On perturbations preserving the immediate norm continuity of semigroups

We show that the Desch–Schappacher perturbation and the Miyadera–Voigt perturbation of an immediately norm continuous semigroup are immediately norm continuous. We also show that a perturbation theorem of C. Batty, C. Kaiser and L. Weis based on a generation theorem of A.M. Gomilko, D.-X. Feng and D.-H. Shi also preserves the immediate norm continuity of semigroups. The novelty of these results is that, contrary to the numerous related results, we obtain the immediate norm continuity of the perturbed semigroup without additional assumptions.