Strong Dunford-Pettis sets and spaces of operators (Monatsh. Math. 144, 275–284 (2005))

Abstract

In a recent paper, Ghenciu and Lewis studied strong Dunford-Pettis sets and made the following two assertions: (1) The Banach space X* contains a nonrelatively compact strong Dunford-Pettis set if and only if l∞ ↪ X*. (2) If c0 ↪ Y and H is a complemented subspace of X so that H* is a strong Dunford-Pettis space, then W(X, Y) is not complemented in L(X, Y). While the statements are correct, the proofs are flawed. The difficulty with the proofs is discussed, and a fundamental result of Elton is used to establish a simple lemma which leads to quick proofs of both (1) and (2).