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The notion of bi-continuous semigroups has recently been introduced to handle semigroups on Banach spaces that are only strongly continuous for a topology coarser than the norm-topology. In this paper, as a continuation of the systematic treatment of such semigroups started in [20-22], we provide a bounded perturbation theorem, which turns out to be quite general in view of various examples.

For a given bi-continuous semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ on a Banach space $X$ we define its adjoint on an appropriate closed subspace $X^{\circ }$ of the norm dual $X^{\text{'}}$. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology $\sigma (X^{\circ },X)$. We give the following application: For $\Omega$ a Polish space we consider operator semigroups on the space ${\mathrm{C}}_{\mathrm{b}}\left(\Omega \right)$ of bounded, continuous functions (endowed with the compact-open topology) and on the space $\mathrm{M}\left(\Omega \right)$ of bounded Baire measures (endowed with the weak${}^{*}$-topology)....