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The compact range property and C0

Published online by Cambridge University Press:  18 May 2009

Neil E. Gretsky  and
Joseph M. Ostroy
Neil E. Gretsky
Affiliation:
Department of Mathematics, University of California, Riverside, California 92521
Joseph M. Ostroy
Affiliation:
Department of Economics, University of California, Los Angeles, California 90024
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The purpose of this short note is to make an observation about Dunford–Pettis operators from L1[0, 1] to C0. Recall that an operator T:E→F (where E and F are Banach spaces) is called Dunford–Pettis if T takes weakly convergent sequences of E into norm convergent sequences of F. A Banach space F has the Compact Range Property (CRP) if every operator T:L1]0, 1]→F is Dunford–Pettis. Talagrand shows in his book [2] that C0 does not have the CRP. It is of interest (especially in mathematical economics [3]) to note that every positive operator from L1[0, 1] to C0 is Dunford–Pettis.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 28 , Issue 1 , January 1986 , pp. 113 - 114
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

REFERENCES

1.Bourgain, J., Dunford–Pettis operator on L1 and the Radon–Nikodyn property, Israel J. Math. 37 (1980), 34–27.CrossRefGoogle Scholar
2.Talagrand, M., The Pettis integral, Mem. Amer. Math. Soc. No. 307, (Rhode Island, 1984).Google Scholar
3.Gretsky, N. E. and Ostroy, J. M., Thick and thin market non-atomic exchange economies, in Advances in Equilibrium Theory, Lecture Notes in Economics and Mathematical Systems No. 244 (1985), 107–130.CrossRefGoogle Scholar