%0 Journal Article
%A Bombal Gordón, Fernando
%T On the Dunford-Pettis property
%D 1988
%@ 0032-5155
%U https://hdl.handle.net/20.500.14352/57885
%X "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."
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