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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Asymptotic periodicity of the iterates of positivity preserving operators


Author: M. Miklavčič
Journal: Trans. Amer. Math. Soc. 307 (1988), 469-479
MSC: Primary 47B55; Secondary 47A35
MathSciNet review: 940213
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Abstract: Assume that

(A1) $ X$ is a real Banach space.

(A2) $ {X^ + }$ is a closed subset of $ X$ with the following properties:

(i) if $ x \in {X^ + }$, $ y \in {X^ + }$, $ \alpha \in [0,\,\infty )$ then $ x + y \in {X^ + }$ and $ \alpha x \in {X^ + }$;

(ii) there exists $ {M_0} \in (0,\,\infty )$ such that for each $ x \in X$ there exist $ {x_ + } \in {X^ + }$ and $ {x_ - } \in {X^ + }$ which satisfy

$\displaystyle x = {x_ + } - {x_ - },\qquad \vert\vert{x_ + }\vert\vert \leqslan... ...t\vert,\qquad \vert\vert{x_ - }\vert\vert \leqslant {M_0}\vert\vert x\vert\vert$

and if $ x = {y_ + } - {y_ - }$ for some $ {y_ + } \in {X^ + }$, $ {y_ - } \in {X^ + }$ then $ {y_ + } - {x_ + } \in {X^ + }$;

(iii) if $ x \in {X^ + }$, $ y \in {X^ + }$ then $ \vert\vert x\vert\vert \leqslant \vert\vert x + y\vert\vert$.

(A3) $ B$ is a bounded linear operator on $ X$.

(A4) $ B{X^ + } \subset {X^ + }$.

(A5) $ {F_0}$ is a nonempty compact subset of $ X$ and $ {\lim _{n \to \infty }}\operatorname{dist} ({B^n}x,\,{F_0}) = 0$ whenever $ x \in {X^ + }$ and $ \vert\vert x\vert\vert = 1$.

Then $ {B^n}x$ is asymptotically periodic for every $ x \in X$. This, and other properties of $ B$, are proven in the paper.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0940213-2
PII: S 0002-9947(1988)0940213-2
Article copyright: © Copyright 1988 American Mathematical Society