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Proceedings of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9939-1992-1123648-3
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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Keywords: Asymptotic periodicity, positive operator, Riesz Decomposition Property
Article copyright: © Copyright 1992 American Mathematical Society