Inverse spectral problems for positive Hankel operators
Alexander Pushnitski, Sergei Treil
TL;DR
This work develops an inverse spectral theory for positive Hankel operators by linking Hankel kernels to Laplace transforms of positive measures and constructing a nonlinear spectral map Ω that sends μ to the spectral measure σ of Γ_μ on the finite-measure class. A key move is reducing Γ_μ to a model operator G_μ on L^2(μ) via the Laplace transform and exploiting a Lyapunov equation to show that the pair (G_μ, X_μ) encodes a duality that makes the map Ω an involution. The paper also treats cofinite measures through a x↔1/x change of variables, establishes scaling and trace-form relations, proves continuity of Ω under weak convergence, and provides explicit Mehler and Rosenblum examples to illustrate the mechanism and spectral data generation. Overall, the results give a bijective, symmetric framework for recovering μ from spectral data σ (and vice versa) and connect control-theoretic balanced realizations to Hankel inverse problems, with implications for understanding the structure and spectra of non-bounded 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}.