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STRONGLY BOUNDED REPRESENTING MEASURES AND CONVERGENCE THEOREMS

Published online by Cambridge University Press:  22 March 2010

IOANA GHENCIU
Affiliation:
Mathematics Department, University of Wisconsin-River Falls, Wisconsin, 54022 e-mail: [email protected]
PAUL LEWIS
Affiliation:
University of North Texas, Department of Mathematics, Box 311430 Denton, Texas, 76203-1430 e-mail: [email protected]
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Abstract

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Let K be a compact Hausdorff space, X a Banach space and C(K, X) the Banach space of all continuous functions f: KX endowed with the supremum norm. In this paper we study weakly precompact operators defined on C(K, X).

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

1.Abbott, C., Weakly precompact and GSP operators on continuous function spaces, Bull. Polish Acad. Sci. Math. 37 (1989), 467476.Google Scholar
2.Abott, C., Bator, E. and Lewis, P., Strictly singular and cosingular operators on spaces of continuous functions, Math. Proc. Camb. Phil. Soc. 110 (1991), 505521.CrossRefGoogle Scholar
3.Abott, C., Bator, E., Bilyeu, R. and Lewis, P., Weak precompactness, strong boundedness, and weak complete continuity, Math. Proc. Camb. Phil. Soc. 108 (1990), 325335.CrossRefGoogle Scholar
4.Bartle, R. G., A general bilinear vector integral, Studia Math. 15 (1956), 337352.CrossRefGoogle Scholar
5.Bator, E., Lewis, P. and Ochoa, J., Evaluation maps, restriction maps, and compactness, Colloq. Math. 78 (1998), 117.CrossRefGoogle Scholar
6.Bator, E. and Lewis, P., Operators having weakly precompact adjoints, Math. Nachr. 157 (1992), 99103.CrossRefGoogle Scholar
7.Batt, J. and Berg, E. J., Linear bounded transformations on the space of continuous functions, J. Funct. Anal. 4 (1969), 215239.CrossRefGoogle Scholar
8.Bello, C. F., On weakly compact and unconditionally converging operators in spaces of vector-valued functions, Revista Real Acad. Madrid 81 (1987), 693706.Google Scholar
9.Bessaga, C., Pelczynski, A., On bases and unconditional convergence of series in Banach spaces, Studia Math. 17 (1958), 151164.CrossRefGoogle Scholar
10.Bombal, F., On (V*) sets and Pelczynski's property (V*), Glasgow Math. J. 32 (1990), 109120.CrossRefGoogle Scholar
11.Bombal, F. and Porras, B., Strictly singular and strictly cosingular operators on C(K, E), Math. Nachr. 143 (1989), 355364.CrossRefGoogle Scholar
12.Bourgain, J., An averaging result for ℓ1 sequences and applications to weakly conditionally compact sets in L 1X, Israel J. Math. 32 (1979), 289298.CrossRefGoogle Scholar
13.Bombal, F., Cembranos, P., Characterizations of some classes of operators on spaces of vector-valued continuous functions, Math. Proc. Camb. Phil. Soc. 97 (1985), 137146.CrossRefGoogle Scholar
14.Brooks, J. K. and Lewis, P., Linear Operators and vector measures, Trans. Amer. Math. Soc. 192 (1974), 139162.CrossRefGoogle Scholar
15.Cembranos, P., Kalton, N., Saab, E. and Saab, P., Pelczyinski's Property (V) on C(Ω, E) spaces, Math. Ann. 271 (1985), 9197.CrossRefGoogle Scholar
16.Diestel, J., Sequences and series in Banach spaces, Grad. texts in math., no. 92 (Springer-Verlag, Berlin, 1984).CrossRefGoogle Scholar
17.Diestel, J. and Uhl, J. J. Jr., Vector measures, math. surveys 15 (American Mathematical Society, Rhode Island, 1977.CrossRefGoogle Scholar
18.Dinculeanu, N., Vector measures (Pergamon Press, Oxford, UK, 1967).CrossRefGoogle Scholar
19.Dobrakov, I., On representation of linear operators on C 0(T, X), Czechoslovak. Math. J. 21 (1971), 1330.Google Scholar
20.Dunford, N. and Schwartz, J.T., Linear operators. Part I: General theory (Wiley-Interscience, New Jersey, 1958).Google Scholar
21.Emmanuele, G., Another proof of a result of N. J. Kalton, E. Saab and P. Saab on the Dieudonné property in C(K, E), Glasgow Math. J. 31 (1989), 137140.CrossRefGoogle Scholar
22.Emmanuele, G., On the Banach spaces with property (V*) of Pelczyinksi. II. Ann. Mat. Pura Appl. 160 (1991), 163170.CrossRefGoogle Scholar
23.Gamlen, J. L. B., On a theorem of Pelczyinski, Proc. Amer. Math. Soc. 44 (1974), 283285.Google Scholar
24.Ghenciu, I. and Lewis, P., Almost weakly compact operators, Bull. Polish. Acad. Sci. Math. 54 (2006), 237256.CrossRefGoogle Scholar
25.Grothendieck, A., Sur les applications linéaires faiblement compactes d'espaces du type C(K), Canad. J. Math. 5 (1953), 129173.CrossRefGoogle Scholar
26.Kalton, N., Saab, E. and Saab, P., On the Dieudonné property for C(Ω, E), Proc. Amer. Math. Soc. 96 (1986), 5052.Google Scholar
27.Pelczyinski, A., Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Acad. Polon. Sci. Math. Astronom. Phys. 10 (1962), 641648.Google Scholar
28.Swartz, C., Unconditionally converging and Dunford–Pettis operators on C X(S), Studia Math. 57 (1976), 8590.CrossRefGoogle Scholar
29.Saab, E. and Saab, P., A stability property of Banach spaces not containing a complemented copy of ℓ1, Proc. Amer. Math. Soc. 84 (1982), 4446.Google Scholar
30.Talagrand, M., Weak Cauchy sequences in L 1(E), Amer. J. Math. 106 (1984), 703724.CrossRefGoogle Scholar
31.Talagrand, M., La proprieté de Dunford-Pettis dans C(K, E) et L 1(E), Israel J. Math. 44 (1983), 317321.CrossRefGoogle Scholar
32.Ülger, A., Continuous linear operators on C(K, X) and pointwise weakly precompact subsets of C(K, X), Math. Proc. Camb. Phil. Soc. 111 (1992), 143150.Google Scholar