On the inversion of infinite moment matrices

Motivated by [8] we study the existence of the inverse of an infinite Hermitian positive definite matrix (in short, HPD matrix) from the point of view of the asymptotic behaviour of the smallest eigenvalues of the finite sections. We prove a sufficient condition to assure the inversion of an HPD matrix with square summable rows. For infinite Toeplitz matrices we introduce the notion of asymptotic Toeplitz matrix and we show that, under certain assumptions, the inverse of an infinite Toeplitz positive definite matrix is asymptotic Toeplitz. Such inverses are computed in terms of the limits of the coefficients of the associated orthogonal polynomials. We apply these results in the context of the theory of orthogonal polynomials. In particular, we show that for measures on the unit circle T verifying that the smallest eigenvalue of the finite sections of the corresponding moment matrix are away from zero in the limit we may assure the existence of all the limits of the coefficients of the orthonormal polynomials with respect to such measures.