  An operator algebra $\mathcal{A}$ acting on a Hilbert space is said to have
the closability property if every densely defined linear transformation
commuting with $\mathcal{A}$ is closable. In this paper we study the
closability property of the von Neumann algebra consisting of the
multiplication operators on $L^2(\mu)$, and give necessary and sufficient
conditions for a normal operator $N$ such that the von Neumann algebra
generated by $N$ has the closability property. We also give necessary and
sufficient conditions for an operator $T$ of class $C_0$ such that the algebra
generated by $T$ in the weak operator topology and the algebra
$H^\infty(T)=\{u(T):u\in H^\infty\}$ have the closability property.
