We consider a strongly continuous semigroup (T (t)) t#0 with generator A on a Banach space X, an A-bounded perturbation B, and the semigroup (S(t)) t#0 generated by A + B. Using the critical spectrum introduced recently, we improve existing spectral mapping theorems for the perturbed semigroup (S(t)) t#0 . The results are applied to a cell equation with age structure and spatial distribution. 1
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