A Weyl Matrix Perspective on Unbounded Non-Self-Adjoint Jacobi Matrices

Abstract

A new way of encoding a non-self-adjoint Jacobi matrix $J$ by a spectral measure of $|J|$ together with a phase function was described by Pushnitski--\v Stampach in the bounded case. We present another perspective on this correspondence, based on Weyl functions instead of moments, which simplifies some proofs and generalizes the correspondence to the unbounded case. In particular, we find a bijection between proper Jacobi matrices with positive off-diagonal elements, and a class of spectral data. We prove that this mapping is continuous in a suitable sense. To prove injectivity of the map, we prove a local Borg--Marchenko theorem for unbounded non-self-adjoint Jacobi matrices in this class that may be of independent interest.