On the domination problem of positive Null almost L-weakly compact operators on Banach lattices

We give a necessary and sufficient conditions for which the domination problem admits a positive solution for the class of positive Null almost L-weakly compact operators, this study ends with an open question which will discussed later. We then consider, the linear span of positive Null almost L-weakly (resp., Null almost M-weakly) compact operators and give results about when they form a Banach lattice and have an order continuous norm.