The main result is the following stability theorem: Let T = (T(t))t 0 be a bounded C₀-semigroup on a reflexive space X. Denote by A the generator of T and by σ (A) the spectrum of A. If σ (A) ∩ iR is countable and no eigenvalue of A lies on the imaginary axis, then lim t → ∞T(t)x = 0 for all x ∈ X.
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Arendt et al. (1988) studied this question.