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 $N \in \mathbb{N} \cup \{ + \infty \}$ 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 $A^{-1}$.