Dunford-Pettis property of the product of some operators (Q2896650)

An operator \(T\) from a Banach lattice \(E\) into a Banach space \(X\) is called almost Dunford-Pettis if \(\| T(x_n)\| \to 0\) for every disjoint sequence \(\{ x_n\}\subset E\) such that \(x_n\to 0\) in the \(\sigma (E,E')\) topology. An operator between two Banach spaces is called Dunford-Pettis if it maps weakly null sequences into norm null sequences. The authors establish a sufficient condition for the product of an order bounded almost Dunford-Pettis operator and an order weakly compact operator to be Dunford-Pettis. Several applications are given, in particular a generalization of a result by \textit{W. Wnuk} [Atti Semin. Mat. Fis. Univ. Modena 42, No. 1, 227--236 (1994; Zbl 0805.46023)].