Order weakly compact and order unconditionally converging operators are consider on the setting of Banach latices. In this paper we characterize the class of Banach lattices on which o-weakly compact operators on it are order unconditionally converging operators. Two class of Banach having the V0 property is showing. We also consider the spaces E on which every ⎸𝜎 ⎸(E′, E)-convergent sequences on E′ are 𝜎(E′, E″) -convergent.