Differentiability of perturbed semigroups and delay semigroups
Suppose that $A$ generates a $C_0$-semigroup $T$ on a Banach space $X$. In 1953 R. S. Phillips showed that, for each bounded operator $B$ on $X$, the perturbation $A+B$ of $A$ generates a $C_0$-semigroup on $X$, and he considered whether certain classes of semigroups are stable under such perturbations. This study was extended in 1968 by A. Pazy who identified a condition on the resolvent of $A$ which is sufficient for the perturbed semigroups to be immediately differentiable. However, M. Renardy showed in 1995 that immediate differentiability is not stable under bounded perturbations. We give a survey account of the partial answers already given to the question of differentiability of perturbed semigroups. Furthermore, we show that Pazy's condition is necessary, as well as sufficient, if one adds a natural requirement of uniformity for the differentiability of the perturbed semigroups. We also present an account of the corresponding theory for delay semigroups associated with $A$, based on an earlier paper of ours but with improved formulation. The necessary and sufficient condition for eventual differentiability of the delay semigroups is that the resolvent of $A$ should have polynomial decay on vertical lines. We also give a brief account of the consequences for asymptotics of individual mild solutions of abstract Cauchy problems and delay differential equations.