We construct a bounded, uniformly continuous function g : [0,infinity) → l 2 with the following properties: This example shows that in the local version of the Arendt-Batty-Lyubich-Vu stability theorem, obtained recently by Batty-van Neerven-Räbiger, the total ergodicity assumption cannot be weakened to ergodicity.
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van Neerven, J. The Vector-valued Loomis Theorem for the Half-line and Individual Stability of C 0-semigroups: A Counterexample . SemiGroup Forum 60, 271–283 (2000). https://doi.org/10.1007/s002339910020
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DOI: https://doi.org/10.1007/s002339910020