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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
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