Some properties of almost L-weakly and almost M-weakly compact operators

In this paper, we investigate necessary and sufficient conditions under which compact operators between Banach lattices must be almost L-weakly compact (resp. almost M-weakly compact). Mainly, it is proved that if X is a non zero Banach space then every compact operator T : X. E (resp. T : E. X) is almost L-weakly compact (resp. almost M-weakly compact) if and only if the norm on E (resp. E ) is order continuous. Moreover, we present some interesting connections between almost Lweakly compact and L-weakly compact operators (resp. almost M-weakly compact and M-weakly compact operators).