The complex moment problem in the exponential form with direct and inverse spectral problems for the block Jacobi type correspondence matrices

Abstract

We present a new generalization of the connection of the classical power moment problem with spectral theory of Jacobi matrices. In the article we propose an analog of Jacobi matrices related to the complex moment problem in the case of exponential form and to the system of orthonormal polynomials with respect to some measure with the compact support on the complex plane. In our case we obtain two matrices that have block three-diagonal structure and acting in the space of $l_2$ type as commuting self-adjoint and unitary operators. With this connection we prove the one-to-one correspondence between the measures defined on a compact set in the complex plane and the couple of block three-diagonal Jacobi type matrices. For simplicity we consider in this article only a bounded self-adjoint operator.