Operators whose adjoints and second adjoints are almost Dunford-Pettis

First we characterize the Banach lattices E whose biduals have the positive Schur property by means of second adjoints of operators on E being almost Dunford-Pettis. Next we extend some known results concerning conditions on the Banach lattices E and F under which the adjoint T* and the second adjoint T** of any positive almost Dunford-Pettis operator T from E to F are almost Dunford-Pettis. Finally, we prove when T* and T** are almost Dunford-Pettis for any (non necessarily almost Dunford-Pettis) T that is either bounded, regular, order bounded or weakly compact.