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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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