RT Journal Article
T1 On the Dunford-Pettis property
A1 Bombal Gordón, Fernando
AB "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."
PB Sociedade Portuguesa de Matematica
SN 0032-5155
YR 1988
FD 1988
LK https://hdl.handle.net/20.500.14352/57885
UL https://hdl.handle.net/20.500.14352/57885
LA eng
NO Bombal, F. and Rodriguez-Salinas, B.- Some classes of operators on C(K, E). Extension and applications, Arch. Math., Vol. 47 (1986), 55-65.Bourgain, J. - An averaging result lor Ll-sequences and applications to weakly conditionally compact sets in L1-, Israel J. of Math., Vol. 32 (1979),289-298.Brooks, J.K. and Lewis, P. W. - Linear operators and vector measures, Trans. Amer. Math. Soc., Vol. 192 (1974), 139-162.Diestel, J. - A survey of resulta related to the Dunford-Pettia property, Proceedings 01 the Con/erence on Integration, Topology and Geometry in Linear Spaces, Contemporary Math., Vol. 2, A.M.S., Providence, R.L, 1980.Diestel, J. and Uhl Jr., J.J. - Vector measures, American Mathematical Society's Mathematical Surveys, Vol. 15, Providence, R.L, 1977.Dinculeanu, N. - Vector measures, Pergamon Press, Oxford, 1967.Grothendieck, A. - Sur les applications linéasires laiblement compacts d'espaces du type C(K), Canad. J. of Math., 5 (1953), 129-173.Rosenthal, H. - A characterization 01 Banach spaces containing L1, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411-2413.Seminaire Maurey-Schwartz, 1972-1973 - Ecole Polytechnique, Paris.
NO CAICYT
DS Docta Complutense
RD 28 abr 2024