# Nilpotent operator

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In [operator theory](/source/Operator_theory), a [bounded operator](/source/Bounded_operator) *T* on a [Banach space](/source/Banach_space) is said to be **[nilpotent](/source/Nilpotent)** if *Tn* = 0 for some positive integer *n*.[1] It is said to be **quasinilpotent** or **topologically nilpotent** if its [spectrum](/source/Spectrum_(functional_analysis)) *σ*(*T*) = {0}.

## Examples

In the finite-dimensional case, i.e. when *T* is a square matrix ([Nilpotent matrix](/source/Nilpotent_matrix)) with complex entries, *σ*(*T*) = {0} if and only if *T* is similar to a matrix whose only nonzero entries are on the superdiagonal[2](this fact is used to prove the existence of [Jordan canonical form](/source/Jordan_canonical_form)). In turn this is equivalent to *Tn* = 0 for some *n*. Therefore, for matrices, quasinilpotency coincides with nilpotency.

This is not true when *H* is infinite-dimensional. Consider the [Volterra operator](/source/Volterra_operator), defined as follows: consider the unit square *X* = [0,1] × [0,1] ⊂ **R**2, with the [Lebesgue measure](/source/Lebesgue_measure) *m*. On *X*, define the [kernel function](/source/Integral_kernel) *K* by

- K ( x , y ) = { 1 , if x ≥ y 0 , otherwise . {\displaystyle K(x,y)=\left\{{\begin{matrix}1,&{\mbox{if}}\;x\geq y\\0,&{\mbox{otherwise}}.\end{matrix}}\right.}

The Volterra operator is the corresponding [integral operator](/source/Integral_operator) *T* on the Hilbert space *L*2(0,1) given by

- T f ( x ) = ∫ 0 1 K ( x , y ) f ( y ) d y . {\displaystyle Tf(x)=\int _{0}^{1}K(x,y)f(y)dy.}

The operator *T* is not nilpotent: take *f* to be the function that is 1 everywhere and direct calculation shows that *Tn f* ≠ 0 (in the sense of *L*2) for all *n*. However, *T* is quasinilpotent. First notice that *K* is in *L*2(*X*, *m*), therefore *T* is [compact](/source/Compact_operator_on_Hilbert_space). By the spectral properties of compact operators, any nonzero *λ* in *σ*(*T*) is an eigenvalue. But it can be shown that *T* has no nonzero eigenvalues, therefore *T* is quasinilpotent.

## References

1. **[^](#cite_ref-1)** Kreyszig, Erwin (1989). "Spectral Theory in Normed Spaces 7.5 Use of Complex Analysis in Spectral Theory, Problem 1. (Nilpotent operator)". *Introductory Functional Analysis with Applications*. Wiley. p. 393.

1. **[^](#cite_ref-2)** [Axler, Sheldon](/source/Sheldon_Axler). ["Nilpotent Operator"](https://linear.axler.net/NilpotentOperators.pdf) (PDF). *Linear Algebra Done Right*.

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