I am trying to understand to what extent the following result of Hille is an extension of the usual theorems on differentiation under the integral sign.

Theorem(Hille). Let $(\Omega,\Sigma,\mu)$ be a measure space and $X$ and $Y$ be Banach spaces. Let $f:\Omega \to X$ be a Bochner-integrable function and $T:X \to Y$ a closed linear operator defined on the domain $\mathscr{D}(T) \subseteq X$. If the image of $f$ is (almost-everywhere) contained in $\mathscr{D}(T)$ and if $Tf:\Omega \to Y$ is Bochner integrable, then $\int_\Omega f \, d\mu \in \mathscr{D}(T)$ and \begin{equation} (1) \, \, \, \, \, \, \, \, \, T \int_\Omega f \, d\mu = \int_\Omega Tf \,d\mu. \end{equation}

While (1) is trivial to show for bounded linear operators, the differential operator $Tf = f'$ defined on $\mathscr{D}(T) = \mathscr{C}^1([0,1]) \subseteq \mathscr{C}([0,1])$ into $\mathscr{C}([0,1])$ is unbounded. As such, an extension to unbounded operators, such as above, is necessary to formulate a differentiation-under-the-integral-sign theorem for the Bochner integral. On the other hand, all linear partial differential operators \begin{equation} D = \sum_{|\alpha| \leq a} c_\alpha(x) D^\alpha \end{equation} with smooth coefficients $c_\alpha \in \mathscr{C}^a(\Omega)$ are closable as operators defined on $\mathscr{D}(D) = \{f \in \mathscr{C}^a(\Omega) \cap L^2(\Omega) : Df \in L^2(\Omega)\}$ into $L^2(\Omega)$ (as Liviu points out in the comment section), and so the above theorem of Hille is a reasonable instance of differentiation-under-the-integral-sign theorems in the finite-dimensional case.

Question. Is there an analogue of the above result on linear differential operators with smooth coefficients for infinite-dimensional differential operators in the sense of Gâteaux derivative / Fréchet derivative? In other words, is (1) applicable for enough differential operators on Banach-valued functions to consider it a differentiation-under-the-integral-sign-type result?