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 −→ 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.2010 Mathematics Subject Classification. 46B25, 46B42, 47B60, 47B65. Key words and phrases. L-weakly compact operator, M-weakly compact operator, order weakly compact operator, order L-weakly compact operator, order M-weakly compact operator, order continuous norm, Banach lattice.
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