We study the relationship between weak* Dunford–Pettis and weakly (resp. M-weakly, order weakly, almost M-weakly, and almost L-weakly) operators on Banach lattices. The following is one of the major results dealing with this matter: If E and F are Banach lattices such that F is Dedekind $\sigma $-complete, then each positive weak* Dunford–Pettis operator $T:E\rightarrow F$ is weakly compact if and only if one of the following assertions is valid: (a) the norms of $E^{\prime }$ and F are order continuous; (b) E is reflexive; and (c) F is reflexive.

中文翻译：

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Banach 格上弱* Dunford-Pettis 算子的各种紧性的一些结果

我们研究了 Banach 格上的弱* Dunford-Pettis 和弱（分别为 M 弱、有序弱、几乎 M 弱和几乎 L 弱）算子之间的关系。以下是处理这个问题的主要结果之一： 如果E和F是 Banach 格子，使得F是 Dedekind $\sigma $完全，则每个正弱* Dunford–Pettis 算子$T:E\rightarrow F$是弱紧当且仅当以下断言之一有效时： (a) $E^{\prime }$和F的范数是阶连续的；(b) E是自反的；(c) F是自反的。