Perturbations of generators of C0-semigroups and resolvent decay

We obtain new stability results for those properties of C0-semigroups which admit characterisation in terms of decay of resolvents of infinitesimal generators on vertical lines, e.g. analyticity, Crandall–Pazy differentiability or immediate norm continuity in the case of Hilbert spaces. As a consequence we get a generalisation of the Kato–Neuberger theorem on approximation of the identity. Finally, we present examples shedding a new light on resolvent characterisation of eventually differentiable C0-semigroups for which differentiability is stable under bounded perturbations.