A Dynamical System Approach To The Inverse Spectral Problem For Hankel Operators: The General Case

TL;DR

This paper develops a dynamical system approach to solve the inverse spectral problem for Hankel operators in the general case, extending previous work on compact operators and introducing new spectral data representations.

Contribution

It introduces a new framework using dynamical systems and complex symmetric operators to analyze the inverse spectral problem for general Hankel operators, including non-compact cases.

Findings

Spectral data characterized by sequences of singular values and probability measures.

Reduction of the inverse problem to asymptotic stability of a contraction.

New translation between spectral data representations using Clark measures.

Abstract

We study the inverse problem for the Hankel operators in the general case. Following the work of G\'erard--Grellier, the spectral data is obtained from the pair of Hankel operators $\Gamma$ and $\Gamma S$, where $S$ is the shift operator. The theory of complex symmetric operators provides a convenient language for the description of the spectral data. We introduce the abstract spectral data for the general case, and use the dynamical system approach, to reduce the problem to asymptotic stability of some contraction, constructed from the spectral data. The asymptotic stability is usually the hard part of the problem, but in the investigated earlier by G\'erard--Grellier case of compact operators we get it almost for free. For the case of compact operators we get a concrete representation of the abstract spectral data as two intertwining sequences of singular values, and two sequences of finitely supported probability measures. This representation is different from one treated by G\'erard--Grellier, and we provide the translation from one language to the other; theory of Clark measures is instrumental there.