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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

Keywords:
Submultiplicative function,
joint spectral radius,
König 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.