Direct and inverse spectral problems for block Jacobi type bounded symmetric matrices related to the two dimensional real moment problem

Abstract

We generalize the connection between the classical power moment problem and the spectral theory of selfadjoint Jacobi matrices. In this article we propose an analog of Jacobi matrices related to some system of orthonormal polynomials with respect to the measure on the real plane. In our case we obtained two matrices that have a block three-diagonal structure and are symmetric operators acting in the space of $l_2$ type. With this connection we prove the one-to-one correspondence between such measures defined on the real plane and two block three-diagonal Jacobi type symmetric matrices. For the simplicity we investigate in this article only bounded symmetric operators. From the point of view of the two dimensional moment problem this restriction means that the measure in the moment representation (or the measure, connected with orthonormal polynomials) has compact support.