非紧性测度的零化度和完全性

On nullity and fullness of measures of noncompactness

Banach空间的非紧性测度(measure of noncompactness,MNC)μ,按照它的零点集kerμ构成的超空间划分成三大类:(1)完全MNC(kerμ恰好是X的所有非空相对紧集构成的超空间K,它蕴涵了在MNC的应用中极为重要的广义Cantor交的性质);(2)不完全MNC,也称为MNC(kerμ是K的一个非空子集,并满足广义Cantor交性质);(3)广义MNC(K是kerμ的一个非空子集,它包括了非弱紧性测度等广义非紧性测度).不要求广义Cantor交性质的MNC称为准MNC,而验证这个性质是否成立往往在技术上又是极为困难的.长期以来,人们不知道这条“额外”的性质是否独立于MNC定义中的其他条件.另一个长期令人困惑的基本问题是,一个MNC是否能够控制一个完全MNC?本文主要研究这两个问题.其结果是通过建立Banach空间准MNC和广义MNC的表示定理,证明广义Cantor交假设是独立的,同时给出MNC能够控制一个完全MNC的特征.

Let X be a Banach space and B(respectively,K)be the super-space of all nonempty bounded(respectively,relatively compact)subsets of X.A measure of noncompactness(MNC)is a nonnegative real-valued function de ned on B satisfying certain conditions that we recognize as established.Generally,an MNCis in one of the following three categories according to properties of its\null set"ker:full MNC(ker=K),MNC(∅̸=ker⊂K with the generalized Cantor's intersection property),and generalized MNC(K$ker).The generalized Cantor's intersection property is extremely important in applications of MNCs,and meanwhile,it is automatically true for every full MNC.But we do not know the answers to the following questions:(1)Is the generalized Cantor's intersection property assumption independent of other conditions in the de nition of MNC?(2)Does every MNC dominate a full MNC?In this paper,we mainly focus on the basic questions mentioned above.As a result,we show that the answer to the former question is\yes",and give two characterizations for an MNC to dominate a full MNC.They are done by establishing a representation theorem of pre-MNCs(hence,of MNCs,full MNCs and generalized MNCs).