Abstract A (not necessarily closed) algebra of operators $$\mathcal{A}$$ A on a Hilbert space is said to have the closability property if every densely defined linear transformation that commutes with $$\mathcal{A}$$ A is closable. We show that an algebra of the form $$\{u(T):u\in H^{\infty}\}$$ { u ( T ) : u ∈ H ∞ } , where T is an operator of class $$C_{0}$$ 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.

Abstract In this paper, we study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, we obtain the following results, Let $$1\leq q\leq p\leq \infty$$ 1 ≤ q ≤ p ≤ ∞ . Then a bounded $$(L^{1, p}, L^{1, q})$$ ( L 1 , p , L 1 , q ) -extension domain is also a $$(W^{1, p}, W^{1, q})$$ ( W 1 , p , W 1 , q ) -extension domain. Let $$1\leq q\leq p<q ^{\star} \leq \infty$$ 1 ≤ q ≤ p < q ⋆ ≤ ∞ or $$n< q \leq p\leq \infty$$ n < q ≤ p ≤ ∞ . Then a bounded domain is a $$(W^{1, p}, W^{1, q})$$ ( W 1 , p , W 1 , q ) -extension domain if and only if it is an $$(L^{1, p}, L^{1, q})$$ ( L 1 , p , L 1 , q ) -extension domain. For $$1\leq q<n$$ 1 ≤ q < n and $$q<nq ^{\star} <p\leq \infty$$ q < n q ⋆ < p ≤ ∞ , there exists a bounded domain $$\Omega\subset\mathbb{R}^n$$ Ω ⊂ R n which is a $$(W^{1, p}, W^{1, q})$$ ( W 1 , p , W 1 , q ) -extension domain but not an $$(L^{1, p}, L^{1, q})$$ ( L 1 , p , L 1 , q ) -extension domain for $$1 \leq q <p\leq n$$ 1 ≤ q < p ≤ n .