Two unbounded operators on a Hilbert space whose domains have empty intersection.

Question

Assume that $A$, $B$, and $C$ are Hilbert spaces. Can we find densely-defined closed unbounded linear operators $S : A \rightarrow B$ and $T : A \rightarrow C$ such that the intersection of their domains $\operatorname{dom}(S) \cap \operatorname{dom}(T)$ is not dense?

Can it happen that the intersection is even zero?