An Inverse Problem for Self-adjoint Positive Hankel Operators

For a sequence {αn}∞n=0, we consider the Hankel operator Γα, realized as the infinite matrix in ℓ2 with the entries αn+m. We consider the subclass of bounded Hankel operators defined by the “double positivity” condition Γα≥0, ΓS*α≥0; here S*α is the shifted sequence {αn+1}∞n=0. We prove that in this class, the sequence α is uniquely determined by the spectral shift function ξα for the pair Γ2α, Γ2S∗α. We also describe the class of all functions ξα arising in this way and prove that the map α↦ξα is a homeomorphism in appropriate topologies.