  Recent work in arXiv:1901.11526 by the author about a class of abstract delay
differential equations (DDEs), as well as earlier work by Diekmann and
Gyllenberg on other classes of delay equations, motivates the introduction of
the general notion of an admissible range and an admissible perturbation for a
given $\mathcal{C}_0$-semigroup $T_0$ on a Banach space $X$ that is not assumed
to be sun-reflexive with respect to $T_0$. We investigate the relationship
between admissible ranges for $T_0$ and the subspace $X^{\odot\times}$ of
$X^{\odot\star}$ introduced by Van Neerven. We answer two questions about
robustness of admissibility with respect to bounded linear perturbations and we
use these answers to study the semilinear problem and its linearization. Partly
as an application of the material developed up to that point, and partly as a
justification of existing work on local bifurcations in models taking the form
of abstract DDEs, we compare the construction of center manifolds in the
non-sun-reflexive case with known results by Diekmann and Van Gils for the
sun-reflexive case. We show that a systematic use of the space
$X^{\odot\times}$ facilitates a generalization of the existing results with
relatively little effort. In this context we also give sufficient conditions
for the existence of appropriate spectral decompositions of $X$ and
$X^{\odot\times}$ without assuming that the linearized semiflow is eventually
compact. A center manifold theorem for the motivating class of abstract DDEs
then follows as a particular case.
