Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: mặc giới hạn của chiếu gần như bất biến. | Copyright by INCREST 1979 J. OPERATOR THEORY 2 1979 79 93 WEAK LIMITS OF ALMOST INVARIANT PROJECTIONS c. FOIA c. PASNICU and D. VOICULESCU In this Note we give a new characterization Corollary of strongly reductive algebras of operators 1 based on a new characterization of the weak limits of orthogonal projections almost invariant with respect to an algebra of operators Theorem . This paper originates in the following question connected to the Note 2 raised privately by c. Apostol If T is an operator on a Hilbert space H such that any operator in the normclosure of the unitary orbit ựJTU ỵ U H H unitary of T is reductive PTP for p p p2 implies PT TP does it follow that T is normal In the sequel we give an affirmative answer to this question see Corollary . below as well as a natural generalization of this answer to the case of operator algebras see Theorem . and Corollary . below . The proof of those results rely on a new characterization given in Theorem . of the weak limits Q of sequences P 1 formed by orthogonal projections such that -P TP H0 n- oo for all T belonging to a norm separable algebra of operators on a Hilbert space. We hope that this characterization connecting dilation theory to quasitriangularity will lead in the future to applications of wider interest. This paper has been circulated as INCREST Preprint No. 23 July 1978. 1 . We begin by recalling the necessary terminology and notation. Let H be a complex Hilbert space with the scalar product . . . The set of all bounded linear operators on H will be denoted by H . The identity operator on any Hilbert space will be loosely denoted by 7 also by 0 we shall denote the null element in any Hilbert space the null operator on any Hilbert space as well as the subspace 0 of any Hilbert space. Each time the Hilbert space involved will be determined from the context. The ideal of compact operators in is denoted by Jf 7 80 c. FOIAS c. PASNICU and D. VOICULESCU the Calkin algebra
