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Banach spaces with property (w)

Published online by Cambridge University Press:  18 May 2009

Denny H. Leung
Denny H. Leung
Affiliation:
Department of Mathematics, National University of Singapore, Singapore0511
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A Banach space E is said to have Property (w) if every operator from E into E' is weakly compact. This property was introduced by E. and P. Saab in [9]. They observe that for Banach lattices, Property (w) is equivalent to Property (V*), which in turn is equivalent to the Banach lattice having a weakly sequentially complete dual. Thus the following question was raised in [9].

Does every Banach space with Property (w) have a weakly sequentially complete dual, or even Property (V*)?

In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type spaces considered in [1]. They both possess properties stronger than Property (w). The first example has the property that every operator from the space into the dual is compact. In the second example, both the space and its dual have Property (w). In the last section we establish some partial results concerning the problem (also raised in [9]) of whether (w) passes from a Banach space E to C(K, E).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 35 , Issue 2 , May 1993 , pp. 207 - 217
Copyright
Copyright © Glasgow Mathematical Journal Trust 1993

References

REFERENCES

1.Bellenot, Steven F., Haydon, Richard and Odell, Edward, Quasi-reflexive and tree spaces constructed in the spirit of R. C. James, Contemporary Math. 85 (1989), 19–43.CrossRefGoogle Scholar
2.Brooks, J. and Lewis, P., Linear operators and vector measures, Trans. Amer. Math. Soc. 192 (1974), 139–162.CrossRefGoogle Scholar
3.Cembranos, P., On Banach spaces of vector valued continuous functions, Bull. Austral. Math. Soc. 28 (1983), 175–186.CrossRefGoogle Scholar
4.Diestel, J. and Uhl, J. J. Jr, Vector measures, Math. Surveys, No. 15 (Amer. Math. Soc, Providence, 1977).CrossRefGoogle Scholar
5.Dobrakov, I., On representation of linear operators in CO(T, X), Czechoslovak. Math. J. 21 (1971), 13–30.CrossRefGoogle Scholar
6.Lindenstrauss, Joram and Tzafriri, Lior, Classical Banach spaces I, Sequence spaces (Springer-Verlag, 1977).Google Scholar
7.Pelczynski, A., On Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Acad. Polon. Sci. 10 (1962), 641–648.Google Scholar
8.Saab, E. and Saab, P., A stability property of a class of Banach spaces not containing a complemented copy of l 1, Proc. Amer. Math. Soc. 84 (1982), 44–46.Google Scholar
9.Saab, E. and Saab, P., Extensions of some classes of operators and applications, preprint.Google Scholar
10.Bombal, F. and Cembranos, P., Characterization of some classes of operators on spaces of vector-valued continuous functions, Math. Proc. Cambridge Philos. Soc. 97 (1985), 137–146.CrossRefGoogle Scholar