关于线性算子的弱统计收敛  

On weak statistical convergence for linear operators

作　　者：鲍玲鑫[1] BAO Lingxin(College of Computer and Information Sciences,Fujian Agricalture and Forestry University,Fuzhou 350002,China)

机构地区：[1]福建农林大学计算机与信息学院,福建福州350002

出　　处：《厦门大学学报（自然科学版）》2021年第1期67-71,共5页Journal of Xiamen University：Natural Science

基　　金：国家自然科学基金(11501108);福建省自然科学基金(2019J01400);福建农林大学杰出青年计划项目(2016011)。

摘　　要：讨论了统计收敛的两个基本问题:1)在第一可数的拓扑空间上,统计收敛和几乎处处收敛等价的,反之,如果统计收敛和几乎处处收敛等价,能否导出这个拓扑空间一定是第一可数的?2)超滤子收敛是否和依统计测度收敛等价?通过构造两个例子,给出了这两个问题以否定的答案.此外,引入有界线性算子序列在弱算子拓扑意义下的统计收敛,证明了一个Connor-Ganichev-Kadets型定理,即证明了对一个可分的Banach空间X,X*可分的当且仅当对任意有界的弱统计收敛的(X)-值序列(Tn),都存在一个弱收敛的有界线性算子序列(Sn)使得{n∈N:Tn=Sn}具有自然密度1.This article discusses two basic problems of statistical convergence:1)On the first countable topological space,statistical convergence is equivalent to almost uaual convergence,so what about the reverse?That is,if statistical convergence is always equivalent to almost usual convergence,can this topology space be derived to be the first countable?2)Is ultrafilter convergence equivalent to statistical measure-based convergence?By constructing two examples,this article gives negative answers to these two questions.On the other hand,we introduce weak statistical convergence for linear operators.We prove a parallel version of Connor-Ganichev-Kadets theorem,that is,for a separable Banach space X,X*is separable if and only if for every bounded statistically convergent(X)-valued sequence(Tn),then there is weakly convergent sequence(Sn)such that{n∈N:Tn=Sn}has natural density 1.

关 键 词：统计收敛 几乎处处收敛 超滤子 可分性 BANACH空间 

分 类 号：O177[理学—数学]