TAILIEUCHUNG - Báo cáo toán học: "Contractions with ($\sigma$, c) defect operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các cơn co thắt với ($ \ sigma $, c) các nhà khai thác khiếm khuyết. | Copyright by INCREST 1984 J. OPERATOR THEORY 12 1984 221- 233 CONTRACTIONS WITH a c DEFECT OPERATORS MITSURU UCHIYAMA Let T be a contraction that is Ill ll 1 on a separable Hilbert space Then DT I T T 1 2 is well defined which is called defect operator of T. In this case we have ơ 7 D where D and D denote the open unit disc and its closure respectively. Contractions which have defect operators of finite ranks have been studied by many mathematicians. For investigations of contraction T with DT e ơ c that is I T Te t c where ơ c and t c denote the Hilbert Schmidt class and the trace class respectively some mathematicians added the condition ơ T D. Such a contraction T was called weak contraction by M. G. Krein. The spectral decomposition for weak contractions T or accretive operators Z T ự T -1 were obtained by and Foiaạ Brodskii and Ginzburg cf. 17 . Since T is a contraction II T x is decreasing for each X. and Foiaẹ defined contractions classes as following Cp 7 lim 7 x 0 for each X Ạ 0 rt- oo co. T lim 117 x 0 for each x H- 0O 7 7 6 Cp -- 7 7 6 co. Cy - c n 0 i j 1 . These formal notations are playing important roles in the study of contractions. In particular they showed that every weak contraction in coo belongs to co about this notation see 7 and that every weak contraction has a co-part and a cu-part. The Jordan models for weak contractions were constructed by 10 . In 9 the author applied the results of Bercovici and Voiculescu s paper 1 to investigate a contraction 7 satisfying ơ 7 D and DT e ơ c in particular to show that 7 belongs to C10 iff there is a quasi-affinity X such that XT SgX 222 . MITSURU UCHIỴAMA where Ổ is a Hilbert space with dim Ỗ index T this index is Fredholm index and Sg is the unilateral shift on if ỹ . From the results of 9 he conjectured that a contraction in coo with ơ c -defect operator belongs to co. In 8 Takahashi and ưchiyama showed that this was true. In this note we study the structure of a

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