In this paper, we introduce and study new concepts of order L-weakly and order M-weakly compact operators. As consequences, we obtain some characterizations of Banach lattices with order continuous norms or whose topological duals have order continuous norms. It is proved that if \(T:E \longrightarrow F\) is an operator between two Banach lattices, then T is order M-weakly compact if and only if its adjoint \(T'\) is order L-weakly compact. Also, we show that if its adjoint \(T'\) is order M-weakly compact, then T is order L-weakly compact. Some related results are also obtained.

中文翻译：

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关于阶L弱和阶M弱紧算子的类

在本文中，我们介绍和研究阶L弱和阶M弱紧凑算子的新概念。结果，我们获得了具有阶连续范数或其拓扑对偶具有阶连续范数的Banach格的某些刻画。证明如果\（T：E \ longrightarrow F \）是两个Banach晶格之间的算子，则且仅当其伴随\（T'\）是L-弱紧凑时，T才是M-弱紧凑。同样，我们表明，如果它的伴随\（T'\）是阶为M弱紧凑，那么T为阶为L弱紧凑。还获得了一些相关结果。