TAILIEUCHUNG - Báo cáo toán học: "On the weakly closed algebra generated by a weak contraction "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên các đại số đóng yếu tạo ra bởi sự co yếu. | J. OPERATOR THEORY 2 1979 141-148 Copyright by INCREST 1979 ON THE WEAKLY CLOSED ALGEBRA GENERATED BY A WEAK CONTRACTION PEI YUAN wu For a bounded linear operator T acting on a complex separable Hilbert space let T T and Alg T denote the commutant double commutant and the weakly closed algebra generated by T and I respectively. Let Lat T and Alg Lat T denote the lattice of invariant subspaces of T and the weakly closed algebra of operators which leave invariant all the subspaces in Lat T. It was shown in 5 and 9 that C0 N contractions and completely non-unitary . cu contractions with finite defect indices satisfy T n Alg Lat T Alg T. Indeed in the latter case we showed that they are even reflexive. In this paper we generalize these to . weak contractions with finite defect indices that is we show that T fl Alg Lat T Alg Talso holds for such more general contractions Theorem 3 . As an application a question raised in 8 is answered. If T and T2 are the contractions considered above we obtain necessary and sufficient conditions for the splitting of Alg Tt T2 Theorem 4 which generalize previous results for Cữ N and cn contractions. Let T be a . weak contraction with finite defect indices defined on H 7 ÃLỈ 0rw Aw w 6 Hi by T f g P è f eizg for f g e H where 1 and denote the standard Lebesgue and Hardy spaces of C -valued functions defined on the unit circle 0r is the characteristic function of T 1 A I 0 0r 2 and p denotes the orthogonal projection onto H. For the definition and properties of weak contractions readers are referred to Chapter VIII of 3 or to our previous paper 8 . Recall that Tis multiplicity-free if it admits a cyclic vector. For other equivalent conditions for T being multiplicity-free compare Theorem 5 of 6 . We start by showing that T n AlgLatT Alg Tholds for multiplicity-free T. Lemma 1. Assume that T is a . multiplicity-free weak contraction with finite defect indices defined on H H ALf Q 0rw Aw w 6 Hf . If QT t is not isometric

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