Confluent operator algebras and the closability property

Dedicated to the memory of our good friends and mentors, Paul R. Halmos and Béla Szőkefalvi-Nagy
https://doi.org/10.1016/j.jfa.2010.03.009Get rights and content
Under an Elsevier user license
open archive

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.

Keywords

Confluent algebra
Closability property
Completely nonunitary contraction
Rationally strictly cyclic vector
Quasisimilarity

Cited by (0)

H.B. and R.G.D. were supported in part by grants from the National Science Foundation.