A note on weak almost limited operators

Let us recall that an operator $T:E\rightarrow F,$ between two Banach lattices, is said to be weak* Dunford-Pettis (resp. weak almost limited) if $f_{n}\left( Tx_{n}\right) \rightarrow 0$ whenever $(x_{n})$ converges weakly to $0$ in $E$ and $(f_{n})$ converges weak* to $0$ in $F^{\prime }$ (resp. $f_{n}\left( Tx_{n}\right) \rightarrow 0$ for all weakly null sequences $\left( x_{n}\right) \subset E$ and all weak* null sequences $\left(f_{n}\right) \subset F^{\prime }$ with pairwise disjoint terms). In this note, we state some sufficient conditions for an operator $R:G\rightarrow E$(resp. $S:F\rightarrow G$), between Banach lattices, under which the product $TR$ (resp. $ST$) is weak* Dunford-Pettis whenever $T:E\rightarrow F$ is an order bounded weak almost limited operator. As a consequence, we establish the coincidence of the above two classes of operators on order bounded operators, under a suitable lattice operations' sequential continuity of the spaces (resp. their duals) between which the operators are defined. We also look at the order structure of the vector space of weak almost limited operators between Banach lattices.