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 ${\displaystyle \lambda \in \mathbb {C} }$ 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 ${\displaystyle P_{\lambda }={\frac {-1}{2\pi \mathrm {i} }}\oint _{\Gamma }(A-zI_{\mathfrak {B}})^{-1}\,dz}$ is finite.

Here ${\displaystyle I_{\mathfrak {B}}}$ is the identity operator in the Banach space ${\displaystyle {\mathfrak {B}}}$ and ${\displaystyle \Gamma \subset \mathbb {C} }$ is a smooth simple closed counterclockwise-oriented curve bounding an open region ${\displaystyle \Omega \subset \mathbb {C} }$ 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 ${\displaystyle \mathrm {dim} \,{\mathfrak {L}}_{\lambda }<\infty }$, ${\displaystyle \mathrm {rank} \,P_{\lambda }=\infty }$. 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

${\displaystyle Q:\,l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} ),\qquad Q:\,(a_{1},a_{2},a_{3},\dots )\mapsto (0,a_{1}/2,a_{2}/2^{2},a_{3}/2^{3},\dots ),}$

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, ${\displaystyle L:\,l^{2}(\mathbb {N} )\to l^{2}(\mathbb {N} ),\quad L:\,(a_{1},a_{2},a_{3},\dots )\mapsto (a_{2},a_{3},a_{4},\dots ).}$ 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:

${\displaystyle \sigma _{\mathrm {p} }(L)=\mathbb {D} _{1},\qquad \sigma (L)={\overline {\mathbb {D} _{1}}};\qquad \sigma _{\mathrm {disc} }(L)=\emptyset .}$

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

1. Reed, M.; Simon, B. (1978). Methods of modern mathematical physics, vol. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich Publishers], New York.
2. Gohberg, I. C; Kreĭn, M. G. (1960). "Fundamental aspects of defect numbers, root numbers and indexes of linear operators". American Mathematical Society Translations. 13: 185–264.
3. Gohberg, I. C; Kreĭn, M. G. (1969). Introduction to the theory of linear nonselfadjoint operators. American Mathematical Society, Providence, R.I.
4. Boussaid, N.; Comech, A. (2019). Nonlinear Dirac equation. Spectral stability of solitary waves. American Mathematical Society, Providence, R.I. ISBN 978-1-4704-4395-5.