlinna and properties of the N -extremal matrices of measures. 1. The classical theory The purpose of this survey is to show the recent extensions to the matrix case of classical results in the theory of the moment problem. The interest that Chicho always showed for these questions makes it interesting for a publication devoted to his memory. For a positive Borel measure ν on R with finite moments of any order sn = ∫ R t dν(t) we denote by V the set of positive Borel measures μ on R satisfying ∫ R t dμ(t) = sn, n ≥ 0, that is, the set of solutions to the Hamburger moment problem defined by ν . By Vn we denote the set of positive Borel measures on R such that ∫ R t dμ(t) = sk, 0 ≤ k ≤ n, that is, the set of solutions to the truncated moment problem defined by ν . Given a sequence of numbers s0, s1, s2, . . . , Hamburger’s theorem from 1920 states that a necessary and sufficient condition for the existence of a positive measure with infinite support having moments s0, s1, s2, . . . is that the sequence s0, s1, s2, . . . is positive definite, or equivalently that all the Hankel matrices Hn = (si+j)0≤i,j≤n =   s0 s1 s2 . . . sn s1 s2 s3 . . . sn+1 s2 s3 s4 . . . sn+2 .. .. .. . . . .. sn sn+1 sn+2 . . . s2n   are positive definite, which is equivalent to detHn > 0, for n ≥ 0. In 1894 Stieltjes had already established the corresponding result for measures supported in [0,∞). We say that the measure ν is determinate if there is no other positive measure having the same moments as those of ν , that is, if V = {ν}, otherwise we say that ν 2000 Mathematics Subject Classification. 42C05, 44A60.