Domination by positive Dunford-Pettis completely continuous operators
An operator \(T\) on a Banach space \(X\) is \(DPcc\) if, and only if, for each Dunford-Pettis set \(A\subset X,\) the set \(T(A)\) is relatively compact. We recall that a norm bounded subset \(A\) of \(X\) is a Dunford-Pettis set whenever every weakly compact operator from \(X\) to an arbitrary Banach space carries \(A\) to a norm totally bounded set. The authors consider the domination problem for the class of positive Dunford-Pettis completely continuous operators, abbreviated DPcc. More precisely, they extend a previous paper of \textit{Y.-M. Wen} and \textit{J.-X. Chen} [Adv. Math., Beijing 45, No. 1, 122--132 (2016; Zbl 1374.47031)]. The main results are Theorems 3.6, 3.11 and Corollary 3.7. We state only Theorem 3.6: Let \(E\) and \(F\) be Banach lattices such that the norm of \(F\) is order continuous. If \(T\) is DPcc and \(0\leq S\leq T:E \to F\), then \(S\) is also DPcc.
Dunford-Pettis completely continuous operator
scientific article; zbMATH DE number 6828496