On eventual regularity properties of operator-valued functions
Volume 265 / 2022
Abstract
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 dependence 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 generalization 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 Bárta follows as a special case of our method.
