Let $T(t)$ be a $C\_0$-semigroup on a Hilbert space $H$ with a generator $A$.


It is well known that for all $x\in H,$ we have: $ \int\_0^t T(s)x ds \in D(A) $ and $ A\int\_0^t T(s)x ds = T(t)x-x$.


How is this formula changed when $x$ is replaced by a continuous function $t \mapsto x(t)$ such that: $x(t)\in D(A),\, \forall t\ge 0?$



On extending a semigroup property
For a function $f\in W^{1,1}(\mathbb R\_+,X)$ (let me stress: only $X$, not $D(A)$) or else for a function $f\in C\_0(\mathbb R\_+,D(A))$ it is well-known that the convolution 
$$
\int\_0^t T(t-s)f(s)ds
$$
is in $D(A)$ for all $t$, see e.g. Corollaries VI.7.6 and VI.7.8 in the book by Engel and Nagel. This holds in general Banach space and not only in Hilbert space, by the way.


