Precise versions of "differential operators are unbounded but closed linear operators" 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?

 A: The question needs to be  formulated more precisely.   First a comment.
Any   unbounded operator
$$ A: D(A)\subset E\to F $$
$E,F$ Banach spaces, can also be viewed  as a bounded operator
$$ D(A)\to F $$
where $D(A)$ is equipped with the graph-norm $\vert-\vert_G$
$$\vert x\vert_G = \vert x\vert_E+\vert Ax\vert_F. $$
The operator is closed iff $(D(A),\vert-\vert_G)$ is Banach.
The above example illustrates the important role played by the domain of the operator. 
Recall what closeness entails: if $(x_n)$ is a sequence in $D(A)$ such that
$$ x_n\to x_\infty\;\;\mbox{in}\;\;\vert-\vert_E, $$
$$ Ax_n \to y_\infty\;\;\mbox{in}\;\;\vert-\vert_F, $$
then $x_\infty \in D(A)$ and $y_\infty=A x_\infty$. Thus, the choice of the norms $\vert-\vert_{E,F}$ is also relevant to your question.
You mention that a  differential operator with constant coefficients is unbounded and closed with respect to the uniform norm, but you  give no suggestion of what its domain ought to be.  To see that this issue is more subtle take a look at the following simple example.  
Consider  the operator $\newcommand{\pa}{\partial}$ $\newcommand{\bR}{\mathbb{R}}$ $A=\pa^2_x-\pa^2_y-\pa_z^2$  whose domain is a subspace $D(A)\subset C^0(\bR^3)$. Here I think that $E=F=C^0(\bR^3)$ with the usual $\sup$-norm. Can you think of a choice of $D(A)$ that will make this a closed operator closed?
Note that if you think of $A$ as an operator $C^2(\bR^3)\to C^0(\bR^3)$ it is obviously  bounded. However, if you  choose $D(A)= C^2(\bR^3)\subset C^0(\bR^3)$, it is not clear to me that the resulting operator
$$ A: D(A)\subset C^0(\bR^3)\to C^0(\bR^3), $$
is closed.
The above concrete operator was not chosen at random. The operator $A$ is a hyperbolic operator, and for such operators strange things could  happen.
