On the Asymptotic Behaviour of Perturbed Semigroups

We give conditions on a strongly continuous semigroup T and a bounded perturbation B such that the perturbed semigroup S preserves asymptotic properties as boundedness, asymptotic almost periodicity, uniform ergodicity and total uniform ergodicity. In the rst systematic treatise on perturbation of semigroups, R.S.Phillips 10] proved in 1953 that, if A is the innnitesimal generator of a strongly continuous semigroup T of linear bounded operators on a Banach space X and B is a linear continuous operator on X, then A + B again generates a strongly continuous semigroup S. This result has been generalized into various directions. On one hand, the hypotheses have been weakened, admitting B to be an unbounded operator. On the other hand, the following question has been studied. Which properties of T are preserved under perturbation? R.S.Phillips proved that immediate norm continuity and immediate compactness of T are inherited by the perturbed semigroup S. This is not the case for eventual norm continuity, eventual compactness and, as shown only in 1995 by M.Renardy 11], for immediate diierentiability. Nonetheless, M.G.Crandall and A.Pazy 4,9] found conditions under which the perturbed semigroup is immediately diierentiable. More recently, R.Nagel and the second author of this paper found in 7] other conditions guaranteeing the permanence of these regularity properties. Their method, based on the representation of S through the Dyson-Phillips series, can also be applied to the study of the asymptotic behaviour of S. This is the main purpose of this paper. In particular, a semigroup T is considered, such that every orbit t 7 ! T(t)x, x 2 X, belongs to a closed, translation-invariant subspace E of the set C ub (R + ; X) of all uniformly continuous bounded functions from R + to X (see section 3 for deenitions). For example, T can be asymptotically almost periodic (also in the sense of Eberlein) or uniformly ergodic or such that t 7 ! T(t)x vanishes at innnity for all x 2 X. In all these cases, the semigroup T is, as a consequence of the uniform boundedness principle, uniformly bounded. Thus, if S inherits the same property, S is uniformly bounded as well. This means that, rst of 1991 Mathematics Subject Classiication. 47D06, 47A55, 34D05. The authors gratefully acknowledge support by INDAM and CNR. They also want to thank Rainer Nagel for many helpful discussions.