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We consider a bounded linear operator $T$ on a complex Banach space $X$ andshow that its spectral radius $r(T)$ satisfies $r(T) < 1$ if all sequences $(<x',T^nx>)_{n \in \mathbb{N}_0}$ ($x \in X$, $x' \in X'$) are, up to a certainrearrangement, contained in a principal ideal of the space $c_0$ of sequenceswhich converge to $0$. From this result we obtain generalizations of theoremsof G. Weiss and J. van Neerven. We also prove a related result on$C_0$-semigroups.

On Weak Decay Rates and Uniform Stability of Bounded Linear Operators