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Semigroups of Operators in C(S)

Published online by Cambridge University Press:  20 November 2018

F. Dennis Sentilles
F. Dennis Sentilles*
Affiliation:
University of Missouri, Columbia, Missouri
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Our study in this paper is two-fold: One is that of a semigroup of linear operators on the space C(S) of bounded continuous functions on a locally compact Hausdorff space S, while the other is that of a transition function of measures in the Banach space M(S) of bounded regular Borel measures on S. It will be seen that an informative and essentially non-restrictive theory of the former can be obtained when C(S) is given the strict topology rather than the usual supremum norm topology and that, in this setting, the natural relationship between semigroups and transition functions obtained when S is compact is maintained, essentially because the dual of C(S) with the strict topology is M(S).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 47 - 54
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Buck, R. C., Bounded continuous junctions on a locally compact space, Michigan Math. J. 5 (1958), 95–104.Google Scholar
2. Conway, J. B., The strict topology and compactness in the space of measures, Trans. Amer. Math. Soc. 126 (1967), 474–486.Google Scholar
3. Dorroh, J. R., Semigroups of maps in a locally compact space, Can. J. Math. 19 (1967), 688–696.Google Scholar
4. Dorroh, J. R., Localization of the strict topology via bounded sets, Proc. Amer. Math. Soc. 20 (1969), 413–414.Google Scholar
5. Dunford, N. and Schwartz, J. T., Linear operators. I (Interscience, New York, 1958).Google Scholar
6. Dynkin, E. B., Markov processes (Springer, Berlin, 1961).Google Scholar
7. Komatsu, H., Semigroups of operators in a locally convex space, J. Math. Soc. Japan 16 (1964), 230–262.Google Scholar
8. Meyer, P. A., Probability and potentials (Blaisdell, Waltham, Massachusetts, 1966).Google Scholar
9. Neveu, J., Mathematical foundations of the theory of probability (Holden-Day, San Francisco, 1965).Google Scholar
10. Sentilles, F. D., Kernel representations of operators and their adjoints, Pacific J. Math. 23 (1967), 153–162.Google Scholar
11. Singbal-Vedak, K., A note on semigroups of operators on a locally convex space, Proc. Amer. Math. Soc. 16 (1965), 696–702.Google Scholar
12. Yosida, K., Functional analysis (Springer, Berlin, 1966).Google Scholar