I have the following question. Let $X$ be a Banach space (you may specify further properties such as reflexivity or Hilbert space structure if needed) and let $A: \mathcal{D}(A) \to X$ be a closed linear operator (densely defined if needed). Let $k \in \mathbb{N}$. Is $A^k$ with natural domain $\mathcal{D}(A^k) = \{x \in \mathcal{D}(A^{k-1}) \mid A^{k-1}x \in \mathcal{D}(A)\}$ a closed operator? I think the answer is yes if we know for example that $\rho(A) \neq \varnothing$. But it should also work for, say, normal operators in Hilbert spaces even if $\sigma(A) = \mathbb{C}$. So, I am in particular interested in the case $\sigma(A) = \mathbb{C}$ for a general closed operator $A$ in a Banach space $X$. Thanks in advance for the help.
Best, Jan