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Continuous linear operators on C(K, X) and pointwise weakly precompact subsets of C(K, X)

Published online by Cambridge University Press:  24 October 2008

A. lger
A. lger
Affiliation:
Department of Mathematics, Bogazici University, 80815 Bebek, Istanbul, Turkey
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Abstract

Let K be a compact Hausdorif space, X a Banach space and C(K, X) the Banach space of all continuous functions : KX equipped with the supremum norm. A subset H of C(K, X) is pointwise weakly precompact if, for each t in K, the set Ht) = {(t):H} is weakly precompact. In this note we study the images of a bounded pointwise weakly precompact subset H of C(K, X) under several classes of linear operators on C(K, X).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 111 , Issue 1 , January 1992 , pp. 143 - 150
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

1Batt, J. and Berg, E. J.. Linear bounded transformations on the space of continuous functions. J. Fund. Anal. 4 (1969), 215239.CrossRefGoogle Scholar
2Bonbal, F. and Cembranos, P.. Characterization of some classes of operators on spaces of vector valued functions. Math. Proc. Cambridge Philos. Soc. 97 (1985), 137146.CrossRefGoogle Scholar
3Bonbal, F. and Cembranos, P.. The Dieudonn property on C(K, X). Trans. Amer. Math. Soc. 285 (1984), 649656.Google Scholar
4Brooks, J. K. and Lewis, P. W.. Linear operators and vector measures. Trans. Amer. Math. Soc. 192 (1974), 139162.CrossRefGoogle Scholar
5Cembranos, P., Kalton, N. J., Saab, E. and Saab, P.. Pelczynski's property (V) on C(K, E) spaces. Math. Ann. 271 (1985), 9197.CrossRefGoogle Scholar
6Diestel, J.. Sequences and Series in Banach Spaces. Graduate Texts in Math. no. 92 (Springer-Verlag, 1984).CrossRefGoogle Scholar
7Diestel, J. and Uhl, J. J. Jr., Vector Measures. Math. Surveys no. 15 (American Mathematical Society, 1977).CrossRefGoogle Scholar
8Dinculeanu, N.. Vector Measures (Pergamon Press, 1967).CrossRefGoogle Scholar
9Dobrakov, I.. On representation of linear operators on C 0(K, X). Czechoslovak Math. J. 21 (1971), 1330.CrossRefGoogle Scholar
10Emmanuele, G.. Another proof of a result of N. J. Kalton, E. Saab and P. Saab on the Dieudonn property in C(K, E). Glasgow Math. J. 31 (1989), 137140.CrossRefGoogle Scholar
11Grothendieck, A.. Sur les applications linaires faiblement compactes d'spaces du type C(K). Canad. J. Math. 5 (1955), 129173.CrossRefGoogle Scholar
12James, R. C.. Weakly compact sets. Trans. Amer. Math. Soc. 13 (1964), 129140.CrossRefGoogle Scholar
13Kalton, N. J., Saab, E. and Saab, P.. On the Dieudonne property for C(,E). Proc. Amer. Math. Soc. 96 (1986), 5052.Google Scholar
14Kalton, N. J., Saab, E. and Saab, P.. L p(X)(1 p < ) has the property (u) whenever X does. Bull. Sci. Math. France, to appear.Google Scholar
15Lindenstrauss, J. and Tzafriri, L.. Classical Banach Spaces, vol. 2. Ergebnisseder Mathematik Grenzgebiete no. 97 (Springer-Verlag, 1979).CrossRefGoogle Scholar
16Pelczynski, A.. Banach spaces on which every unconditionally converging operator is weakly compact. Bull. Acad. Pol. Sci. 10 (1962), 641648.Google Scholar
17Riddle, L. H., Saab, E. and Uhl, J. J. Jr., Sets with the weak Radon-Nikodym property in dual Banach spaces. Indiana Univ. Math. J. 32 (1983), 527541.CrossRefGoogle Scholar
18Rosenthal, H. P.. A characterization of Banach spaces containing l 1. Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 24112413.CrossRefGoogle Scholar
19Talagrand, M.. Weak Cauchy sequences in L 1(E). Amer. J. Math. 106 (1984), 703724.CrossRefGoogle Scholar
20lger, A.. Weak compactness in L l(,X). Proc. Amer. Math. Soc., to appear.Google Scholar