This is the abstract of the paper
  "On the asymptotic behaviour of perturbed semigroups"
by Valentina Casarino and Susanna Piazzera.

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too.

Abstract: We give conditions on a strongly continuous semigroup $\T$
and a bounded perturbation $B$ such that the
perturbed semigroup $\Se$ preserves asymptotic properties as
boundedness, asymptotic almost periodicity, uniform ergodicity
and total uniform ergodicity.