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An elementary proof of the Birkhoff-Hopf theorem

Published online by Cambridge University Press:  24 October 2008

Simon P. Eveson  and
Roger D. Nussbaum
Simon P. Eveson
Affiliation:
Department of Mathematics, University of York, Heslington, York, YO1 5DD
Roger D. Nussbaum
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903, U.S.A.
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Extract

In important work some thirty years ago, G. Birkhoff[2, 3] and E. Hopf [16, 17] showed that large classes of positive linear operators behave like contraction mappings with respect to certain ‘almost’ metrics. Hopf worked in a space of measurable functions and took as his ‘almost’ metric the oscillation ω(y/x) of functions y and x with x(t) > 0 almost everywhere, defined by

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 1 , January 1995 , pp. 31 - 55
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

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