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Remarks on the range of a vector measure

Published online by Cambridge University Press:  18 May 2009

Jesús M. F. Castilo  and
Fernando Sánchez
Jesús M. F. Castilo
Affiliation:
Departamento de Matematicas, Universidad de Extremadura, Avda. de Elvas, S/N. 06071 Badajoz, España.
Fernando Sánchez
Affiliation:
Departamento de Matematicas, Universidad de Extremadura, Avda. de Elvas, S/N. 06071 Badajoz, España.
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A long-standing problem is the characterization of subsets of the range of a vector measure. It is known that the range of a countably additive vector measure is relatively weakly compact and, in addition, possesses several interesting properties (see [2]). In [6] it is proved that if m: Σ → Χ is a countably additive vector measure, then the range of m has not only the Banach–Saks property, but even the alternate Banach-Saks property. A tantalizing conjecture, which we shall disprove in this article, is that the range of m has to have, for some p > 1, the p-Banach–Saks property. Another conjecture, which has been around for some time (see [2]) and is also disproved in this paper, is that weakly null sequences in the range of a vector measure admit weakly-2-summable sub-sequences. In fact, we shall show a weakly null sequence in the range of a countably additive vector measure having, for every p < ∞, no weakly-p-summable sub-sequences.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 36 , Issue 2 , May 1994 , pp. 157 - 161
Copyright
Copyright © Glasgow Mathematical Journal Trust 1994

References

REFERENCES

1.Altshuler, Z., Casazza, P. G. and Lin, Bor-Luh, On symmetric basic sequences in Lorentz sequence spaces Israel J. Math. 15 (1973), 140–155.Google Scholar
2.Anantharaman, R. and Diestel, J., Sequences in the range of a vector measure, Comment. Math. Prace. Mat., XXX (1991), 221–235.Google Scholar
3.Casazza, P. G. and Shura, T. J., Tsirelson's space. (Springer-Verlag, LNM 1363, 1989).CrossRefGoogle Scholar
4.Castillo, J. M. F. and Sánchez, F., Remarks on some basic properties of Tsirelson's space, Note di Mat., (to appear).Google Scholar
5.Castillo, J. M. F. and Sánchez, F., Weakly-p-compact, p-Banach-Saks and super-reflexive Banach spaces, J. Math. Anal, and Appl., (to appear).Google Scholar
6.Diestel, J. and Seifert, C. J., an averaging property of the range of a vector measure, Bull. Amer. Math. Soc., 82 (1976), 907–909.CrossRefGoogle Scholar
7.Figiel, T., Johnson, W. B. and Tzafriri, L., On Banach lattices and spaces having local unconditional structure, with applications to Lorentz function spaces, J. Approximation Theory, 13 (1975), 395–412.CrossRefGoogle Scholar
8.Rakov, S. A., Banach-Saks property of a Banach space, Math Notes, 26 (1979) n.6, (1980), 909–916.CrossRefGoogle Scholar
9.Rosenthal, H. P., Some applications of p-summing operators to Banach space theory, Studia Math., 58 (1976), 21–43.CrossRefGoogle Scholar
10.Wojtaszczyk, P., Banach Spaces for Analysts. (Cambridge Studies 25, 1991).CrossRefGoogle Scholar