An inverse problem for self-adjoint positive Hankel operators

Abstract

For a sequence $\{α_n\}_{n=0}^\infty$, we consider the Hankel operator $Γ_α$, realised as the infinite matrix in $\ell^2$ with the entries $α_{n+m}$. We consider the subclass of such Hankel operators defined by the "double positivity" condition $Γ_α\geq0$, $Γ_{S^*α}\geq0$; here $S^*α$ is the shifted sequence $\{α_{n+1}\}_{n=0}^\infty$. We prove that in this class, the sequence $α$ is uniquely determined by the spectral shift function $ξ_α$ for the pair $Γ_α^2$, $Γ_{S^*α}^2$. We also describe the class of all functions $ξ_α$ arising in this way and prove that the map $α\mapstoξ_α$ is a homeomorphism in appropriate topologies.