Abstract
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.
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