We establish necessary and sufficient conditions under which b-weakly compact operators between Banach lattices have b-weakly compact adjoints or operators with b-weakly compact adjoints are themselves b-weakly compact. Also, we give some consequences. Mathematics Subject Classification (2000). 46A40, 46B40, 46B42.

We study some properties of almost Dunford-Pettis operators and we characterize pairs of Banach lattices for which the adjoint of an almost DunfordPettis operator inherits the same property and look at conditions under which an operator is almost Dunford-Pettis whenever its adjoint is.

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We introduce and study the class of weak almost limited operators. We establish a characterization of pairs of Banach lattices E, F for which every positive weak almost limited operator T : E → F is almost limited (resp. almost Dunford-Pettis). As consequences, we will give some interesting results.