  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.
