  In this paper we study the preservation of strong stability of strongly
continuous semigroups on Hilbert spaces. In particular, we study a situation
where the generator of the semigroup has a finite number of spectral points on
the imaginary axis and the norm of its resolvent operator is polynomially
bounded near these points. We characterize classes of finite rank perturbations
preserving the strong stability of the semigroup. In addition, we improve
recent results on preservation of polynomial stability of a semigroup under
finite rank perturbations of its generator. Theoretic results are illustrated
with an example where we consider the preservation of the strong stability of a
multiplication semigroup.
