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On the class of order Dunford-Pettis operators. (English) Zbl 1289.46029
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 $$T': F'\longrightarrow E'$$ which is Dunford-Pettis if and only if the norm of $$E'$$ is order continuous or $$F'$$ has the Schur property.

##### MSC:
 46B42 Banach lattices 47B60 Linear operators on ordered spaces 47B07 Linear operators defined by compactness properties
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