A (not necessarily closed) algebra of operators \(\mathcal{A}\) on a Hilbert space is said to have the closability property if every densely defined linear transformation that commutes with \(\mathcal{A}\) is closable. We show that an algebra of the form \(\{u(T):u\in H^{\infty}\}\) , where T is an operator of class \(C_{0}\) , has the closability property precisely when T has a certain finiteness property, usually know as property (P). An analogous result is proved for commutative von Neumann algebras.
