Remarks on weak compactness of operators defined on certain injective tensor products

Abstract

We show that if X X is a L ∞ {\mathcal {L}_\infty } -space with the Dieudonné property and Y Y is a Banach space not containing l 1 {l_1} , then any operator T : X ⊗ ε Y → Z T:X{ \otimes _\varepsilon }Y \to Z , where Z Z is a weakly sequentially complete Banach space, is weakly compact. Some other results of the same kind are obtained.