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

 

Strongly asymptotically stable Frobenius-Perron operators


Author: Radu Zaharopol
Journal: Proc. Amer. Math. Soc. 128 (2000), 3547-3552
MSC (2000): Primary 47A35; Secondary 28D99, 37A30, 37A40, 47B38, 47B65
Published electronically: May 18, 2000
MathSciNet review: 1691011
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Abstract:

Let $(X,\Sigma,\mu)$ be a $\sigma$-finite measure space and let $T : L^1(X,\Sigma,\mu) \to L^1(X,\Sigma,\mu)$ be a Frobenius-Perron operator.

In 1997 Bartoszek and Brown proved that if $T$ overlaps supports and if there exists $h \in L^1(X,\Sigma,\mu)$, $h > 0$ on $X$, such that $Th = h$, then $T$ is (strongly) asymptotically stable.

In the note we prove that instead of assuming that $h > 0$ on $X$, it is enough to assume that $h\geq 0$ and $h\neq 0$. More precisely, we prove that $T$ is asymptotically stable if and only if $T$ overlaps supports and there exists $h\in L^1(X,\Sigma,\mu)$, $h\geq 0$, $h\neq 0$, such that $Th=h$.


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Additional Information

Radu Zaharopol
Affiliation: Department of Mathematical Sciences, Binghamton University (S.U.N.Y. at Binghamton), Binghamton, New York 13902-6000
Email: radu@math.binghamton.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05473-3
PII: S 0002-9939(00)05473-3
Received by editor(s): November 11, 1997
Received by editor(s) in revised form: January 29, 1999
Published electronically: May 18, 2000
Dedicated: Dedicated to Professor Alexandra Bellow in celebration of her achievements in all the aspects of being that involve mathematics
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society