Banach格上b-AM-紧算子的性质  

The Properties of b-AM-compact Operators on Banach Lattice

作　　者：程娜 CHENG Na(School of Science,Xihua University,Chengdu 610039 China)

机构地区：[1]西华大学理学院,四川成都610039

出　　处：《西华大学学报（自然科学版）》2020年第6期19-25,共7页Journal of Xihua University:Natural Science Edition

基　　金：国家自然科学基金资助项目(11801454)。

摘　　要：本文首先介绍Banach格上的b-AM-紧算子的基本性质:1)如果Banach格F是无限维的,则E是K B-空间当且仅当每个从E到F的AM-紧算子是b-AM-紧算子;2)Banach格E是离散的K B-空间当且仅当每个从E到F的连续算子是b-AM-紧算子;3)如果E′是离散的,则每个从E到X的b-弱紧算子是b-AM-紧算子。其次给出了b-AM-紧算子的控制性质:1)如果E和F是Banach格,算子S,T:E→F满足0≤S≤T且T是b-AM-紧算子,则算子S是b-AM-紧算子当且仅当F具有序连续范数或者E′是离散空间;2)如果S,T是从E到F的算子满足0≤S≤T,若T是b-AM-紧算子,则S2也是bAM-紧算子。接着给出了b-AM-紧算子的共轭性质:1)若E′具有序连续范数,T:E→F是正则b-AM-紧算子,则T′:F′→E也是b-AM-紧算子;2)如果T:E→F是正算子,T′:F′→E′是b-AM-紧算子,有T:E→F也是b-AM-紧算子,则E′是离散的或者F具有序连续范数。最后,给出了正则b-AM-紧算子空间是Dedekindσ-完备子格的充要条件:Kb-AMr(E,F)是Dedekindσ-完备向量格当且仅当F具有序连续范数或者E′是离散的且F是Dedekindσ-完备的。Several characterizations of b-AM-compact operators are considered in this paper,and we show that:1)If F is an infinite-dimensional Banach lattice,then E is a KB-space if and only if every AMcompact operator from E into F is b-AM-compact.2)The Banach lattice E is a discrete KB-space if and only if every continuous operator from E into Banach lattice F is b-AM-compact.3)If the topological dual E′is discrete,then every b-weakly compact operator from Banach E into Banach space X is b-AM-compact.Moreover,following properties about the problems of domination in the class of positive b-AM-compact operators are established:1)If E and F are two Banach lattices,then for all operators S,T:E→F such that 0≤S≤T and T is b-AM-compact,the operator S is b-AM-compact if and only if the norm of F is order continuous or E′is discrete.2)If S,T are two operators from E into F with 0≤S≤T,if T is b-AMcompact,then S2 is likewise b-AM-compact.Then we give some necessary conditions and some sufficient conditions on Banach lattices E and F for the duality properties for b-AM-compact operators:(i)If T:E→F is a regular b-AM-compact operator,and the norm on E′is order continuous,then T′:F′→E is also bAM-compact operator(ii)If for every positive operator T:E→F with T′:F′→E′b-AM-compact,the operator T is b-AM-compact operator,then either E′is discrete or F has order continuous norm.Last,we give several equivalent conditions characterizing the case when Kb-AMr(E,F)is Dedekindσ-complete.

关 键 词：BANACH格 b-AM-紧算子 序连续范数 离散空间 共轭算子 Dedekindσ-完备 

分 类 号：O177[理学—数学]