For $x\in X$ and $t>0$, form the <Bochner integral>
$$
x_t=\frac1t\int_0^tU(s)x\,ds.
$$
The semigroup property gives, for $h>0$,
$$
\frac{U(h)x_t-x_t}{h}
=\frac1{th}
\left(\int_t^{t+h}U(s)x\,ds-
\int_0^hU(s)x\,ds\right).
$$
Strong continuity lets $h\downarrow0$, yielding
$$
x_t\in D(A),
\qquad
Ax_t=\frac{U(t)x-x}{t}.
$$
Also,
$$
\|x_t-x\|
\leq\frac1t\int_0^t\|U(s)x-x\|\,ds\longrightarrow0
$$
by strong continuity. Every $x\in X$ is therefore a norm limit of elements of $D(A)$, so
$$
\boxed{\overline{D(A)}=X}.
$$
This approximation is the basic <Yosida averaging of a semigroup>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 319 / 1 / b / Solution

{c}

For $x$ in the underlying Banach space,
$$
x_t=\frac1t\int_0^tU(s)x\,ds
$$
belongs to the generator domain, satisfies $Ax_t=[U(t)x-x]/t$, and converges to $x$ as $t\downarrow0$.


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