Local spectra and individual stability of uniformly bounded $C_0$-semigroups

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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