tailieunhanh - Báo cáo toán học: "Transitive algebra problem and local resolvent techniques "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Transitive đại số vấn đề và kỹ thuật chỉ nhưng thuốc làm tiêu độc địa phương. | J. OPERATOR THEORY 1 1979 273 285 Copyright by INCREST 1979 TRANSITIVE ALGEBRA PROBLEM AND LOCAL RESOLVENT TECHNIQUES A. A. JAFARIAN and M. RADJABALIPOUR 0. INTRODUCTION Throughout the paper by an operator we mean a bounded linear transformation acting in a general Banach space s. When refering to the terms normal or adjoint it is understood that the underlying space is a Hilbert space. The algebra of all operators on is denoted by By a subsp ace of 3C we always mean a closed linear manifold in sc. A subalgebra .aZ of is called transitive if the only invariant subspaces of sđ are 0 and ỉí. The results of this paper are contained in the next two sections. In the first section we generalize some results due to Foias 7 Lomonosov 13 Fong-Nordgren-Radjabalipour-Radjavi-Rosenthal 9 and Jafarian 12 as follows If sđ is a uniformly closed algebra of operators if K and L are two operators such that L sZ ơ Ẩ 0 and Oe ơ L and if K respectively L is in the following class I resp. class II then is not transitive. Class I a Decomposable operators. b Adjoints of subdecomposable operators. c Adjoints of M-hyponormal operators. Class II a Subdecomposable operators. P Hyponormal operators. See below for the definitions. In 9 it is shown thát if sđK c Lsi for a compact operator K and a certain operator L then jaf is not transitive. Their proof is based on Lomonosov s theorem 13 24 pages. 156 - 159 and cannot be regarded as a local resolvent technique while our proof is heavily based on the properties and existence of certain local resolvents. Even the techniques used in 7 and 12 are not applicable here though their results can be obtained as corollaries of ours. We would like to mention that our results are essentially disjoint from those of 9 and cannot be obtained as corollaries of each other. However we acknowledge some motivations from 9 and 12 . 274 A. A. JAFARIAN and M. RADJABALIPOUR In the course of our study we obtained some results of possibly independent interest on local .