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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

46H05 General theory of topological algebras
40E05 Tauberian theorems
47D03 Groups and semigroups of linear operators
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