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′,Tnx⟩)n∈N0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\langle x',T^{n}x\rangle)_{n \in \mathbb{N}_0}}$$\end{document} (x∈X,x′∈X′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x \in X, x' \in X'}$$\end{document}) are, up to a certain rearrangement, contained in a principal ideal of the space c0 of sequences which converge to 0. From this result we obtain generalizations of theorems of Weiss and van Neerven. We also prove a related result on C0-semigroups.