An inverse spectral problem for non-compact Hankel operators with simple spectrum

TL;DR

This paper investigates an inverse spectral problem for a class of non-compact Hankel operators with simple spectrum, establishing uniqueness of spectral data determination but showing the spectral map is not surjective, with applications to the cubic Szegő equation.

Contribution

It proves the injectivity of the spectral map for non-compact Hankel operators with simple spectrum and explores the limitations of spectral data representation.

Findings

Spectral data uniquely determine the Hankel operator.

The spectral map is not surjective for the class studied.

Applications show not all initial data lead to almost periodic solutions.

Abstract

We consider an inverse spectral problem for a class of non-compact Hankel operators $H$ such that the modulus of $H$ (restricted onto the orthogonal complement to its kernel) has simple spectrum. Similarly to the case of compact operators, we prove a uniqueness result, i.e. we prove that a Hankel operator from our class is uniquely determined by the spectral data. In other words, the spectral map, which maps a Hankel operator to the spectral data, is injective. Further, in contrast to the compact case, we prove the failure of surjectivity of the spectral map, i.e. we prove that not all spectral data from a certain natural set correspond to Hankel operators. We make some progress in describing the image of the spectral map. We also give applications to the cubic Szeg\H{o} equation. In particular, we prove that not all solutions with initial data in BMOA are almost periodic; this is in a sharp contrast to the known result for initial data in VMOA.