## König chains for submultiplicative functions and infinite products of operators

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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 $C[0,1]$.## References

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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
- Received by editor(s): October 5, 2007
- Published electronically: June 23, 2009
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2529921