Editing Spectrum (functional analysis) (section)

==Spectra of particular classes of operators==

===Compact operators===

If ''T'' is a [[compact operator]], or, more generally, an [[strictly singular operator|inessential operator]], then it can be shown that the spectrum is countable, that zero is the only possible [[accumulation point]], and that any nonzero ''λ'' in the spectrum is an eigenvalue.

===Quasinilpotent operators===

A bounded operator <math>A:\,X\to X</math> is '''quasinilpotent''' if <math>\lVert A^n\rVert^{1/n} \to 0</math> as <math>n\to\infty</math> (in other words, if the spectral radius of ''A'' equals zero). Such operators could equivalently be characterized by the condition

:<math>\sigma(A)=\{0\}.</math>

An example of such an operator is <math>A:\,l^2(\N)\to l^2(\N)</math>, <math>e_j\mapsto e_{j+1}/2^j</math> for <math>j\in\N</math>.

===Self-adjoint operators===

If ''X'' is a [[Hilbert space]] and ''T'' is a [[self-adjoint operator]] (or, more generally, a [[normal operator]]), then a remarkable result known as the [[spectral theorem]] gives an analogue of the diagonalisation theorem for normal finite-dimensional operators (Hermitian matrices, for example).

For self-adjoint operators, one can use [[spectral measure]]s to define a [[decomposition of spectrum (functional analysis)|decomposition of the spectrum]] into absolutely continuous, pure point, and singular parts.