Banach格上序弱紧算子的序Dunford-Pettis性质

柳雅朋, 陈滋利, 王雅娟

（西南交通大学数学学院, 四川成都 610031）

Banach格上序弱紧算子的序Dunford-Pettis性质

柳雅朋, 陈滋利, 王雅娟

（西南交通大学数学学院, 四川成都 610031）

根据序Dunford–Pettis算子和序弱紧算子的有关性质, 主要研究Banach格中任意的序弱紧算子是序Dunford–Pettis算子的空间必要条件. 得到了一些相关的结果.

序Dunford-Pettis算子; 序弱紧算子; Dunford-Pettis算子; Banach格

1 引言

设E和F是Banach格,T:E→F 是有界线性算子, 若T将E中弱紧集映为F中的范数相对紧集, 则T是Dunford-Pettis算子; 若T映E中序区间中不交序列为F中范收敛于0的序列, 那么T就是序弱紧算子. 文献[5]、[6]、[7]、[11]已经深入研究了上述两个算子基本性质和等价刻画, 以及与别的算子之间的关系. 同时, 有关它们的延伸和推广是现在该领域研究的热点. J. A. Sanchz在文献[12]首次提出了几乎Dunford-Pettis算子, 而弱几乎Dunford- Pettis算子在文献[13]中被K. Bouras 和M. Moussa引入.

在文献[16] 中作者引入了“Banach格上序Dunford-Pettis算子”, 并建立了其基本性质和一些刻画. 在文献[17]中主要研究了序Dunford-Pettis算子与Dunford-Pettis算子、弱紧算子以及与AM-紧算子的关系. 本文将研究序弱紧和序Dunford-Pettis这两类算子等价时, 空间具有的性质.

那些没有被注释的有关正算子和Banach格中的定义、符号和术语详见文献 [1]、[2]、[4].

2 主要结果

推论 2.6设E,F为Banach格, 任序弱紧算子T:E→F为序Dunford-Pettis算子, 则至少以下结论之一成立

1）E的格运算为序有界弱序列连续;

2）F为KB空间.

定理 2.7设E、F为Banach格,T:E→F, 则有以下结论：

1)若E有序连续范数且离散, 则算子T :E→F是序Dunford-Pettis算子

2)若F离散并且有序连续范数, 则任序有界的算子T:E→F都是序Dunford-Pettis算子

3)若F=E则下列等价：

i）任算子T:E→E都为序Dunford-Pettis算子;

iii）E有序连续范数且离散.

证明1）设W ⊂E 序有界弱紧集, 根据文[3]中定理1知W是紧的, 则T(W)是紧的. 故T是序Dunford-Pettis算子.

(1. 取{xn}⊂E 序有界弱零序列, 则{T(xn)}为序有界弱紧集, 根据文[3]中定理1知{T(xn)}是紧集, 则‖T(xn)‖→0. 即T是序Dunford-Pettis算子.

(2. i）⇒ii）显然.

iii）⇒i）由1）可得.

定理 2.8设E为有序连续范数Banach格，则以下叙述等价：

1)T:E→c0AM紧算子;

2)T:E→c0序Dunford-Pettis算子;

3)E有弱序列格运算;

4)E是离散的.

证明1）⇒2）取W为E中序有界弱紧集,T是AM紧算子 则T（W）为F中的全有界集, 故T是序Dunford-Pettis算子

2）⇒3）由推论2.5可知

3)⇒4）易知

4）⇒1）T:E→c0,E是离散的且有序连续范数, 根据文[3]中定理1知[-x,x]为全有界集. 则T[-x,x]为相对紧集, 故T为AM紧算子.

[1] C D ALIPRANTIS, O BURKINSHAW. Positive Operators[M]. New York: Academic Press, 1985.

[2] P MEYER-NIEBERG. Banach Lattices[M]. Berlin: Universitext, Springer-Verlag, 1991.

[3] A W WICKSTEAD, Converses for the Dodds-Fremlin and Kalton-Saab theorems[J] Math Proc Camb Phil Soc, 1996, 120: 175- 179.

[4] A C ZAANEN. Riesz spaces II[M]. North Holland Publishing Company, 1983.

[5] B AQZZOUZ, J HMICHANE. The Duality Problem for the Class of order Weakly Compact Operators[J]. Glasgow Math, 2009, 51: 101-108.

[6] B AQZZOUZ, J HMICHANE. Some New Results on Order Weakly Operators[J]. Bohemica Math, 2009, 134: 395-367.

[7] B AQZZOUZ, R NOUIRA, L ZRAOILA. On the Duality Problem of Positive Dunford-Pettis Operators on Banach lattices[J]. Rendiconti del Circolo Matematico di Palemo, 2008, 57: 287-294.

[8] B AQZZOUZ, J HMICHANE. Complement on order weakly compact operators[J]. Mathods of Functional Analysis and Topology, 2011, 17: 112-117

[9] B AQZZOUZ, A ELBOUR, ANTHONY W WICKSTEAD. Positive almost Dunford-Pettis operators and their duality[J]. Positivity,(15): 185-197.

[10] B AQZZOU, R NOUIRA, L ZRAOILA. Semi-compactness of positive Dunford-Pettis operators on Banach Lattices[J]. Amer Math Soc, 2008, 136: 1997-2006.

[11] B AQZZOU, L ZRAOILA. AM-compactness of positive Dunford-Pettis operators on Banach lattice[J]. Rendiconti del circolo Matematico di palermo, 2007, 41:305-316

[12] J A SANCHZ. Operators on Banach lattices (Spanish)[D]. Madrid: Completeness University, 1985.

[13] K BOURAS, M MOUSSA. Banach lattices with weak Dunford-Pettis property[J]. International Journal of Information and Mathematical Sciences, 2010(6): 203-207.

[14] J A DIESTEL. Survey of results related to the Dunford-Pettis property[J]. Contemporary Math, 1980(2): 15-60.

[15] B AQZZOUZ, SALAELJADIDA A. Elbour,Some Charaterizations of Order Weakly Operators[J]. Bohemica, Math, 2011, 136(1): 105-112.

[16] 孙文涛. Banach格上序Dunford-Pettis算子［D]. 成都: 西南交通大学, 2011.

[17] 黄怀香, 孙文涛, 陈滋利. Banach格上序Dunford-Pettis算子的相关性质[J]. 西南民族大学学报: 自然科学版, 2013, 39: 47-50.

The order Dunford–Pettis property of order weak compact operators on Banach lattices

LIU Ya-peng, CHEN Zi-li, WANG Ya-juan
(School of Mathematics, Southwest Jiaotong University, Chengdu 610031, P.R.C.)

Based on the related properties of order weak compact operators and order Dunford–Pettis operators, a research is conducted on some necessary properties of the space on which each order weak compact operator is order Dunford-Pettis operators. Some related results are also obtained.

order Dunford-Pettis operator; order weak compact operator; Dunford-Pettis operator; Banach lattice

O177.2

A

1003-4271(2014)02-0244-05

10.3969/j.issn.1003-4271.2014.02.15

2013-11-13

柳雅朋(1988-), 女, 河南许昌人, 硕士研究生, 研究方向:泛函分析; 陈滋利(1961-), 男, 教授, 博士生导师.