Diagonalizations of two classes of unbounded Hankel operators

TL;DR

This paper establishes a unitary equivalence between Hankel operators and pseudo-differential operators, enabling spectral analysis of specific classes with polynomial kernels and singularities, revealing their spectra and eigenvalue asymptotics.

Contribution

It introduces a method to diagonalize certain unbounded Hankel operators by relating them to differential operators, advancing spectral analysis techniques.

Findings

Essential spectrum is the entire axis for odd-degree kernels.

Essential spectrum is the positive half-axis for even-degree kernels.

Eigenvalues of singular Hankel operators accumulate at infinity with known asymptotics.

Abstract

We show that every Hankel operator $H$ is unitarily equivalent to a pseudo-differential operator $A$ of a special structure acting in the space $L^2 ({\Bbb R}) $. As an example, we consider integral operators $H$ in the space $L^2 ({\Bbb R}_{+}) $ with kernels $P (\ln (t+s)) (t+s)^{-1}$ where $P(x)$ is an arbitrary real polynomial of degree $K$. In this case, $A$ is a differential operator of the same order $K$. This allows us to study spectral properties of Hankel operators $H$ with such kernels. In particular, we show that the essential spectrum of $H$ coincides with the whole axis for $K$ odd, and it coincides with the positive half-axis for $K$ even. In the latter case we additionally find necessary and sufficient conditions for the positivity of $H$. We also consider Hankel operators whose kernels have a strong singularity at some positive point. We show that spectra of such operators consist of the zero eigenvalue of infinite multiplicity and eigenvalues accumulating to $+\infty$ and $-\infty$. We find the asymptotics of these eigenvalues.