Weak Almost Grothendieck Operators in Banach Lattices

We introduce and study a weak version of almost Grothendieck operators that we will call weak almost Grothendieck operators, these are operators between Banach space and Banach lattice which exactly carry Dunford-Pettis sets into almost limited ones. We establish some characterizations of this class of operators, and we compare with some other classes in Banach lattice. Furthermore, we establish some connections between weak almost Grothendieck operators and whose adjoint carries positive weak*null sequences into weakly null ones are obtained.