Inverses of moment Hermitian matrices

Motivated by [9] we study the existence of the inverse of infinite Hermitian moment matrices associated with measures with support on the complex plane. We relate this problem to the asymptotic behaviour of the smallest eigenvalues of finite sections and we study it from the point of view of infinite transition matrices associated to the orthogonal polynomials. For Toeplitz matrices we introduce the notion of weakly asymptotic Toeplitz matrix and we show that, under certain assumptions, the inverse of a Toeplitz moment matrix is weakly asymptotic Toeplitz. Such inverses are computed in terms of some limits of the coefficients of the associated orthogonal polynomials. We finally show that the asymptotic behaviour of the smallest eigenvalue of a moment Toeplitz matrix only depends on the absolutely part of the associated measure.
Comment: 21 pages