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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Mỗi C_ {00} co lại với nhà điều hành Schmidt Hilbert-IS khiếm khuyết của lớp c_0. | Copyright by INCREST 1983 J. OPERATOR THEORY 10 1983 . 331-335 EVERY Coo CONTRACTION WITH HILBERT-SCHMIDT DEFECT OPERATOR IS OF CLASS Co KATSUTOSHI TAKAHASHI and MITSURU UCHIYAMA 1. INTRODUCTION According to the theory of Sz.-Nagy and Foias 5 the most natural genera lization of the theory of the minimal polynomial for a matrix of finite order is developed for a contraction T of class co there exists a non-zero analytic function m ẳ in the Hardy space such that u T - 0 and the spectrum of T in the open unit disc coincides with the set of zeros of the function m 2 . Each contraction T of class Co is of class Coo that is T - 0 and T - 0 strongly as n - co. An important example of a contraction of class Co is any contraction T of class Coo which is a weak contraction in the sense that it satisfies the following two conditions see 5 Chapter VIII i the spectrum of T does not fill the unit disc and ii the defect operator Dt I T T 1 2 is of Hilbert-Schmidt class i.e. tr Z - T-T co. The main result of the present paper is that for a contraction of class Coo i is a consequence of ii . In this connection we remark that tr Z T T oo can not be replaced by tr Z T T ir oo for any p 1. Then we apply the result to obtain various characterizations for a contraction T for which T - 0 and trự T T co to be of class co. As another application we present a characterization for a hyponormal i.e. T T TT contraction to have no non-trivial normal direct summand. 2. CONTRACTION OF CLASS coo A contraction T i.e. T 1 on a separable Hilbert space is said to be completely non-unitary if it has no non-trivial unitary direct summand. Sz.-Nagy and Foias 5 developed a - -functional calculus for each completely non-unitary contraction that is there is a weak -weak continuous multiplicative homomorphism u Ả - n T from the Hardy space Z on the open disc D p ẤỊ 1 to the 332 KATSUTOSHI TAKAHASHI and MITSURU UCHIYAMA weakly closed algebra generated by T that is an extension of the usual functional .
