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Keywords:
Dunford-Pettis operator; weak Dunford-Pettis operator; order Dunford-Pettis operator; order continuous norm; Schur property
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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