almost order-weakly compact operators on banach lattices

A continuous operator t between two banach lattices e and f is called almost order-weakly compact, whenever for each almost order bounded subset a of e, t(a) is a relatively weakly compact subset of f. we show that the positive operator t from e into a dedekind complete banach lattice f is almost order-weakly compact iff t(xn) ∥.∥ −−→ 0 in f for each disjoint almost order bounded sequence {xn} in e. in this manuscript, we study some properties of this class of operators and its relationships with the others known classes of operators.