Adjoints and second adjoints of almost Dunford-Pettis operators

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.Comment: 12 page