On Weak Decay Rates and Uniform Stability of Bounded Linear Operators

We consider a bounded linear operator $T$ on a complex Banach space $X$ and show 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 certain rearrangement, contained in a principal ideal of the space $c_0$ of sequences which converge to $0$. From this result we obtain generalizations of theorems of G. Weiss and J. van Neerven. We also prove a related result on $C_0$-semigroups.