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Analytic semigroups in Banach algebras and a theorem of Hille. (English) Zbl 0632.46043
Let A be a Banach algebra. An analytic semigroup in A is a function $$t\to y_ t$$ from the right open half plane H into A which is analytic and $$y_{m+t}=y_ my_ t$$ for all s,t$$\in H$$. The author proves the following theorem:
Theorem 1: Assume that an element y in a Banach algebra satisfies $$\| u(ny)\| =O(| n|^ r)$$ as $$| n| \to \infty$$ where $$u(y)=(\exp -1)(y)$$. Then there exists an analytic semigroup $$(y_ t)$$ in A, such that $$\sup \{| t|^{-k}\| y_ t\|$$, Re $$t\geq 1\}<+\infty$$ for some $$k=k(r)$$ and $$\lim_{t\to 0,t>0}\| y_ ty^ k-y^ k\| =0.$$
Using a theorem of J. Esterle [see Theorem 2.1 in Ann. Institut Fourier, Grenoble 30, 91-96 (1980; Zbl 0419.40005)], the author obtains thus a characterization of nilpotent elements in radical algebras. Some corollaries of this type, all closely related to a theorem of E. Hille [see E. Hille and R. S. Phillips, Functional analysis and semigroups (1957; Zbl 0078.100), theorem 4.10.1] are discussed in the last part of the paper.
Reviewer: J.Ludwig

##### MSC:
 46H05 General theory of topological algebras 40E05 Tauberian theorems 47D03 Groups and semigroups of linear operators
##### Citations:
Zbl 0419.40005; Zbl 0078.100
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