# Article

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Keywords:
Dunford-Pettis operator; weak Dunford-Pettis operator; order Dunford-Pettis operator; order continuous norm; Schur property
Summary:
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 Dunford-Pettis 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'\colon F'\longrightarrow E'\$ which is Dunford-Pettis if, and only if, the norm of \$E'\$ is order continuous or \$F'\$ has the Schur property.
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