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Báo cáo toán học: "On the reflexivity of C_{1.} contractions and weak contractions "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên phản xạ của C_ {1.} Co bóp và co bóp yếu. | Copyright by INCREST 1982 J. OPERATOR THEORY 8 1982 209 - 217 ON THE REFLEXIVITY OF Q CONTRACTIONS AND WEAK CONTRACTIONS PEI YUAN wu For a bounded linear operator T on a complex separable Hilbert space let T T and AlgT denote respectively its commutant double commutant and the weakly closed algebra generated by T and I. Let AlgLatT be the weakly closed algebra consisting of those operators which leave invariant every invariant subspace of T. Recall that T is reflexive if AlgLatT AlgT. It has been shown in 10 and 14 that any cu contraction with finite defect indices is reflexive. In Section 1 of this paper we generalize this to cx. contractions. More precisely we show that any Cỵ. contraction with at least one defect index finite is always reflexive. The problem of the reflexivity of weak contractions is taken up in Section 2. We are able to characterize reflexive weak contractions in terms of their characteristic functions and co parts. In particular this problem is reduced to that of the reflexivity of C0 7V contractions which was studied before in 12 Our proof will be based on the dilation theory of contractions as developed by Sz.-Nagy and Foias. The main reference is their book 5 Let T be a contraction on the Hilbert space We use 0T to denote its characteristic function. T is completely non-unitary c.n.u. if there is no non-trivial reducing subspace on which the restriction of Tis unitary. The defect indices of T are dT rank 7 - T T V2 and dT rank 7 - 7T 1 2. T is of class cv resp. c.x if T x - 0 resp. T nx - 0 as n - oofor any x 0 in Y . T is of class co. resp. c.o if Tnx 0 resp. T x - 0 as n oo for any .vcJf. Ctf c . n c.p a p 0 1. For operators 7 x and To on J x and .5 2 respectively 7 - T2 denotes that 7 is a quasi-affine transform of T2 that is there exists an injection Y .Yx - 2 with dense range called quasi-affinity such that ĩfx - X7 . 7 - T2 denotes that there exists a family YJ of injections 210 PEI YUAN wu Xx X - Xf. such that 2 -- V X7 d1 and T2Xx