tailieunhanh - Báo cáo toán học: "$C_{\cdot 0}$ contractions: cyclic vectors, commutants and Jordan models "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: $ C_ {\ cdot 0} co bóp: chu kỳ vectơ, commutants và các mô hình Jordan. | Copyright by INCREST 1981 J. OPERATOR THEORY 5 1981 53-62 CONTRACTIONS CYCLIC VECTORS COMMUTANTS AND JORDAN MODELS PEI YUAN wu The study of contractions was initiated by and Foias 8 . There they obtained the Jordan model for Co contractions whose defect indices are finite. Later on this was generalized to contractions with at least one finite defect index cf. 4 . Using this model Uchiyama was able to characterize the hyperinvariant subspaces of such operators and prove that if the defect indices are not equal they are always reflexive cf. 10 and 11 resp. . In this paper we will also use this model to explore other properties of contractions. In Section 1 we are mainly concerned with the following question Are the properties of being cyclic and having a commutative commutant equivalent for contractions Note that for the more restrictive class of C0 N contractions the answer is affirmative cf. 5 and 6 . We obtain necessary and sufficient conditions for each of these properties. It turns out that for general contractions these two properties are not equivalent. The main result in Section 2 is that if two contractions with finite defect indices are such that one is a quasi-affine transform of the other then they have the same Jordan model. This is a generalization of the corresponding results for C0 A contractions and C10 contractions cf. 6 and 15 resp. . From this we can derive other results for contractions when they are intertwined by operators which are one-to-one or have dense ranges. i d For operators Tj r2 on Hilbert spaces 2 71 - T2 resp. Tz - T2 denotes that there exists an operator X intertwining Tz T2 which is one-to- one resp. has dense range . - T2 denotes that there exists a family Xa of intertwining operators x . x z - XC2 such that each x is one-to-one and Jf2 VUjJCj. If there is only one operator called a quasi-affinity in this family then we say that Tz is a quasi-affine transform of T-z and denote this by Tj