Quasi-Carleman operators and their spectral properties

TL;DR

This paper investigates a class of self-adjoint Hankel operators called quasi-Carleman operators, providing explicit spectral formulas and conditions for positivity, extending the classical Carleman operator case.

Contribution

It introduces quasi-Carleman operators as a generalization of the Carleman operator and derives explicit spectral properties and positivity criteria.

Findings

Explicit formulas for the number of negative eigenvalues

Necessary and sufficient conditions for positivity

Extension of spectral analysis techniques to generalized Hankel operators

Abstract

The Carleman operator is defined as integral operator with kernel $(t+s)^{-1}$ in the space $L^2 ({\Bbb R}_{+}) $. This is the simplest example of a Hankel operator which can be explicitly diagonalized. Here we study a class of self-adjoint Hankel operators (we call them quasi-Carleman operators) generalizing the Carleman operator in various directions. We find explicit formulas for the total number of negative eigenvalues of quasi-Carleman operators and, in particular, necessary and sufficient conditions for their positivity. Our approach relies on the concepts of the sigma-function and of the quasi-diagonalization of Hankel operators introduced in the preceding paper of the author.