Criteria for Hankel operators to be sign-definite

TL;DR

This paper establishes criteria for when self-adjoint Hankel operators are sign-definite by relating their spectra to a sign-function, providing explicit formulas and spectral multiplicity results.

Contribution

It introduces explicit criteria linking the spectra of Hankel operators to a sign-function, including formulas and multiplicity results, applicable to various classes.

Findings

Negative and positive spectra multiplicities coincide

Sign-definiteness characterized by the sign of the function s(x)

Explicit formulas for kernels and spectral multiplicities

Abstract

We show that total multiplicities of negative and positive spectra of a self-adjoint Hankel operator $H$ with kernel $h(t)$ and of an operator of multiplication by some real function $s(x)$ coincide. In particular, $\pm H\geq 0$ if and only if $\pm s(x)\geq 0$. The kernel $h(t)$ and its "sign-function" $s(x)$ are related by an explicit formula. An expression of $h(t)$ in terms of $s(x)$ leads to an exponential representation of $h(t)$. Our approach directly applies to various classes of Hankel operators. In particular, for Hankel operators of finite rank, we find an explicit formula for the total multiplicity of their negative and positive spectra.