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A note on Dunford-Pettis operators

Published online by Cambridge University Press:  18 May 2009

J. R. Holub
J. R. Holub*
Affiliation:
Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061
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Talagrand has shown [4, p. 76] that there exists a continuous linear operator from L1[0, 1] to c0 which is not a Dunford-Pettis operator. In contrast to this result, Gretsky and Ostroy [2] have recently proved that every positive operator from L[0, 1] to c0 is a Dunford-Pettis operator, hence that every regular operator between these spaces (i.e. a difference of positive operators) is Dunford-Pettis.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 29 , Issue 2 , July 1987 , pp. 271 - 273
Copyright
Copyright © Glasgow Mathematical Journal Trust 1987

References

1. Bourgain, J., Dunford–Pettis operators on L 1 and the Radon–Nikodym property, Israel J. Math. 37 (1980), 27–34.10.1007/BF02762866CrossRefGoogle Scholar
2. Gretsky, N. and Ostroy, J., The compact range property and c 0 , Glasgow Math. J. 28 (1986), 113–114.CrossRefGoogle Scholar
3. Schaefer, H., Topological vector spaces (Macmillan, 1966).Google Scholar
4. Talagrand, M., Pettis integral and measure theory, Mem. Amer. Math. Soc. 307 (1984).Google Scholar