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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Riesz decomposition property implies asymptotic periodicity of positive and constrictive operators


Author: Wojciech Bartoszek
Journal: Proc. Amer. Math. Soc. 116 (1992), 101-111
MSC: Primary 47B65; Secondary 46B40, 47A35, 47B60
MathSciNet review: 1123648
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Abstract: Consider a linear and positive operator $ {\mathbf{T}}$ acting on an ordered, $ F$-normed linear space $ {\mathbf{X}}$. Assume that there exists an open neighborhood $ {\mathbf{U}} \ni {\mathbf{0}}$ such that the trajectory $ \left\{ {{{\mathbf{T}}^n}({\mathbf{x}})} \right\}$ is attracted to a compact set $ {{\mathbf{F}}_{\mathbf{U}}}$ whenever $ {\mathbf{x}}$ is taken from $ {\mathbf{U}}$ and that the positive cone $ {{\mathbf{X}}_ + }$ is closed, proper, and reproducing. It is shown that if $ ({\mathbf{X}},{{\mathbf{X}}_ + })$ has the Riesz Decomposition Property then $ {\mathbf{T}}$ has asymptotically periodic iterates.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1123648-3
PII: S 0002-9939(1992)1123648-3
Keywords: Asymptotic periodicity, positive operator, Riesz Decomposition Property
Article copyright: © Copyright 1992 American Mathematical Society