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 $\lambda \in ℂ$ in the spectrum $\sigma (A)$ of a closed linear operator $A:𝔅\to 𝔅$ in the Banach space $𝔅$ with domain $𝔇(A)\subset 𝔅$ is said to belong to discrete spectrum ${\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(A)$ of $A$ if the following two conditions are satisfied:

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

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

Relation to normal eigenvalues

The discrete spectrum ${\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(A)$ coincides with the set of normal eigenvalues of $A$:

${\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(A)=\{\text{normal eigenvalues of }A\}.$

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 ${𝔏}_{\lambda }$ of the corresponding eigenvalue, and in particular it is possible to have $\mathrm{d}\mathrm{i}\mathrm{m}{𝔏}_{\lambda }<\mathrm{\infty }$, $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}{P}_{\lambda }=\mathrm{\infty }$. So, there is the following inclusion:

${\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(A)\subset \{\text{isolated points of the spectrum of }A\text{ with finite algebraic multiplicity}\}.$

In particular, for a quasinilpotent operator

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

one has ${𝔏}_{\lambda }(Q)=\{0\}$, $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}{P}_{\lambda }=\mathrm{\infty }$, $\sigma (Q)=\{0\}$, ${\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(Q)=\mathrm{\varnothing }$.

Relation to the point spectrum

The discrete spectrum ${\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(A)$ of an operator $A$ is not to be confused with the point spectrum ${\sigma }_{\mathrm{p}}(A)$, which is defined as the set of eigenvalues of $A$. While each point of the discrete spectrum belongs to the point spectrum,

${\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(A)\subset {\sigma }_{\mathrm{p}}(A),$

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

${\sigma }_{\mathrm{p}}(L)={𝔻}_1,\sigma (L)=\overline{{𝔻}_1};{\sigma }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}(L)=\mathrm{\varnothing }.$

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