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Tauberian theorems and stability of one-parameter semigroups. (English) Zbl 0652.47022
Let $\{T(t):t\ge 0\}$ be a bounded $C\sb 0$-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 $A\sp*$. Then, for every vector x, we have $T(t)x\to 0$ 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 {\it Yu. I. Lyubich} and {\it 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 {\it G. R. Allan} and {\it T. J. Ransford} “Power- dominated elements in a Banach algebra” [Stud. Math. 94 (1989), to appear].
Reviewer: J.Zemánek

47D03(Semi)groups of linear operators
47A10Spectrum and resolvent of linear operators
44A10Laplace transform
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