Criteria for Hankel operators to be sign-definite

Yafaev, Dimitri

doi:10.2140/apde.2015.8.183

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

ISSN 2157-5045 (print)

Author index

To appear

Other MSP journals

Criteria for Hankel operators to be sign-definite

Dimitri R. Yafaev

Vol. 8 (2015), No. 1, 183–221

DOI: 10.2140/apde.2015.8.183

Abstract

We show that the total multiplicities of negative and positive spectra of a self-adjoint Hankel operator $H$ in $L^{2} (ℝ_{+})$ with integral kernel $h (t)$ and of the operator of multiplication by the inverse Laplace transform of $h (t)$ , the distribution $σ (λ)$ , coincide. In particular, $\pm H \geq 0$ if and only if $\pm σ (λ) \geq 0$ . To construct $σ (λ)$ , we suggest a new method of inversion of the Laplace transform in appropriate classes of distributions. Our approach directly applies to various classes of Hankel operators. For example, for Hankel operators of finite rank, we find an explicit formula for the total numbers of their negative and positive eigenvalues.

Keywords

Hankel operators, convolutions, necessary and sufficient conditions for positivity, sign function, operators of finite rank, the Carleman operator and its perturbations

Mathematical Subject Classification 2010

Primary: 47A40

Secondary: 47B25

Milestones

Received: 13 April 2014

Accepted: 26 November 2014

Published: 15 April 2015

Authors

Dimitri R. Yafaev
Institut de Recherche Mathématique de Rennes Université de Rennes I Campus de Beaulieu 35042 Rennes Cedex France

Vol. 8, No. 1, 2015