柳雅朋, 陳滋利, 王雅娟

（西南交通大學(xué)數(shù)學(xué)學(xué)院, 四川成都 610031）

Banach格上序弱緊算子的序Dunford-Pettis性質(zhì)

柳雅朋, 陳滋利, 王雅娟

（西南交通大學(xué)數(shù)學(xué)學(xué)院, 四川成都 610031）

根據(jù)序Dunford–Pettis算子和序弱緊算子的有關(guān)性質(zhì), 主要研究Banach格中任意的序弱緊算子是序Dunford–Pettis算子的空間必要條件. 得到了一些相關(guān)的結(jié)果.

序Dunford-Pettis算子; 序弱緊算子; Dunford-Pettis算子; Banach格

1 引言

設(shè)E和F是Banach格,T:E→F 是有界線性算子, 若T將E中弱緊集映為F中的范數(shù)相對(duì)緊集, 則T是Dunford-Pettis算子; 若T映E中序區(qū)間中不交序列為F中范收斂于0的序列, 那么T就是序弱緊算子. 文獻(xiàn)[5]、[6]、[7]、[11]已經(jīng)深入研究了上述兩個(gè)算子基本性質(zhì)和等價(jià)刻畫, 以及與別的算子之間的關(guān)系. 同時(shí), 有關(guān)它們的延伸和推廣是現(xiàn)在該領(lǐng)域研究的熱點(diǎn). J. A. Sanchz在文獻(xiàn)[12]首次提出了幾乎Dunford-Pettis算子, 而弱幾乎Dunford- Pettis算子在文獻(xiàn)[13]中被K. Bouras 和M. Moussa引入.

在文獻(xiàn)[16] 中作者引入了“Banach格上序Dunford-Pettis算子”, 并建立了其基本性質(zhì)和一些刻畫. 在文獻(xiàn)[17]中主要研究了序Dunford-Pettis算子與Dunford-Pettis算子、弱緊算子以及與AM-緊算子的關(guān)系. 本文將研究序弱緊和序Dunford-Pettis這兩類算子等價(jià)時(shí), 空間具有的性質(zhì).

那些沒(méi)有被注釋的有關(guān)正算子和Banach格中的定義、符號(hào)和術(shù)語(yǔ)詳見(jiàn)文獻(xiàn) [1]、[2]、[4].

2 主要結(jié)果

推論 2.6設(shè)E,F為Banach格, 任序弱緊算子T:E→F為序Dunford-Pettis算子, 則至少以下結(jié)論之一成立

1）E的格運(yùn)算為序有界弱序列連續(xù);

2）F為KB空間.

定理 2.7設(shè)E、F為Banach格,T:E→F, 則有以下結(jié)論：

1)若E有序連續(xù)范數(shù)且離散, 則算子T :E→F是序Dunford-Pettis算子

2)若F離散并且有序連續(xù)范數(shù), 則任序有界的算子T:E→F都是序Dunford-Pettis算子

3)若F=E則下列等價(jià)：

i）任算子T:E→E都為序Dunford-Pettis算子;

iii）E有序連續(xù)范數(shù)且離散.

證明1）設(shè)W ?E 序有界弱緊集, 根據(jù)文[3]中定理1知W是緊的, 則T(W)是緊的. 故T是序Dunford-Pettis算子.

(1. 取{xn}?E 序有界弱零序列, 則{T(xn)}為序有界弱緊集, 根據(jù)文[3]中定理1知{T(xn)}是緊集, 則‖T(xn)‖→0. 即T是序Dunford-Pettis算子.

(2. i）?ii）顯然.

iii）?i）由1）可得.

定理 2.8設(shè)E為有序連續(xù)范數(shù)Banach格，則以下敘述等價(jià)：

1)T:E→c0AM緊算子;

2)T:E→c0序Dunford-Pettis算子;

3)E有弱序列格運(yùn)算;

4)E是離散的.

證明1）?2）取W為E中序有界弱緊集,T是AM緊算子 則T（W）為F中的全有界集, 故T是序Dunford-Pettis算子

2）?3）由推論2.5可知

3)?4）易知

4）?1）T:E→c0,E是離散的且有序連續(xù)范數(shù), 根據(jù)文[3]中定理1知[-x,x]為全有界集. 則T[-x,x]為相對(duì)緊集, 故T為AM緊算子.

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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, 河南許昌人, 碩士研究生, 研究方向:泛函分析; 陳滋利(1961-), 男, 教授, 博士生導(dǎo)師.