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

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 is an element of N boolean OR{ +infinity} copies of the unilateral shift if A has spectral multiplicity at most N. We further show that the set of all isometrics, V, so that A is Hankel with respect to V, are in bijection with the set of all closed, symmetric restrictions of A(-1).