Weak and almost Grothendieck operators in Banach lattices

In this paper, we introduce and study a weak version of Grothendieck operators that we will call weak Grothendieck operators, these are operators between Banach spaces which exactly carry Dunford-Pettis sets into limited ones. We establish some characterizations of this class of operators. After that, we look for some conditions on the starting space under which this class of operators and that of Grothendieck operators coincide. Furthermore, we study the weak compactness of almost Grothendieck operators. Besides, we present some results concerning the domination property of positive Grothendieck operators. Finally, some connections between almost Grothendieck operators and those whose adjoint carries positive weak* null sequences into weakly null ones are obtained.