König chains for submultiplicative functions and infinite products of operators

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 $C[0,1]$.