Let $\mathcal{A}$ be a $W^{*}$-algebra, $\omega$ a normal, positive linear functional on $\mathcal{A}$, $\mathbf{p}$ the support projection of $\omega$, and $(\mathcal{H},\pi,\psi)$ the GNS triple associated with $\omega$.
If $S$ is a densely-defined self-adjoint operator on $\mathcal{H}$ with $\psi$ in its domain, then the linear functional $\xi$ on $\mathcal{A}$ defined by $$ \xi(\mathbf{a}):=\langle\phi|\pi(\mathbf{a})|\psi\rangle + \langle \psi|\pi(\mathbf{a})|\phi\rangle $$ with $\phi=S\psi$, is a bounded, normal, self-adjoint linear functional on $\mathcal{A}$.
Clearly, $\xi(\mathbf{a})=0$ for all $\mathbf{a}\in\,\mathbf{q}\mathcal{A}\mathbf{q}$ with $\mathbf{q}=\mathbb{I}-\mathbf{p}$.
I am wondering if the converse is also true, and/or there are other assumptions needed for the converse to hold.
In some sense, this would be a sort of (linear) Radon-Nikodym theorem along the lines of the paper by Maltese-Niestegge.