An inverse problem for self-adjoint positive Hankel operators

Abstract: For a sequence ${\alpha_n}{n=0}^\infty$, we consider the Hankel operator $\Gamma\alpha$, realised as the infinite matrix in $\ell^2$ with the entries $\alpha_{n+m}$. We consider the subclass of such Hankel operators defined by the "double positivity" condition $\Gamma_\alpha\geq0$, $\Gamma_{S^*\alpha}\geq0$; here $S^*\alpha$ is the shifted sequence ${\alpha_{n+1}}{n=0}^\infty$. We prove that in this class, the sequence $\alpha$ is uniquely determined by the spectral shift function $\xi\alpha$ for the pair $\Gamma_\alpha^2$, $\Gamma_{S^*\alpha}^2$. We also describe the class of all functions $\xi_\alpha$ arising in this way and prove that the map $\alpha\mapsto\xi_\alpha$ is a homeomorphism in appropriate topologies.