Criteria for Hankel operators to be sign-definite

Abstract

We show that the total multiplicities of negative and positive spectra of a self-adjoint Hankel operator $H$ in $L^2\left({ℝ}_{+}\right)$ with integral kernel $h\left(t\right)$ and of the operator of multiplication by the inverse Laplace transform of $h\left(t\right)$, the distribution $\sigma \left(\lambda \right)$, coincide. In particular, $\pm H\ge 0$ if and only if $\pm \sigma \left(\lambda \right)\ge 0$. To construct $\sigma \left(\lambda \right)$, 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.