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Closability property of operator algebras generated by normal operators and operators of class $C_0$

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 $L^2(μ)$, 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.