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 \({(\langle x',T^{n}x\rangle)_{n \in \mathbb{N}_0}}\) (\({x \in X, x' \in X'}\)) 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 Weiss and van Neerven. We also prove a related result on C 0-semigroups.
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During his work on this article the author was supported by a scholarship of the “Landesgraduierten-Förderung Baden-Württemberg”.
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Glück, J. On weak decay rates and uniform stability of bounded linear operators. Arch. Math. 104, 347–356 (2015). https://doi.org/10.1007/s00013-015-0746-5
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DOI: https://doi.org/10.1007/s00013-015-0746-5