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.
Categories: math.SP