Some results on the class of weakly sequentially precompact sets and operators

The paper contains some results on weakly sequentially precompact sets and operators. In particular, we establish some relationships between weakly sequentially precompat operators and those whose the adjoint map (L) sets into relatively norm compact ones. Besides, we characterize the class of weak* Dunford-Pettis operators through weakly sequentially precompact operators and deduce in the sequel a new characterization of Dunford-Pettis* property. Moreover, we generalize [9, Theorem 2.5.9] and show that order weakly compact operators carry almost order Dunford-Pettis sets into weakly sequentially precompact ones. Furthermore, we prove that the product of order weakly compact operators and b-weakly compact ones maps weakly sequentially precompact sets into relatively weakly compact ones. Finally, we present some results about the positive Schur property.