An inverse problem for self-adjoint positive Hankel operators

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For a sequence {αₙ}ₙ₌₀∞, we consider the Hankel operator Γα, realised as the infinite matrix in ℓ² with the entries αₙ₊ₘ. We consider the subclass of such 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 Γα², Γₛ*α². We also describe the class of all functions ξα arising in this way and prove that the map αξα is a homeomorphism in appropriate topologies.