Inverse spectral problems for positive Hankel operators

Abstract

A Hankel operator $\hat{I}“$ 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 $\hat{I}“$ is positive semi-definite if and only if $h$ is the Laplace transform of a positive measure $\hat{I}¼$ on ${\mathbb{R}}_{+}$. Thus, positive semi-definite Hankel operators $\hat{I}“$ are parameterised by measures $\hat{I}¼$ on ${\mathbb{R}}_{+}$. We consider the class of $\hat{I}“$ corresponding to \emph{finite} measures $\hat{I}¼$. In this case it is possible to define the (scalar) spectral measure $\ddot{I}ƒ$ of $\hat{I}“$ in a natural way. The measure $\ddot{I}ƒ$ is also finite on ${\mathbb{R}}_{+}$. This defines the \emph{spectral map} $\hat{I}¼\mapsto \ddot{I}ƒ$ 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 $\hat{I}¼$ 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}.

final version (minor updates). To appear in Analysis & PDE