ON THE CLASS OF ORDER DUNFORD-PETTIS OPERATORS

We characterize Banach lattices E and F on which the adjoint of each operator from E into F which is order Dunford-Pettis and weak Dunford-Pettis, is Dunford-Pettis. More precisely, we show that if E and F are two Banach lattices then each order DunfordPettis and weak Dunford-Pettis operator T from E into F has an adjoint Dunford-Pettis operator T' from F' into E' if, and only if, the norm of E' is order continuous or F' has the Schur property. As a consequence we show that, if E and F are two Banach lattices such that E or F has the Dunford-Pettis property, then each order Dunford-Pettis operator T from E into F has an adjoint T': F' -> E' which is Dunford-Pettis if, and only if, the norm of E' is order continuous or F' has the Schur property.