# Boundedness of Riemann integral of generator of a semigroup

I'm stuck with the following claim:

Let $(T(t))_{t \geq 0}$ be a $C_0$ semigroup with generator $A$. The generator $A$ is not necessarily bounded. Let $s>0$ such that the range of $T(s)$ is dense in $D(A^\infty)$. Define an operator $S_n$ for $x \in D(A^\infty)$ by $$S_n x = \int_0^t (t-s)^n A^{n+1} T(s) x ds$$ Then $S_n$ is bounded: $\Vert S_n \Vert \leq (n+1)^{-1} \Vert t^{n+1} A^{n+1} T(s)\Vert$.

I guess the whole claim reduces to showing that $\Vert A^{n+1} T(u) \Vert \leq \Vert A^{n+1} T(s) \Vert$ for all $u \in [0,t]$, which then yields the bound. Is it necessary to assume that the semigroup is analytic to get there? What is the significance of the range condition?

• Do you have $s$ and $t$ mixed up in the statement? – Keith McClary Jun 17 '17 at 4:15