Nilpotent operator
In operator theory, a bounded operator T on a Banach space is said to be nilpotent if Tn = 0 for some positive integer n. It is said to be quasinilpotent or topologically nilpotent if its spectrum σ =.
Examples
In the finite-dimensional case, i.e. when T is a square matrix with complex entries, σ = if and only ifT is similar to a matrix whose only nonzero entries are on the superdiagonal. 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 = × ⊂ R2, with the Lebesgue measure m. On X, define the kernel function K by
The Volterra operator is the corresponding integral operator T on the Hilbert space L2 given by
The operator T is not nilpotent: take f to be the function that is 1 everywhere and direct calculation shows that
Tn f ≠ 0 for all n. However, T is quasinilpotent. First notice that K is in L2, therefore T is compact operator on [Hilbert space|compact]. By the spectral properties of compact operators, any nonzero λ in σ is an eigenvalue. But it can be shown that T has no nonzero eigenvalues, therefore T is quasinilpotent.