On unitary equivalence to a self-adjoint or doubly–positive Hankel operator

Abstract Let A be a bounded, injective and self-adjoint linear operator on a complex separable Hilbert space. We prove that there is a pure isometry, V, so that AV > 0 and A is Hankel with respect to V, i.e. V*A = AV, if and only if A is not invertible. The isometry V can be chosen to be isomorphic to N ∈ ℕ ∪ {+∞} copies of the unilateral shift if A has spectral multiplicity at most N. We further show that the set of all isometries, V, so that A is Hankel with respect to V, are in bijection with the set of all closed, symmetric restrictions of A−1.