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König chains for submultiplicative functions and infinite products of operators


Author: Jacek Jachymski
Journal: Trans. Amer. Math. Soc. 361 (2009), 5967-5981
MSC (2000): Primary 47A35, 46H05, 47B38; Secondary 15A60, 26B35, 54D30, 54D20.
DOI: https://doi.org/10.1090/S0002-9947-09-04909-5
Published electronically: June 23, 2009
MathSciNet review: 2529921
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Abstract: We generalize the so-called Weighted König Lemma, due to Máté, for a submultiplicative function on a subset of the union $ \bigcup_{n\in\mathbb{N}}\Sigma^n$, where $ \Sigma$ is a set and $ \Sigma^n$ is the Cartesian product of $ n$ copies of $ \Sigma$. Instead of a combinatorial argument as done by Máté, our proof uses Tychonoff's compactness theorem to show the existence of a König chain for a submultiplicative function. As a consequence, we obtain an extension of the Daubechies-Lagarias theorem concerning a finite set $ \Sigma$ of matrices with right convergent products: Here we replace matrices by Banach algebra elements, and we substitute compactness for finiteness of $ \Sigma$. The last result yields new generalizations of the Kelisky-Rivlin theorem on iterates of the Bernstein operators on the Banach space $ C[0,1]$.


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Additional Information

Jacek Jachymski
Affiliation: Institute of Mathematics, Technical University of Łódź, Wólczańska 215, 93-005 Łódź, Poland
Email: jachym@p.lodz.pl

DOI: https://doi.org/10.1090/S0002-9947-09-04909-5
Keywords: Submultiplicative function, joint spectral radius, K\"onig chain, Tychonoff's compactness theorem, infinite products of operators, Hutchinson system, Bernstein operators
Received by editor(s): October 5, 2007
Published electronically: June 23, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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