Inverse problem for semi-infinite Jacobi matrices and associated Hilbert spaces of analytic functions

TL;DR

This paper develops a method to connect semi-infinite Jacobi matrices with de Branges spaces, advancing inverse spectral theory and linking it to classical moment problems and dynamical systems.

Contribution

It extends the association procedure between Jacobi matrices and Hilbert spaces of analytic functions to semi-infinite cases, enriching inverse spectral analysis.

Findings

Established a procedure linking semi-infinite Jacobi matrices with de Branges spaces.

Compared properties of matrices from moment problems with those from dynamical systems.

Extended previous methods to semi-infinite matrices in inverse spectral theory.

Abstract

We consider the dynamic problems for the discrete systems with discrete time associated with finite and semi-infinite Jacobi matrices. The result of the paper is a procedure of association of special Hilbert spaces of functions, namely de Branges space, playing an important role in the inverse spectral theory, with these systems. %Thus the procedure, offered by the authors in the %previous papers now is extended to the case of semi-infinite %Jacobi matrices. We point out the relationships with the classical moment problems theory and compare properties of classical Hankel matrices associated with moment problems with properties of matrices of connecting operators associated with dynamical systems.