Local spectra and individual stability of uniformly bounded 𝐶₀-semigroups

Batty, Charles; van Neerven, Jan; Räbiger, Frank

doi:10.1090/S0002-9947-98-01919-9

Local spectra and individual stability of uniformly bounded $C_0$-semigroups
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by Charles J. K. Batty, Jan van Neerven and Frank Räbiger

Trans. Amer. Math. Soc. 350 (1998), 2071-2085

DOI: https://doi.org/10.1090/S0002-9947-98-01919-9

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Abstract:

We study the asymptotic behaviour of individual orbits $T(\cdot )x$ of a uniformly bounded $C_{0}$-semigroup $\{T(t) \}_{t\ge 0}$ with generator $A$ in terms of the singularities of the local resolvent $(\lambda -A)^{-1} x$ on the imaginary axis. Among other things we prove individual versions of the Arendt-Batty-Lyubich-Vũ theorem and the Katznelson-Tzafriri theorem.

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