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On the reciprocal Dunford-Pettis property in projective tensor products

Published online by Cambridge University Press:  24 October 2008

G. Emmanuele
Affiliation:
Department of Mathematics, University of Catania, Italy

Abstract

We prove the following result: if a Banach space E does not contain l1 and F has the (RDPP), then EnF has the same property, provided that L(E, F*) = K(E, F*). Hence we prove that if En F has the (RDPP) then at least one of the spaces E and F must not contain l1. Some corollaries are then presented as well as results concerning the necessity of the hypothesis L(E, F*) = K(E, F*).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Cembranos, P.. On Banach spaces of vector valued continuous functions. Bull. Austr. Math. Soc. 28 (1983), 175186.CrossRefGoogle Scholar
[2]Dubinski, E., Pelczynski, A. and Rosenthal, H. P.. On Banach spaces X for which Π2(ℒ, X) = B(ℒ, X). Studia. Math. 44 (1972), 617648.CrossRefGoogle Scholar
[3]Emmanuele, G.. A dual characterization of Banach spaces not containing l 1. Bull. Polish Acad. Sci. Math. 34 (1986), 155160.Google Scholar
[4]Emmanuele, G.. On the Banach spaces with the property (V*) of Pelczynski. II. Ann. Mat. Pura Appl. (To appear.)Google Scholar
[5]Emmanuele, G.. On the containment of c 0 by spaces of compact operators. Bull. Sci. Math. (To appear.)Google Scholar
[6]Emmanuele, G.. Dominated operators on C[0, 1] and the (CRP). Collect. Math. (To appear.)Google Scholar
[7]Godefroy, G. and Saphar, P.. Duality in spaces of operators and smooth norms on Banach spaces. Illinois J. Math. 32 (1988), 672695.CrossRefGoogle Scholar
[8]Grothendieck, A.. Sur les applicationes lineaires faiblement compactes d'éspace du type C(K). Canad. J. Math. 5 (1953), 129173.CrossRefGoogle Scholar
[9]Kalton, N. J., Saab, E. and Saab, P.. On the Dieudonné property for C(Ω,E). Proc. Amer. Math. Soc. 96 (1986), 5052.Google Scholar
[10]Leavelle, T.. The Reciprocal Dunford–Pettis Property. (To appear.)Google Scholar
[11]Lewis, D. R.. Conditional weak compactness in certain inductive tensor products. Math. Ann. 201 (1973), 201209.CrossRefGoogle Scholar
[12]Niculescu, C.. Weak compactness in Banach lattices. J. Operator Theory 9 (1981), 217231.Google Scholar
[13]Pelczynski, A.. Banach spaces on which every unconditionally converging operator is weakly compact. Bull. Polish Acad. Sci. Math. 10 (1962), 641648.Google Scholar
[14]Pisier, G., Factorization of Linear Operators and Geometry of Banach Spaces. CBMS Regional Conf. Series in Math. no. 60 (American Mathematical Society, 1986).CrossRefGoogle Scholar
[15]Ruess, W.. Duality and geometry of spaces of compact operators. In Functional Analysis: Surveys and Recent Results 3, Studies in Math, and its Applications no. 90 (North-Holland, 1984).Google Scholar
[16]Tzafriri, L.. Reflexivity in Banach lattices and their subspaces. J. Funct. Anal. 10 (1972), 118.CrossRefGoogle Scholar