# zbMATH — the first resource for mathematics

Power-dominated elements in a Banach algebra. (English) Zbl 0705.46021
Authors’ abstract: Let $$(A,\|\|)$$ be a Borel algebra. Using only elementary estimates and a simple renorming technique, we give short new proofs of theorems of Gelfand and of Katznelson and Tzafriri on power- bounded elements of A. Combining these ideas with a version of the theorem of support leads to the following quantitative generalization, valid for a wide range of functions f:
if $$x\in A$$ and $$\{\mu (n)\}_{n\geq 0}$$ is a sequence such that $$\| x^ n\| \leq \mu (n)$$ (n$$\geq 0)$$ and $$\lim_{n\to \infty}\mu (n+1)/\mu (n)=1,$$ then $$\limsup_{n\to \infty}\| f(x)x^ n\| /\mu (n)$$ is majorized by a bound depending only on f and the spectrum of x (not on A). Some examples are then considered and we finish by proving a partial converse.
Reviewer: J.B.Cooper

##### MSC:
 46H05 General theory of topological algebras
##### Keywords:
renorming; power-bounded elements; theorem of support
Full Text: