Let

$\left\{T\right(t):t\ge 0\}$ 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].