TAILIEUCHUNG - Báo cáo toán học: On the structure of (BCP)-operators and related algebras

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Về cấu trúc của (BCP)-nhà khai thác và đại số có liên quan. | J. OPERATOR THEORY 12 1984 235-245 Copyright by INCREST 1984 ON THE STRUCTURE OF BCP -OPERATORS AND RELATED ALGEBRAS. II GREG ROBEL 1. INTRODUCTION In this paper we extend the methods and results of Part I 6 to a context first explored by B. Chevreau c. Pearcy and A. Shields in 3 . The results of Part 1 may be regarded as statements about the ultraweakly closed algebra generated by a BCP -operator together with the identity operator . Here we study certain analogous algebras generated by several operators. This paper proceeds from the general to the particular. In the next section we show how the techniques from Part I can be adapted to the study of a class of representations of jFỈ G ỉor G a bounded domain in the complex plane. Since the development proceeds very much as in Part I most proofs are only sketched or omitted. We also prove that the range of such a representation is a reflexive algebra by adapting the argument given by H. Bercovici c. Foias J. Langsam and c. Pearcy in 1 to establish the reflexivity of operators of class BCP . In Section 3 we specialize to the case of certain particular domains G which we call circular domains. For an operator suitably related to such a domain G Chevreau Pearcy and Shields constructed in 3 an CCj-functional calculus to which the analysis of Section 2 applies. Moreover the range of this functional calculus is precisely the ultraweakly closed algebra generated by the operator together with several specific rational functions of it. We point out that two special cases of this situation are the ultraweakly closed algebras generated by certain polynomially bounded operators in which case G is the open unit disc D and the ultraweakly closed algebras generated by T and T-1 for certain invertible operators T in which case G is an annular region contained in D . We give an example of a weighted bilateral shift operator of the latter type. In the final Section 4 we return our attention to the class of BCP -operators and make some

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