Nilpotent operator

Summary

A nilpotent operator is a bounded linear operator $T$ on a Banach space with $T^n = 0$ for some positive integer $n$. It is the operator version of a nilpotent endomorphism and of a nilpotent matrix. On a nonzero space its spectrum is $\set{0}$. Operators with spectrum $\set{0}$ are called quasinilpotent; every nilpotent operator is quasinilpotent, but the Volterra operator is quasinilpotent without being nilpotent.