Inverse spectral problems for positive Hankel operators

Abstract

A Hankel operator $Γ$ in $L^2(\mathbb{R}_+)$ is an integral operator with the integral kernel of the form $h(t+s)$, where $h$ is known as the kernel function. It is known that $Γ$ is positive semi-definite if and only if $h$ is the Laplace transform of a positive measure $μ$ on $\mathbb{R}_+$. Thus, positive semi-definite Hankel operators $Γ$ are parameterised by measures $μ$ on $\mathbb{R}_+$. We consider the class of $Γ$ corresponding to \emph{finite} measures $μ$. In this case it is possible to define the (scalar) spectral measure $σ$ of $Γ$ in a natural way. The measure $σ$ is also finite on $\mathbb{R}_+$. This defines the \emph{spectral map} $μ\mapstoσ$ on finite measures on $\mathbb{R}_+$. We prove that this map is an involution; in particular, it is a bijection. We also consider a dual variant of this problem for measures $μ$ that are not necessarily finite but have the finite integral \[ \int_0^\infty x^{-2}\mathrm{d}μ(x); \] we call such measures \emph{co-finite}.