Banach空间中有界算子的非紧性测度

Measure of Non-compactness of Bounded Operators in Banach Spaces

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).