Perturbations of the unilateral shift and transitive operator algebras

An operator T on a Hubert space ℋ is said to have the transitive algebra property if ℒ(ℋ) is the only transitive operator algebra which contains T. It was shown by Arveson that the unilateral shift has this property. It is the purpose of the present paper to show that perturbations of the unilateral shift by a large class of finite rank operators have the transitive algebra property. Our results are partial solutions of the well-known transitive algebra problem.