tailieunhanh - Báo cáo toán học: "Toward a characterization of reflexive contractions "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hướng tới một đặc tính của các cơn co thắt phản xạ. | J. OPERATOR THEORY 13 1985 73-86 Copyright by INCREST 1985 TOWARD A CHARACTERIZATION OF REFLEXIVE CONTRACTIONS PEI YUAN wu In memory of James p. Williams 1938 1983 Throughout this paper we consider bounded linear operators on complex separable Hilbert spaces. For an operator T let T T and AlgT denote respectively its commutant double commutant and the weakly closed algebra generated by T and I. Let Lat T denote the invariant subspace lattice of T and AlgLatT S. LatT c LatS . Recall that an operator T is reflexive if AlgLatT AlgT. The study of reflexive operators was initiated by Sarason 2 . He showed the reflexivity of normal operators and analytic Toeplitz operators. Since then various classes of operators are known to be reflexive. Among contractions it is now known that contractions with unequal defect indices and Cỵ. contractions with at least one finite defect index are reflexive cf. 6 and 12 resp. . Moreover the characterization of reflexive operators among c0 contractions and completely non-unitary weak contractions with finite defect indices has been reduced to that of S p the compression of the shift on H2 pH2 p inner cf. 1 and 12 . It is generally agreed that reflexivity is difficult to characterize unless we have a rather deep understanding of the structure of the operators under study. In this paper we propose a characterization of reflexive contractions with at least one finite defect index which summarizes all the above-mentioned results at least when restricted to the class of operators we consider . We have not been successful with the most general case. What we can handle is under the additional condition that the outer factor of the characteristic function of the contraction admits a right outer scalar multiple see Section 1 for the definition . However the results obtained together with our previous experiences seem to indicate that this is the appropriate criterion. Our proofs for the more restricted case depend heavily on the scalar .