An inverse problem for self-adjoint positive Hankel operators

For a sequence $\{{\alpha }_n{\}}_{n=0}^{\mathrm{\infty }}$, we consider the Hankel operator ${\mathrm{\Gamma }}_{\alpha }$, realised as the infinite matrix in ${\mathrm{\ell }}^2$ with the entries ${\alpha }_{n+m}$. We consider the subclass of such Hankel operators defined by the “double positivity” condition ${\mathrm{\Gamma }}_{\alpha }\ge 0$, ${\mathrm{\Gamma }}_{S^{\ast }\alpha }\ge 0$; here $S^{\ast }\alpha$ is the shifted sequence $\{{\alpha }_{n+1}{\}}_{n=0}^{\mathrm{\infty }}$. We prove that in this class, the sequence $\alpha$ is uniquely determined by the spectral shift function ${\xi }_{\alpha }$ for the pair ${\mathrm{\Gamma }}_{\alpha }^2$, ${\mathrm{\Gamma }}_{S^{\ast }\alpha }^2$. 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.