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I quasi-nilpotent operator $T \in B(X)$ is nilpotent if and only if $0$ is a pole of its resolvent $(\cdot - T)^{-1}$, which means that there exists a number $M \ge 0$ and an integer $n \ge 1$ such that $$ \|(\lambda-T)^{-1}\| \le \frac{M}{\rvert\lambda\lvert^n} $$ for all $0 \not= \lambda \in \mathbb{C}$ close to $0$. This follows from the Laurent series expansion of the resolvent about the point $0$.