We study the operator $(\mathcal{L} u)(t) := u'(t) - A(t) u(t)$ on $L^p (\mathbb{R}; X)$ for sectorial operators $A(t)$, with $t \in \mathbb{R}$, on a Banach space $X$ that are asymptotically hyperbolic, satisfy the Acquistapace–Terreni conditions, and have the property of maximal $L^p$-regularity. We establish optimal regularity on the time interval $\mathbb{R}$ showing that $\mathcal{L}$ is closed on its minimal domain. We further give conditions for ensuring that $\mathcal{L}$ is a semi-Fredholm operator. The Fredholm property is shown to persist under $A(t)$-bounded perturbations, provided they are compact or have small $A(t)$-bounds. We apply our results to parabolic systems and to generalized Ornstein–Uhlenbeck operators.