Confluent operator algebras and the closability property

TL;DR

This paper investigates operator algebras with the closability property, especially those generated by nonunitary contractions, revealing new classes and conditions under which the property holds, impacting the transitive algebra problem.

Contribution

It introduces new classes of algebras with the closability property generated by nonunitary contractions and links this property to strict cyclicity and confluence.

Findings

Closability property follows from strict cyclicity.

Identifies classes of contractions satisfying the property.

Provides detailed analysis of confluence in contractions.

Abstract

Certain operator algebras A on a Hilbert space have the property that every densely defined linear transformation commuting with A is closable. Such algebras are said to have the closability property. They are important in the study of the transitive algebra problem. More precisely, if A is a two-transitive algebra with the closability property, then A is dense in the algebra of all bounded operators, in the weak operator topology. In this paper we focus on algebras generated by a completely nonunitary contraction, and produce several new classes of algebras with the closability property. We show that this property follows from a certain strict cyclicity property, and we give very detailed information on the class of completely nonunitary contractions satisfying this property, as well as a stronger property which we call confluence.