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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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König chains for submultiplicative functions and infinite products of operators
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by Jacek Jachymski PDF
Trans. Amer. Math. Soc. 361 (2009), 5967-5981 Request permission

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
  • 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