# 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 of the corresponding Riesz projector is finite.

## Definition

A point $\displaystyle{ \lambda\in\C }$ in the spectrum $\displaystyle{ \sigma(A) }$ of a closed linear operator $\displaystyle{ A:\,\mathfrak{B}\to\mathfrak{B} }$ in the Banach space $\displaystyle{ \mathfrak{B} }$ with domain $\displaystyle{ \mathfrak{D}(A)\subset\mathfrak{B} }$ is said to belong to discrete spectrum $\displaystyle{ \sigma_{\mathrm{disc}}(A) }$ of $\displaystyle{ A }$ if the following two conditions are satisfied:[1]

1. $\displaystyle{ \lambda }$ is an isolated point in $\displaystyle{ \sigma(A) }$;
2. The rank of the corresponding Riesz projector $\displaystyle{ P_\lambda=\frac{-1}{2\pi\mathrm{i}}\oint_\Gamma(A-z I_{\mathfrak{B}})^{-1}\,dz }$ is finite.

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

## Relation to normal eigenvalues

The discrete spectrum $\displaystyle{ \sigma_{\mathrm{disc}}(A) }$ coincides with the set of normal eigenvalues of $\displaystyle{ A }$:

$\displaystyle{ \sigma_{\mathrm{disc}}(A)=\{\mbox{normal eigenvalues of }A\}. }$[2][3][4]

## 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 $\displaystyle{ \mathfrak{L}_\lambda }$ of the corresponding eigenvalue, and in particular it is possible to have $\displaystyle{ \mathrm{dim}\,\mathfrak{L}_\lambda\lt \infty }$, $\displaystyle{ \mathrm{rank}\,P_\lambda=\infty }$. So, there is the following inclusion:

$\displaystyle{ \sigma_{\mathrm{disc}}(A)\subset\{\mbox{isolated points of the spectrum of }A\mbox{ with finite algebraic multiplicity}\}. }$

In particular, for a quasinilpotent operator

$\displaystyle{ Q:\,l^2(\N)\to l^2(\N),\qquad Q:\,(a_1,a_2,a_3,\dots)\mapsto (0,a_1/2,a_2/2^2,a_3/2^3,\dots), }$

one has $\displaystyle{ \mathfrak{L}_\lambda(Q)=\{0\} }$, $\displaystyle{ \mathrm{rank}\,P_\lambda=\infty }$, $\displaystyle{ \sigma(Q)=\{0\} }$, $\displaystyle{ \sigma_{\mathrm{disc}}(Q)=\emptyset }$.

## Relation to the point spectrum

The discrete spectrum $\displaystyle{ \sigma_{\mathrm{disc}}(A) }$ of an operator $\displaystyle{ A }$ is not to be confused with the point spectrum $\displaystyle{ \sigma_{\mathrm{p}}(A) }$, which is defined as the set of eigenvalues of $\displaystyle{ A }$. While each point of the discrete spectrum belongs to the point spectrum,

$\displaystyle{ \sigma_{\mathrm{disc}}(A)\subset\sigma_{\mathrm{p}}(A), }$

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, $\displaystyle{ L:\,l^2(\N)\to l^2(\N), \quad L:\,(a_1,a_2,a_3,\dots)\mapsto (a_2,a_3,a_4,\dots). }$ 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:

$\displaystyle{ \sigma_{\mathrm{p}}(L)=\mathbb{D}_1, \qquad \sigma(L)=\overline{\mathbb{D}_1}; \qquad \sigma_{\mathrm{disc}}(L)=\emptyset. }$