Abstract
Consider a \(C_0\)-semigroup \((e^{tA})_{t \ge 0}\) on a function space or, more generally, on a Banach lattice E. We prove a sufficient criterion for the operators \(e^{tA}\) to be positive for all sufficiently large times t, while the semigroup itself might not be positive. This complements recently established criteria for the individual orbits of the semigroup to become eventually positive for all positive initial values. We apply our main result to study the qualitative behaviour of the solutions to various partial differential equations.
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Acknowledgements
This paper was initiated during a very pleasant stay of the second author at the University of Sydney. Part of the work was done while the second author was financially supported by a scholarship within the scope of the Landesgraduiertenförderung Baden–Württemberg (Grant No. 130l LGFG-E).
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JG was supported by a scholarship within the scope of the LGFG Baden-Württemberg, Germany.
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Daners, D., Glück, J. A Criterion for the Uniform Eventual Positivity of Operator Semigroups. Integr. Equ. Oper. Theory 90, 46 (2018). https://doi.org/10.1007/s00020-018-2478-y
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DOI: https://doi.org/10.1007/s00020-018-2478-y
Keywords
- One-parameter semigroups of linear operators
- Semigroups on Banach lattices
- Eventually positive semigroup