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Báo cáo toán học: "Transitive algebra containing triangular operator matrices "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Transitive đại số có chứa ma trận điều hành hình tam giác. | Copyright by INCREST 1985 J. OPERATOR THEORY 14 1985 173-180 TRANSITIVE ALGEBRA CONTAINING TRIANGULAR OPERATOR MATRICES MOHAMAD A. ANSARI 1. INTRODUCTION Arveson 1 introduced the notion of transitive operator algebra that is an algebra with no nontrivial invariant subspaces. He proved that a transitive operator algebra containing either the unilateral shift or a maximal abelian von Neumann algebra is strongly dense in the algebra of all operators. A number of other authors using Arveson s Lemma obtained similar results for a transitive operator algebra satisfying some additional hypothesis 2 4 5 6 7 8 9 10 and 12 Theorem 6 . By presenting a new technique in this paper we generalize the results of 1 and 10 to prove that if a transitive operator algebra ứ z either contains a perturbation of the unilateral shift by certain rank one operators or contains a certain triangular operator matrix then ứu is strongly dense in the algebra of all operators. We will conclude by listing a number of open question which arise from our work. 2. MAIN RESULTS Let denote an infinite dimensional separable complex Hilbert space and let denote the algebra of all operators on Ji . Definition a set .0 of operators on w is said to have the transitive algebra property TAP if whenever fc is a transitive operator algebra such that sẩ a ty 4Z is strongly dense in The operator A e is said to have the TAP if the set A has the TAP. it is easy to prove that if S J u 1 has the TAP then s has the TAP. If z is an operator algebra on we will write ữU for the closure of in the strong operator topology of 174 MOHAMAD A. ANSARI Theorem 2.1. LetYd . be two Hilbert spaces and let si be a subset of consisting of operators of the form A .0 0 with the following condition t 0 OJ J If T is an operator in ỹẸ if A e Lat T and if T yd has the TAP then the set sđ u T has the TAP. Proof. Let be a transitive operator algebra such that u T 7 . Define the set A - x IPfXA there are operators Y e 6 and z e Sh O such that