Unbounded operators: product of adjoints strictly extended by the adjoint of product

Question

It is well known that, if $T,S$ and $ST$ are densely defined operators on a Hilbert space $H$, then $T^* S^* \subset (ST)^*$. The proof of this is easy. Moreover, it's readily seen that equality certainly holds if $S$ is bounded. What it's more difficult is to find an example which shows that in general equality does not hold. I'm not aware of an immediate prototype. I've tried with position and momentum operators on $L^2(\mathbb R)$ using a few essential self-adjointness domains (even self-adjointness ones) without reaching a clear conclusion. Have you some ready-to-hand example?

Answer 1

Let $T=0$ and let $S$ be a densely-defined selfadjoint operator with $\mathcal{D}(S)\ne X$, where $X$ is the underlying space. Then $ST=0$ is defined everywhere and, hence, $(ST)^{\star}=0$ is also defined everywhere. However, $T^{\star}S^{\star} \ne 0$ because $\mathcal{D}(T^{\star}S^{\star})=\mathcal{D}(S^{\star})=\mathcal{D}(S)\ne X$.