From Wikipedia:
Closed linear operators are a class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still retain nice enough properties that one can define the spectrum and (with certain assumptions) functional calculus for such operators. Many important linear operators which fail to be bounded turn out to be closed, such as the derivative and a large class of differential operators.
Let X Banach space, and $D(A) ⊂ X$ the domain of operator A.
A linear operator A : $D(A) ⊂ X → X$ is closed if for every sequence $\{u_k\}$ in $D(A)$ converging to $u$ in X such that $A u_k → v ∈ X$ as $k → ∞$ one has $u ∈ D(A)$ and $A u = v$.
Prove that, if A is closed, then
$$A\int_0^\infty S(t) u dt=\int_0^\infty S(t) A u dt$$
where $S: X\rightarrow X$ is any function such that the integral is defined, i.e. $u\in D(A)$, S such that $S(t)u\in D(A)$ for all $t>0$.