Abstract
The purpose of this paper is to revisit previous works of the author with Helffer and Sjöstrand (arXiv:1001.4171v1. 2010; Int Equ Op Theory 93(3):36, 2021) on the stability of semigroups which were proved in the Hilbert case by considering the Banach case at the light of a paper by Latushkin and Yurov (Discrete Contin Dyn Syst 33:5203–5216, 2013).
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Notes
The authors consider the case when \(m(t):= Le^{\lambda t} \) and prove instead a consequence of this theorem as in [6].
By a density argument, we can replace \(C^1([0,+\infty [)\) in (6) by the space of locally Lipschitz functions on \([0,+\infty [\).
We thank J. Rozendahl for this remark.
The more common notation is \({\mathcal {B}}'\).
In [7], we were using instead Cauchy–Schwarz.
A scaling can be used in order to have this additional property. See also the last paragraph of Sect. 1.
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Acknowledgements
The author was motivated by a question of L. Boulton at the Banff conference (2022) and encouraged later by discussions with Y. Latushkin during the Aspect Conference in Oldenburg (2022) organized by K. Pankrashkin. We thank J. Rozendaal for comments on a previous version of the manuscript and Y. Latushkin for his careful reading and suggestions of simplification. Finally, we thank the referee for his useful remarks.
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Helffer, B. Stability estimates for semigroups in the Banach case. J. Evol. Equ. 24, 29 (2024). https://doi.org/10.1007/s00028-024-00958-7
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DOI: https://doi.org/10.1007/s00028-024-00958-7