König chains for submultiplicative functions and infinite products of operators

Jachymski, Jacek

doi:10.1090/S0002-9947-09-04909-5

König chains for submultiplicative functions and infinite products of operators
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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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