On the admissibility of observation operators for evolution families

This paper is concerned with unbounded observation operators for non-autonomous evolution equations. Fix $\tau > 0$ and let $\left(A(t)\right)_{t \in [0,\tau]} \subset \mathcal{L}(D,X)$, where $D$ and $X$ are two Banach spaces such that $D$ is continuously and densely embedded into $X$. We assume that the operator $A(t)$ has maximal regularity for all $t \in [0,\tau]$ and that $A(\cdot) : [0,\tau] \to \mathcal{L}(D,X)$ satisfies a regularity condition (viz. relative $p$-Dini for some $p \in (1,\infty)$). At first sight, we show that there exists an evolution family on $X$ associated to the problem $$\dot{u}(t) + A(t) u(t) = 0 \quad t\text{ a.e. on } [0,\tau], \qquad u(0) = x \in X.$$ Then we prove that an observation operator is admissible for $A(\cdot)$ if and only if it is admissible for each $A(t)$ for all $t \in [0,\tau)$.