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A Local Ratio Theorem

Published online by Cambridge University Press:  20 November 2018

M. A. Akcoglu and
R. V. Chacon
M. A. Akcoglu
Affiliation:
University of Toronto, Toronto, Ontario
R. V. Chacon
Affiliation:
University of Minnesota, Minneapolis, Minnesota
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Let Tt, t > 0, be a strongly continuous semigroup of positive linear contractions on the L 1-space of a σ-finite measure space . We denote the integral ∫0 t T s ƒ ds, ƒ ∈ L 1, by S 0t ƒ, which is denned as the limit of Riemann sums, in the norm topology of L 1. It is easy to see that, given ƒ ∈ L 1 +, there exists a function F on the product space X× (0, ∞), measurable with respect to the usual product σ-field, such that for every t ≧ 0, ∫0tF(·, s) ds gives a representation of S 0t ƒ. We write S 0t ƒ(x) for ∫0t F(x, S) ds, with a fixed choice of F.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970, pp. 545 - 552
Copyright
Copyright © Canadian Mathematical Society 1970

References

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