operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operat ...

, 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 In mathematics, in the area of functional analysis and operator theory, the Volterra operator, named after Vito Volterra, is a bounded linear operator on the space ''L''2 ,1of complex-valued square-integrable functions on the interval ,1 On the su ...

, 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 An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol * Integral linear operators, which are linear operators induced by bilinear forms invol ...

''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