  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.
