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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: $x(t)\in D(A),\, \forall t\ge 0?$

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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.

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