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

TL;DR

This paper characterizes when a bounded, injective, self-adjoint operator is unitarily equivalent to a Hankel operator via a pure isometry, linking this to the operator's invertibility and spectral properties.

Contribution

It provides necessary and sufficient conditions for such operators to be Hankel with respect to a pure isometry, including spectral multiplicity considerations.

Findings

Existence of a pure isometry V making A Hankel if and only if A is not invertible.

The isometry V can be chosen based on the spectral multiplicity of A.

The set of all such isometries V corresponds bijectively to closed, symmetric restrictions of A^{-1}.

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