Proving that $T(t)x$ is in the domain

Question

$(T(t))_{t\ge 0}$ is a $C_{0}$-semigroup on a Banach space $X$ with generator $A:D(A)\subset X\to X$. For $k\ge 2$, define

$$D(A^{k}):=\{x\in D(A^{k-1})|A^{k-1}x\in D(A)\}$$

I want to show that for any $x\in X$ such that $x\in D(A^{k})$ for some integer $k\ge 2$

$$T(t)x\in D(A^{k})$$

for all $t\ge 0$.

I know that if $x\in D(A)$, then $T(t)x\in D(A)$ for all $t\ge 0$, with $\frac{d}{dt}T(t)x=T(t)Ax=AT(t)x$ for all $t\ge 0$.

Is it simply a case of proving that property for $A^{k}$?

Answer 1

Hint: Use induction on $k\mbox{}$.