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Tauberian theorems and stability of one-parameter semigroups. (English) Zbl 0652.47022
Let $\left\{T\left(t\right):t\ge 0\right\}$ be a bounded ${C}_{0}$-semigroup, with generator A, on a Banach space. Assume that the intersection of the spectrum of A with the imaginary axis is at most countable, containing no eigenvalues of the adjoint ${A}^{*}$. Then, for every vector x, we have $T\left(t\right)x\to 0$ as $t\to \infty$. The authors’ proof is based on Tauberian techniques involving the Laplace transform. A shorter proof of the same result was obtained a year earlier by Yu. I. Lyubich and Vũ Quôc Phóng [Stud. Math. 88, No.1, 37-42 (1988; Zbl 0639.34050)]. On the other hand, the present paper contains some interesting examples and discrete analogues for power bounded operators. A very simple proof of the result of Y.Katznelson and L. Tzafriri, quoted in Theorem 5.6, can be found in a forthcoming paper by G. R. Allan and T. J. Ransford “Power- dominated elements in a Banach algebra” [Stud. Math. 94 (1989), to appear].
Reviewer: J.Zemánek

##### MSC:
 47D03 (Semi)groups of linear operators 47A10 Spectrum and resolvent of linear operators 44A10 Laplace transform