Continuity of the Restriction of C0-Semigroups to Invariant Banach Subspaces

A linear semigroup in a Banach space induces a linear semigroup on a Banach space that can be continuously embedded in the former such that its image is invariant. This restriction need not be strongly continuous, although the original semigroup is strongly continuous. We show that norm or weak compactness of partial orbits is a necessary and sufficient condition for strong continuity of the restriction of a C0-semigroup. We then show that if the embedded Banach space is reflexive and the norms of the restricted semigroup operators are bounded near the initial time, then the restricted semigroup is strongly continuous.