# Power of the infinitesimal generator

Let $$A$$ be the infinitesimal generator of a $$C_0$$ semigroup of linear operators in a Banach space. Let $$n$$ be a positive integer $$n \geq 2$$? Is the power operator $$A^n$$ closed?

Here (setting $$A^1$$ $$:=$$ $$A$$, and denoting the domain of $$A$$ by $$\cal{D}(A)$$), the operator $$A^n$$ has been defined inductively for $$n=2,3...,$$, by $${\cal{D}}(A^n):=\{f: f\in {\cal{D}}(A^{n-1})\; and \; A^{n-1}f \in {\cal{D}}(A) \},$$ $$A^{n}f:=A (A^{n-1} f).$$