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 $A:\,\mathfrak{B}\to\mathfrak{B}$ in the Banach space $\mathfrak{B}$ with domain $\mathfrak{D}(A)\subset\mathfrak{B}$ is said to belong to discrete spectrum $\sigma_{\mathrm{disc}}(A)$ of $A$ if the following two conditions are satisfied:{{ cite book

|author1=Reed, M.

|author2=Simon, B.

|title=Methods of modern mathematical physics, vol. IV. Analysis of operators

|year=1978

|publisher = Academic Press [Harcourt Brace Jovanovich Publishers], New York

}}

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

Here $I_{\mathfrak{B}}$ is the identity operator in the Banach space $\mathfrak{B}$ 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, $\sigma(A)\cap\overline{\Omega}=\{\lambda\}.$

Relation to normal eigenvalues

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

: $\sigma_{\mathrm{disc}}(A)=\{\mbox{normal eigenvalues of }A\}.$ {{ cite journal

|author1=Gohberg, I. C

|author2=Kreĭn, M. G.

|title=Fundamental aspects of defect numbers, root numbers and indexes of linear operators

|journal=American Mathematical Society Translations

|volume=13

|year=1960

|pages=185–264

|url=http://mi.mathnet.ru/umn7581

}}{{ cite book

|author1=Gohberg, I. C

|author2=Kreĭn, M. G.

|title=Introduction to the theory of linear nonselfadjoint operators

|year=1969

|publisher = American Mathematical Society, Providence, R.I.

|url=http://gen.lib.rus.ec/book/index.php?md5=9CE2F03854312C3E29ED684CD84D8CA3

}}

{{ cite book

|author1=Boussaid, N.

|author2=Comech, A.

|title=Nonlinear Dirac equation. Spectral stability of solitary waves

|year=2019

|publisher = American Mathematical Society, Providence, R.I.

|isbn=978-1-4704-4395-5

|url=https://bookstore.ams.org/surv-244

}}

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

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

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

$\mathfrak{L}_\lambda(Q)=\{0\}$ , $\mathrm{rank}\,P_\lambda=\infty$ ,

$\sigma(Q)=\{0\}$ ,

$\sigma_{\mathrm{disc}}(Q)=\emptyset$ .

Relation to the point spectrum

The discrete spectrum $\sigma_{\mathrm{disc}}(A)$ 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,

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

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,

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

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:

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

References

Category:Spectral theory