Analysis & PDE

  • Anal. PDE
  • Volume 8, Number 1 (2015), 183-221.

Criteria for Hankel operators to be sign-definite

Dimitri Yafaev

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Abstract

We show that the total multiplicities of negative and positive spectra of a self-adjoint Hankel operator H in L2(+) 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, ± H 0 if and only if ± σ(λ) 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.

Article information

Source
Anal. PDE, Volume 8, Number 1 (2015), 183-221.

Dates
Received: 13 April 2014
Accepted: 26 November 2014
First available in Project Euclid: 28 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1511895961

Digital Object Identifier
doi:10.2140/apde.2015.8.183

Mathematical Reviews number (MathSciNet)
MR3336924

Zentralblatt MATH identifier
1312.47035

Subjects
Primary: 47A40: Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx]
Secondary: 47B25: Symmetric and selfadjoint operators (unbounded)

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

Citation

Yafaev, Dimitri. Criteria for Hankel operators to be sign-definite. Anal. PDE 8 (2015), no. 1, 183--221. doi:10.2140/apde.2015.8.183. https://projecteuclid.org/euclid.apde/1511895961


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