On the spectral mapping theorem for perturbed strongly continuous semigroups

We consider a strongly continuous semigroup $(T(t))_{t \geqq 0}$ with generator A on a Banach space X, an A-bounded perturbation B, and the semigroup $(S(t))_{t \geqq 0}$ generated by A + B. Using the critical spectrum introduced recently, we improve existing spectral mapping theorems for the perturbed semigroup $(S(t))_{t \geqq 0}$ .