TITLE: Spectrum of a nilpotent operator
QUESTION [5 upvotes]: Let $X$ be a Banach space and $A:X\to X$ be a bounded operator such that $A^n=0$ for some $n\in \mathbb{N}$. Is the spectrum of $A$ finite, countable ?

REPLY [3 votes]: For any polynomial $p$ we know $\sigma(p(A)) = p(\sigma(A))$.