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

TL;DR

This paper investigates the conditions under which certain operator algebras, generated by normal operators and operators of class C_0, possess the closability property, focusing on multiplication operators and functional calculus.

Contribution

It provides necessary and sufficient conditions for the closability property of von Neumann algebras generated by normal operators and C_0 class operators.

Findings

Characterization of closability property for multiplication operator algebras

Necessary and sufficient conditions for normal operators

Criteria for operators of class C_0

Abstract

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.