---
tag: P971S9AX0E
kind: definition
name: Nilpotent operator
msc: [47A05, 47B01]
defines: DK3Q4RPRZJ
uses: [06GZ956657, 03TM135G0N]
sources:
  - url: https://en.wikipedia.org/wiki/Nilpotent_operator
---

## Definition

Let $X$ be a [[Banach space|06GZ956657]] and let $T \colon X \to X$ be a bounded [[linear operator|03TM135G0N]].
The operator $T$ is *nilpotent* when there is an integer $n \ge 1$ with
$$T^n = 0,$$
where $T^n$ is the $n$-fold composition of $T$ with itself.
The least such $n$ is the *index* of nilpotency of $T$.

## Remarks

Example: on $X = \ell^2 \oplus \ell^2$ let $T(x, y) = (0, x)$.
Then $T^2(x, y) = T(0, x) = (0, 0)$, so $T$ is nilpotent of index $2$, although its range is infinite-dimensional.

Non-example: the Volterra operator $(Vf)(x) = \int_0^x f(t) \, dt$ on $L^2[0, 1]$ has spectrum $\set{0}$, but $(V^n 1)(x) = x^n / n!$ is not the zero function for any $n$, so $V$ is not nilpotent.

If $T$ is nilpotent and $X \ne \set{0}$, then $\sigma(T) = \set{0}$: for $\lambda \ne 0$ the series $\lambda^{-1} \sum_{k=0}^{n-1} (T/\lambda)^k$ is a finite sum that inverts $\lambda I - T$.