On the Dunford-Pettis property

Abstract

"A Banach space E has the Dunford-Pettis property (DPP) if every weakly compact operator on E is a Dunford-Pettis operator, that is, takes weakly convergent sequences into norm convergent sequences. For many years it remained an open question whether the Banach space of all continuous E -valued functions on a compact Hausdorff space K has the DPP if E has. This question was answered in the negative in 1983 by M. Talagrand [Israel J. Math. 44 (1983), no. 4, 317–321;] who constructed a Banach space E with the DPP and a weakly compact operator from C([0,1],E) into c 0 that is not a Dunford-Pettis operator. The author and B. Rodríguez-Salinas introduced [Arch. Math. (Basel) 47 (1986), no. 1, 55–65;] a more general class of operators that they called almost Dunford-Pettis. An operator T from C(K,E) into X whose representing measure has a semivariation continuous at ∅ said to be almost Dunford-Pettis if, for every weakly null sequence (x n ) in E and every bounded sequence (φ n ) in C(K) , we have lim n→∞ T(φ n x n )=0 . In that same paper they posed the problem of characterizing those Banach spaces E such that, for all compact Hausdorff spaces K , every weakly compact operator on C(K,E) is almost Dunford-Pettis. In the paper under review the author shows that such spaces are precisely those with the Dunford-Pettis property. In particular, the main result of the paper is that the following conditions are equivalent for a Banach space E : (a) For any compact Hausdorff space K , every weakly compact operator on C(K,E) is almost Dunford-Pettis; (b) every weakly compact operator on C([0,1],E) is almost Dunford-Pettis; (c) every weakly compact operator from C([0,1],E) into c 0 is almost Dunford-Pettis; (d) E has the Dunford-Pettis property."