Adjoints and second adjoints of almost Dunford-Pettis operators

Abstract

We investigate when the adjoint T * and the second adjoint T * * of an operator T betweeen Banach lattices is almost Dunford-Pettis. First we show when T * * is almost Dunford-Pettis for every T . Then we prove when T * or T * * is almost Dunford-Pettis whenever T is positive and almost Dunford-Pettis. Finally we prove when T * * is an almost Dunford-Pettis operator whenever T is order weakly compact.