Positive almost Dunford-Pettis operators and their duality
The authors study positive almost Dunford-Pettis operators and in particular their duality properties. Recall that the linear operator \(T: E \to F\), where \(E\) is a Banach lattice and \(F\) a Banach space, is an almost Dunford-Pettis operator if \(\| Tx_n\| \to 0\) as \(n \to \infty\) for every weak-null sequence \((x_n) \subset E\) consisting of pairwise disjoint elements. Let \(E\) and \(F\) be Banach lattices. Some sample results: (i) \(T^*: F^* \to E^*\) is almost Dunford-Pettis for every positive almost Dunford-Pettis operator \(T: E \to F\) if and only if one of the following conditions hold: (a) \(E^*\) has order continuous norm, or (b) \(F^*\) has the positive Schur property (that is, \(\| f_n\| \to 0\) as \(n \to \infty\) for any positive weak-null sequence \((f_n) \subset F^*_+\)), (ii) every positive operator \(E \to c_0\) is almost Dunford-Pettis if and only if \(\lim_n f_n(x_n) = 0\) for every weak-null disjoint sequence \((x_n) \subset E\) and positive weak\(^*\)-null sequence \((f_n) \subset E^*_+\). In addition the authors provide (separate) necessary and sufficient conditions for the converse property that for positive operators \(T: E \to F\) one has that \(T\) is almost Dunford-Pettis whenever \(T^*: F^* \to E^*\) is almost Dunford-Pettis.
scientific article; zbMATH DE number 5913989