K. Bouras

In this paper, we establish conditions under which each positive Null almost L-weakly compact operator is Null almost M-weakly compact and conversely. Moreover, we provide the necessary and sufficient conditions under which any positive Null almost L-weakly compact operator $T: E\rightarrow F$ admits a Null almost M-weakly compact adjoint $T': F'\rightarrow E'$. Finally, we give some connections between the class of Null almost L-weakly compact (resp. Null almost M-weakly compact) operators and the class of L-weakly compact (resp. M-weakly compact).

This paper is devoted to investigation of conditions on a pair of Banach lattices $E; F$ under which every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited. Mainly, it is proved that if every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited, then the norm on $E'$ is order continuous or $F$ is finite dimensional. Also, it is proved that every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited, if one of the following statements is valid:
1) The norm on $E^{\prime }$ is order continuous, and $F^{\prime }$ has weak$^{\ast }$ sequentially continuous lattice operations.
2) The topological dual $E^{\prime }$is discrete and its norm is order continuous.
3) The norm of $E^{\prime }$ is order continuous and the lattice operations in $E^{^{\prime }}$ are weak$^{\ast }$ sequentially continuous.
4) The norms of $E$ and of $E^{\prime }$ are order continuous.