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  "body": "\n## Summary\n\nA nilpotent operator is a bounded linear operator $T$ on a Banach space with $T^n = 0$ for some positive integer $n$.\nIt is the operator version of a [[nilpotent endomorphism|063D5Z2C70]] and of a [[nilpotent matrix|07BSPZPSMF]].\nOn a nonzero space its spectrum is $\\set{0}$.\nOperators with spectrum $\\set{0}$ are called quasinilpotent; every nilpotent operator is quasinilpotent, but the Volterra operator is quasinilpotent without being nilpotent.\n\n## Explanations\n\n### L1 human\n\nA nilpotent operator dies after a fixed number of applications.\nOn pairs of sequences, the map $(x, y) \\mapsto (0, x)$ is an example: applied twice it gives $(0, 0)$.\nIn infinite dimensions one must be careful: an operator can shrink everything faster and faster under powers without ever reaching $0$, and such an operator is not nilpotent.\n",
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