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

Abstract

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 $$T{:}X\rightarrow E$$ (resp. $$T{:}E\rightarrow X$$) is almost L-weakly compact (resp. almost M-weakly compact) if and only if the norm on E (resp. $$E^{\prime }$$) is order continuous. Moreover, we present some interesting connections between almost L-weakly compact and L-weakly compact operators (resp. almost M-weakly compact and M-weakly compact operators).