Inverse problem for semi-infinite Jacobi matrices and associated Hilbert spaces of analytic functions
Alexander Mikhaylov, Victor Mikhaylov
TL;DR
The paper develops a framework to attach de Branges analytic-function spaces to inverse spectral problems for semi-infinite Jacobi matrices by linking discrete-time dynamical systems to infinite-dimensional de Branges spaces. It constructs finite-dimensional de Branges spaces from reachable states via boundary-control methods and spectral transforms, then analyzes the infinite-limit using moment problems, Hankel/connecting operators, and reproducing-kernel structures. A core contribution is showing how the positivity and closability properties of the Hankel and connecting operators determine when a limit de Branges space $B_A^\infty$ exists and how its kernel $J^\\infty_z$ encodes the spectral data, with distinct behaviors in limit-point versus limit-circle cases. The work unifies discrete dynamics, moment problems, and analytic function spaces, providing a cohesive route to extend de Branges-space constructions to semi-infinite Jacobi systems and clarifying the roles of Krein-type equations and reproducing kernels in inverse problems.
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.