Publication: Copies of l∞ in Lp(μ;X).
Loading...
Official URL
Full text at PDC
Publication Date
1990-05
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
American Mathematical Society
Citations
Exportar
Abstract
Let (Ω,Σ,μ) be any measure space, X a Banach space and for 1≤p<+∞ let Lp(μ,X) be the Banach space of all X-valued Bochner "pth power integrable'' functions on Ω, with the usual "Lp''-norm. A natural question is: do properties enjoyed by Lp(μ,X) "descend'' to X? In the present paper it is proved that Lp(μ,X) contains l∞ isomorphically (if and) only if X does.
In a sense, the author's result completes earlier ones [e.g., S. Kwapien, Studia Math. 52 (1974), 187–188; G. Pisier, C. R. Acad. Sci. Paris Sér. A 286 (1978), no. 17, 747–749; ; L. Drewnowski, "Copies of l∞ in the operator spaces Kω∗(X∗,Y)'', to appear].
The proof of the theorem is achieved by applying three earlier results; one is from the paper of Drewnowski [op. cit.], and the other two from a paper by H. P. Rosenthal [Studia Math. 37 (1970), 13–36].
Another recent paper by Drewnowski ["When does ca(Σ,X) contain a copy of l∞ or c0?'', Proc. Amer. Math. Soc., to appear] is also relevant to the present paper.
Description
UCM subjects
Unesco subjects
Keywords
Citation
J. Diestel and J. J. Uhl, Jr., Vector measures, Math. Surveys No. 15, Amer. Math. Soc., Providence, RI, 1977.
L. Drewnowski, Copies of l∞ in the operator space Kw∗(X∗,Y), (to appear).
N. Dunford and J. T. Schwartz, Linear operators, vol. I. New York, Interscience, 1955.
N. J. Kalton, Spaces of compact operators, Math. Ann. 208 (1974), 267-278.
S. Kwapien, Sur les espaces de Banach contenant c0, Studia Math. 52 (1974), 187-188.
S. Lang, Analysis II, Addison-Wesley, Reading, MA, 1969.
G. Pisier, Une propriété de stabilité de la classe des espaces ne contenant pas l1, C. R. Acad. Sci. Paris Sér. A 286 (1978), 747-749.
H. P. Rosenthal, On relatively disjoint families of measures with some applications to Banach space theory, Studia Math. 37 (1970), 13-16