On finite rank Hankel operators

TL;DR

This paper derives an explicit formula for the total multiplicity of positive and negative spectra of finite rank self-adjoint Hankel operators and shows their spectral properties are stable under certain strong perturbations.

Contribution

It provides a new explicit formula for spectral multiplicities and demonstrates spectral stability under strong perturbations for finite rank Hankel operators.

Findings

Explicit formula for spectral multiplicities

Spectral stability under Carleman operator perturbation

Negative eigenvalues remain unchanged under strong perturbations

Abstract

For self-adjoint Hankel operators of finite rank, we find an explicit formula for the total multiplicity of their negative and positive spectra. We also show that very strong perturbations, for example, a perturbation by the Carleman operator, do not change the total number of negative eigenvalues of finite rank Hankel operators.