Confluent operator algebras and the closability property. (English) Zbl 1190.47086
Summary: Certain operator algebras \(\mathcal A\) on a Hilbert space have the property that every densely defined linear transformation commuting with \(\mathcal A\) 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 \(\mathcal A\) is a two-transitive algebra with the closability property, then \(\mathcal A\) 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.
MSC:
| 47L30 | Abstract operator algebras on Hilbert spaces |
Keywords:
confluent algebra; closability property; completely nonunitary contraction; rationally strictly cyclic vector; quasisimilarityReferences:
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