A Dynamical System Approach to the Inverse Spectral Problem for Hankel Operators: A Model Case

TL;DR

This paper provides an alternative proof for the inverse spectral problem of Hankel operators using a dynamical systems approach, establishing uniqueness and existence of operators from spectral data.

Contribution

It introduces a new proof method based on dynamical systems, simplifying the stability analysis and extending the classical inverse spectral results for Hankel operators.

Findings

Existence and uniqueness of Hankel operators from spectral data proven.

Simplified proof leveraging dynamical systems and stability analysis.

Extension of classical inverse spectral results for Hankel operators.

Abstract

We present an alternative proof of the result by P. Gerard and S. Grellier, stating that given two real sequences $(\lambda_n)_{n=1}^\infty$, $(\mu_n)_{n=1}^\infty$ satisfying the intertwining relations \[ |\lambda_1| > |\mu_1| > |\lambda_2| > |\mu_2| > ...> |\lambda_n| > |\mu_n|>\ldots >0 , \qquad \lambda_n\to 0, \] there exists a unique compact Hankel operator $\Gamma$ such that $\lambda_n$ are the (simple) eigenvalues of $\Gamma$ and $\mu_n$ are the simple eigenvalues of its truncation $\Gamma_1$ obtained from $\Gamma$ by removing the first column. We use the dynamical systems approach originated in a paper by A. V. Megretski, V.V. Peller. S. R. Treil in 1995, and the proof is split into three independent parts. The first one, which is a slight modification of a result in that paper, is an abstract operator-theoretic statement reducing the problem to the asymptotic stability of some operators. The second one is the proof of the asymptotic stability, which is usually the hardest part, but in our case of compact operators it is almost trivial. And the third part is an abstract version of the Borg's two spectra theorem, which is essentially a simple exercise in graduate complex analysis.