Assume that $T:V\rightarrow V$ is a linear map from the $\mathbb{C}$ vector space $V$ to itself. Is it true that if $0$ is the only eigenvalue of $T$ then for all $v\in V$ there exists $n$ such that $T^n(v)=0$?
Edit : $V$ is not assumed to be finite dimensional, so $T$ itself may not be nilpotent.