Strongly Irreducible Operators on Banach Spaces  

作　　者：Yun Nan ZHANG Huai Jie ZHONG 

机构地区：[1]School of Mathematics and Computer Science,Fujian Normal University [2]Mathematics and Information Science College,Hebei Normal University

出　　处：《Acta Mathematica Sinica,English Series》2012年第4期727-740,共14页数学学报（英文版）

基　　金：Supported by National Natural Science Foundation of China(Grant Nos.10926173,11171066 and 10771034);Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.2010350311001);Natural Science Foundation of Fujian Province of China(Grant No.2009J05002)

摘　　要：This paper firstly discusses the existence of strongly irreducible operators on Banach spaces. It shows that there exist strongly irreducible operators on Banach spaces with w＊-separable dual. It also gives some properties of strongly irreducible operators on Banach spaces. In particular, if T is a strongly irreducible operator on an infinite-dimensional Banach space, then T is not of finite rank and T is not an algebraic operator. On Banach spaces with subsymmetric bases, including infinite-dimensional separable Hilbert spaces, it shows that quasisimilarity does not preserve strong irreducibility. In addition, we show that the strong irreducibility of an operator does not imply the strong irreducibility of its conjugate operator, which is not the same as the property in Hilbert spaces.This paper firstly discusses the existence of strongly irreducible operators on Banach spaces. It shows that there exist strongly irreducible operators on Banach spaces with w＊-separable dual. It also gives some properties of strongly irreducible operators on Banach spaces. In particular, if T is a strongly irreducible operator on an infinite-dimensional Banach space, then T is not of finite rank and T is not an algebraic operator. On Banach spaces with subsymmetric bases, including infinite-dimensional separable Hilbert spaces, it shows that quasisimilarity does not preserve strong irreducibility. In addition, we show that the strong irreducibility of an operator does not imply the strong irreducibility of its conjugate operator, which is not the same as the property in Hilbert spaces.

关 键 词：Banach spaces strongly irreducible operators w＊-separable quasisimilar 

分 类 号：O177.2[理学—数学] O177.1[理学—基础数学]