Banach空间中有界算子的非紧性测度  

Measure of Non-compactness of Bounded Operators in Banach Spaces

作　　者：沈钦锐 孙俊俊 SHEN Qinrui;SUN Junjun(School of Mathematics and Statistics,Minnan Normal University,Zhangzhou,Fujian,363000,P.R.China)

机构地区：[1]闽南师范大学数学与统计学院,福建漳州363000

出　　处：《数学进展》2024年第4期875-882,共8页Advances in Mathematics（China）

基　　金：国家自然科学基金青年基金项目(No.11801255);福建省自然科学基金面上项目(No.2020J01798);高校博士启动科研基金项目(No.L21704)。

摘　　要：本文利用经典的Hausdorff非紧性测度理论,在Banach空间X中讨论了有界算子的非紧性测度β(T)与几个算子半范数之间的关系:设B(X)为X到自身的有界线性算子全体,则■T∈B(X),β(T)=‖T‖0.当X的基常数为1时,有β(T)=‖T‖K.在X中,有β(T**)≤β(T)≤2β(T**),特别地当X为Hilbert空间时,有β(T**)=β(T*)=β(T)=‖T‖0=‖T*‖0=‖T‖K.In this paper,we use the classical Hausdorff measure of noncompactness to discuss the relationship between measure of non-compactness of operatorβ(T)and several operator seminorms in Banach space X:Denoting by B(X)the set of bounded linear operators from X to itself,we haveβ(T)=|T‖_(0),■T∈B(X).When the base constant of X is 1,there isβ(T)=|T‖_(K).In X,we obta.in thatβ(T^(**))≤β(T)≤2β(T^(**));especially when X is a Hilbert space,we haveβ(T^(**))=β(T^(*))=β(T)=|T‖_(0)=‖T^(*)‖_(0)=‖T‖_(K).

关 键 词：非紧性测度 算子非紧性测度 BANACH空间 

分 类 号：O177.92[理学—数学]