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Fixed Point Theorem for Positive Operators on KB Spaces

Published online by Cambridge University Press:  20 November 2018

Ling Fai Lam
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The existence of nonzero fixed points of positive contractions in L1 spaces has received considerable attention in recent years. In 1966, Dean and Sucheston [1] and independently, Neveu [5] showed that a positive contraction has a strictly positive invariant function if and only if for any measurable subset A with positive measure, where 1 is the constant function of value one.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 6 , 01 December 1976 , pp. 1146 - 1152
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Dean, D. W. and Sucheston, L., On invariant measures for operators, Z. Wahrscheinlichkeitstheorie Verw. Geb. 6 (1966), 1–9.Google Scholar
2. Dunford, N. and Schwartz, J. T., Linear operators I (Interscience Publishers, 1958).Google Scholar
3. Fong, H., On invariant functions for positive operators, Colloquium Math. 22 (1970), 75–84.Google Scholar
4. Lam, L. F., Invariant functions for positive operators on normed Kothe spaces, to appear. Colloquium Math. 80 (1974), 259–267.Google Scholar
5. Neveu, J., Existence of bounded invariant measures in ergodic theory, Proc. Fifth Berkeley Sympos. Math. Statist. Probability II, Part II (1967), 461–472.Google Scholar
6. Sato, R., Ergodic properties of bounded L\ operators, Proceedings Amer. Math. Soc. 30 (1973), 540–546.Google Scholar
7. Vulikh, B. Z., Introduction to the theory of partially ordered spaces (Wolters-N∞rdhoff Scientific Publications Ltd. 1967).Google Scholar