On Weak Decay Rates and Uniform Stability of Bounded Linear Operators

Abstract

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 $(<x^{\prime },T^{n}x>{)}_{n\in {\mathbb{N}}_0}$ ($x\in X$, $x^{\prime }\in X^{\prime }$) 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.