In operator theory, a bounded operator ''T'' on a

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

is said to be

nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the class ...

if ''Tⁿ'' = 0 for some ''n''. It is said to be quasinilpotent or topologically nilpotent if its

spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...

''σ''(''T'') = .

Examples

In the finite-dimensional case, i.e. when ''T'' is a square matrix with complex entries, ''σ''(''T'') = if and only if ''T'' is similar to a matrix whose only nonzero entries are on the superdiagonal, by the

Jordan canonical form In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...

. 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, defined as follows: consider the unit square ''X'' = ,1× ,1⊂ R², with the Lebesgue measure ''m''. On ''X'', define the (kernel) function ''K'' by :

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 ''T'' on the Hilbert space ''L''²(0,1) given by :

T f(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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

. 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. {{DEFAULTSORT:Nilpotent Operator Operator theory