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  "body": "\n## Definition\n\nLet $X$ be a [[Banach space|06GZ956657]] and let $T \\colon X \\to X$ be a bounded [[linear operator|03TM135G0N]].\nThe operator $T$ is *nilpotent* when there is an integer $n \\ge 1$ with\n$$T^n = 0,$$\nwhere $T^n$ is the $n$-fold composition of $T$ with itself.\nThe least such $n$ is the *index* of nilpotency of $T$.\n\n## Remarks\n\nExample: on $X = \\ell^2 \\oplus \\ell^2$ let $T(x, y) = (0, x)$.\nThen $T^2(x, y) = T(0, x) = (0, 0)$, so $T$ is nilpotent of index $2$, although its range is infinite-dimensional.\n\nNon-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.\n\nIf $T$ 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 $\\lambda I - T$.\n",
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