Discrete Spectrum (mathematics)

In mathematics, specifically in spectral theory, a discrete spectrum of a closed linear operator is defined as the set of isolated points of its spectrum such that the rank (linear algebra), rank of the corresponding Riesz projector is finite.

Definition

A point $\lambda\in\C$
in the spectrum $\sigma(A)$ of a closed linear operator $A:\,\mathfrak\to\mathfrak$ in the Banach space $\mathfrak$ with domain $\mathfrak(A)\subset\mathfrak$ is said to belong to ''discrete spectrum'' $\sigma_(A)$ of $A$ if the following two conditions are satisfied:
# $\lambda$ is an isolated point in $\sigma(A)$;
# The rank (linear algebra), rank of the corresponding Riesz projector $P_\lambda=\frac\oint_\Gamma(A-z I_)^\,dz$ is finite.

Here $I_$ is the identity operator in the Banach space $\mathfrak$ and $\Gamma\subset\C$ is a smooth simple closed counterclockwise-oriented curve bounding an open region $\Omega\subset\C$ such that $\lambda$ is the only point of the spectrum of $A$ in the closure of $\Omega$; that is, $\sigma(A)\cap\overline=\.$

Relation to normal eigenvalues

The discrete spectrum $\sigma_(A)$ coincides with the set of normal eigenvalues of $A$:
:$\sigma_(A)=\.$

Relation to isolated eigenvalues of finite algebraic multiplicity

In general, the rank of the Riesz projector can be larger than the dimension of the root lineal $\mathfrak_\lambda$ of the corresponding eigenvalue, and in particular it is possible to have $\mathrm\,\mathfrak_\lambda<\infty$, $\mathrm\,P_\lambda=\infty$. So, there is the following inclusion:
:$\sigma_(A)\subset\.$
In particular, for a quasinilpotent operator
:$Q:\,l^2(\N)\to l^2(\N),\qquad Q:\,(a_1,a_2,a_3,\dots)\mapsto (0,a_1/2,a_2/2^2,a_3/2^3,\dots),$
one has
$\mathfrak_\lambda(Q)=\$, $\mathrm\,P_\lambda=\infty$,
$\sigma(Q)=\$,
$\sigma_(Q)=\emptyset$.

Relation to the point spectrum

The discrete spectrum $\sigma_(A)$ of an operator $A$ is not to be confused with the point spectrum $\sigma_(A)$, which is defined as the set of eigenvalues of $A$.
While each point of the discrete spectrum belongs to the point spectrum,
:$\sigma_(A)\subset\sigma_(A),$
the converse is not necessarily true: the point spectrum does not necessarily consist of isolated points of the spectrum, as one can see from the example of the ''left shift operator'',
$L:\,l^2(\N)\to l^2(\N), \quad L:\,(a_1,a_2,a_3,\dots)\mapsto (a_2,a_3,a_4,\dots).$
For this operator, the point spectrum is the unit disc of the complex plane, the spectrum is the closure of the unit disc, while the discrete spectrum is empty:
:$\sigma_(L)=\mathbb_1, \qquad \sigma(L)=\overline; \qquad \sigma_(L)=\emptyset.$

See also

* Spectrum (functional analysis)
* Decomposition of spectrum (functional analysis)
* Normal eigenvalue
* Essential spectrum
* Spectrum of an operator
* Resolvent formalism
* Riesz projector
* Fredholm operator
* Operator theory

References

{{Functional analysis

Spectral theory