tailieunhanh - Báo cáo toán học: "C_{1.}-contractions with Hilbert-Schmidt defect operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: C_ {1} co bóp với những hoạt động bị dị tật bẩm Hilbert-Schmidt. | Copyright by INCREST 1984 J. OPERATOR THEORY 12 1984 331-347 WITH HILBERT-SCHMIDT DEFECT OPERATORS KATSUTOSH1 TAKAHASHI 1. INTRODUCTION According to their theory of characteristic functions and functional models for contractions and Foia investigated a contraction T whose defect operator DT J is of Hilbert-Schmidt class and whose spectrum does not fill the unit disc see 5 Chapter VIII . Such a contraction was called a weak contraction and proved to possess a good structure. In this paper we investigate a contraction T of class Cl. whose defect operator DT is of Hilbert-Schmidt class. We note that such a contraction is a weak contraction if and only if it is of class cn see 5 Chapter VIII . Recall that a contraction Tis of class Cl. if lim T x i 0 for every non-zero X and T is of class Co. if lim 0 for every X. The classes and are defined by using Tif instead of T and Cxp Ca. n Tor a p - 0 1. For a contraction T of class Cl. there is an injection X with dense range such that XT vx for some isometry V see 5 pp. 71 72 and 4 . In the recent paper 12 Uchiyama proved for a C10-contraction T with Hilbert-Schmidt defect operator Dt that there exist an injection X with dense range and an injection Y such that XT sx and TY YS for a unilateral shift s with ind5 indr for a semiFredholm operator A indX denotes Fredholm index . In Section 3 we show for a T with Hilbert-Schmidt defect operator DT that there exist an injection X with dense range and a sequence of injections y n I 2 . . such that XT vx TY Y v n 1 2 . and the span of ran y w 1 2 . is the space on which Tacts where Vis an isometry with ind V ind T and if T is of class C10 then the isometry V is a unilateral shift. Then we treat three natural weakly closed algebras of operators associated with such T the first one is the weakly closed algebra generated by T and the identity the second is the double commutant T and the third is AlgLat T where Lat Tdenotes .