Recently, the concept of 𝐿−limited subsets of order 𝑝 (1≤𝑝<∞) in dual Banach spaces and limited p-convergent operators are introduced. The paper is devoted to some results of almost 𝐿−limited subsets of order p in dual Banach lattices and disjoint limited 𝑝−convergent operators. In particular, we derive the following result: Banach lattices with the property that 𝐿−limited subsets (or 𝐿−limited subsets of order 𝑝) in their dual are relatively weakly compact, are precisely the Grothendieck spaces. Moreover, it is established that a Banach lattice 𝑀 of some operator spaces has the disjoint limited 𝑝−Schur property if and only if all evaluation operators on 𝑀 is disjoint limited p-convergent.
