On dominant contractions and a generalization of the zero–two law

On dominant contractions and a generalization of the zero–two law Zaharopol proved the following result: let $${T,S:L^1(X,{\mathcal{F}},\mu)\to L^1(X, {\mathcal{F}},\mu)}$$ be two positive contractions such that T ≤ S. If $${\|S-T\| <1 }$$ then $${\left\|S^n-T^n\right\| <1 }$$ for all $${n\in\mathbb{N}}$$ . In the present paper we generalize this result to multi-parameter contractions acting on L 1. As an application of that result we prove a generalization of the “zero–two” law. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

On dominant contractions and a generalization of the zero–two law

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Publisher
SP Birkhäuser Verlag Basel
Copyright
Copyright © 2010 by Springer Basel AG
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Econometrics; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-010-0102-8
Publisher site
See Article on Publisher Site

Abstract

Zaharopol proved the following result: let $${T,S:L^1(X,{\mathcal{F}},\mu)\to L^1(X, {\mathcal{F}},\mu)}$$ be two positive contractions such that T ≤ S. If $${\|S-T\| <1 }$$ then $${\left\|S^n-T^n\right\| <1 }$$ for all $${n\in\mathbb{N}}$$ . In the present paper we generalize this result to multi-parameter contractions acting on L 1. As an application of that result we prove a generalization of the “zero–two” law.

Journal

PositivitySpringer Journals

Published: Nov 17, 2010

References

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