On Eventual Regularity Properties of Operator-Valued Functions

Let $\mathcal{L}(X;Y)$ be the space of bounded linear operators from a Banach space $X$ to a Banach space $Y$. Given an operator-valued function $u:\mathbb{R}_{\geq 0}\rightarrow \mathcal{L}(X;Y)$, suppose that every orbit $t\mapsto u(t)x$ has a regularity property (e.g. continuity, differentiability, etc.) on some interval $(t_x,\infty)$ in general depending on $x\in X$. In this paper we develop an abstract set-up based on Baire-type arguments which allows, under certain conditions, removing the dependency on $x$ systematically. Afterwards, we apply this theoretical framework to several different regularity properties that are of interest also in semigroup theory. In particular, a generalisation of the prior results on eventual differentiability of strongly continuous functions $u:\mathbb{R}_{\geq 0} \rightarrow \mathcal{L}(X;Y)$ obtained by Iley and B\'arta follows as a special case of our method.