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 operato ...

, a

bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...

''T'' on a

Banach space In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...

is said to be

nilpotent In mathematics, an element x of a ring (mathematics), 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, along with its sister Idempotent (ring theory), idem ...

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

spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...

''σ''(''T'') = .

Examples

In the finite-dimensional case, i.e. when ''T'' is a square matrix (

Nilpotent matrix In linear algebra, a nilpotent matrix is a square matrix ''N'' such that :N^k = 0\, for some positive integer k. The smallest such k is called the index of N, sometimes the degree of N. More generally, a nilpotent transformation is a linear trans ...

) with complex entries, ''σ''(''T'') = if and only if ''T'' is similar to a matrix whose only nonzero entries are on the superdiagonal (this fact is used to prove the existence of

Jordan canonical form \begin \lambda_1 1\hphantom\hphantom\\ \hphantom\lambda_1 1\hphantom\\ \hphantom\lambda_1\hphantom\\ \hphantom\lambda_2 1\hphantom\hphantom\\ \hphantom\hphantom\lambda_2\hphantom\\ \hphantom\lambda_3\hphantom\\ \hphantom\ddots\hphantom\\ ...

). 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 In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...

''m''. On ''X'', define the

kernel function In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solvi ...

''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 involvi ...

''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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

. 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

{{DEFAULTSORT:Nilpotent Operator Operator theory