tailieunhanh - Báo cáo toán học: "Remarks on Lebesgue-type decomposition of positive operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nhận xét về Lebesgue loại phân hủy của các nhà khai thác tích cực. | J. OPERATOR THEORY 11 1984 137- 143 p Copyright by INCREST 1984 REMARKS ON LEBESGUE-TYPE DECOMPOSITION OF POSITIVE operators HIDEKI KOSAKI 0. INTRODUCTION The purpose of the paper is to give alternative proofs to Ando s results 1 based on a different technique which will be explained below. Also this technique is shown to provide quite a natural viewpoint to the subject matter. Let a b be positive bounded operators on a Hilbert space. As a generalization of absolute continuity in measure theory we say that b is a-absohttely continuous if there exists a sequence b of positive operators such that bn T b strongly as n T oo and bn l a for some positive number . We also say that b is a-singular if c 0 follows whenever an operator c satisfies 0 c c b and O c a. In the paper we consider a Lebesgue decomposition b Ò1 Ờ2 with an a-absolutely continuous operator bt and an a-singular operator b2 In 1 Ando introduced the positive operator ữ ố ft as the strong limit of a certain sequence of positive operators see 1 . Among other results he showed that i a b resp. b ữ Z is a-absolutely continuous resp. a-singular ii a ố is maximal in the sense that b a b whenever 0 b b and b is a-absolutely continuous. In the paper we further assume that a is non-singular Kerữ 0 . Applications of this subject to the theory of operator algebras will appear in subsequent papers. And in this context the case when Ker 0 is most interesting. We show that the subject matter is closely related almost equivalent to recent theories 4 7 of decompositions of unbounded operators and quadratic forms into their closable parts and singular parts . Use of unbounded operators and forms actually gives quite a powerful tool although involved arguments are simple. In fact based on this technique we obtain certain simple and more importantly explicit expressions of a b. 138 KOSAK1 1. PRELIMINARIES In this section we collect some basic definitions as well as results. We fix positive bounded operators b on a