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 23 -45 Copyright by INCREST 1984 ON THE STRUCTURE OF BCP -OPERATORS AND RELATED ALGEBRAS. I GREG ROBEL 1. INTRODUCTION This paper is concerned with a class of bounded linear operators on Hilbert space which we call BCP -operators and which were first studied by s. Brown B. Chevreau and c. Pearcy in 7 where the existence of invariant subspaces for such operators was established. The class BCP together with various related classes of operators has been further studied in a number of works including 1 2 3 4 8 and 13 . In particular recent work of c. Foia c. Pearcy and B. 10 11 has given strong reason to hope that a sufficient understanding of BCP -operators would yield the existence of invariant subspaces for a large class of operators containing for example all Toeplitz operators and all hyponormal operators. For this reason and others results about the structure of BCP --operators are of considerable interest. This paper is divided into six sections. In Section 2 we recall some preliminary material and establish our notational conventions. We also give here our definition of the class BCP which has the advantage that the class BCP is now both self-adjoint and more general than the class studied in 7 . We are able to work in this broader context by virtue of Lemma Corollary and Lemma B . Operators of class BCP together with some related classes of operators were studied in 3 in the context of the Foias functional model for contraction operators. In Section 3 we obtain many of the key results of an early version of 3 for our more general BCP -operators. Unlike 3 our proofs make no use of the Foias functional model. In addition to giving a more accessible approach to these results our techniques admit certain extensions which will be taken up in the forthcoming Part II of this paper. In Section 4 we use the results of Section 3 to obtain Theorem which concerns the realization of certain operators as

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