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

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A point  λ ∈ C {\displaystyle \lambda \in \mathbb {C} } in the spectrum σ ( A ) {\displaystyle \sigma (A)} of a closed linear operator A : B -> B {\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}} in the Banach space B {\displaystyle {\mathfrak {B}}} with domain D ( A ) ⊂ B {\displaystyle {\mathfrak {D}}(A)\subset {\mathfrak {B}}} is said to belong to discrete spectrum σ d i s c ( A ) {\displaystyle \sigma _{\mathrm {disc} }(A)} of  A {\displaystyle A} if the following two conditions are satisfied:[1]

1. λ {\displaystyle \lambda } is an isolated point in  σ ( A ) {\displaystyle \sigma (A)} ;
2. The rank of the corresponding Riesz projector P λ = − 1 2 π i ∮ Γ ( A − z I B ) − 1 d z {\displaystyle P_{\lambda }={\frac {-1}{2\pi \mathrm {i} }}\oint _{\Gamma }(A-zI_{\mathfrak {B}})^{-1}\,dz} is finite.

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

## Relation to normal eigenvalues

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

σ d i s c ( A ) = { normal eigenvalues of A } . {\displaystyle \sigma _{\mathrm {disc} }(A)=\\{{\mbox{normal eigenvalues of }}A\\}.} [2][3][4]

## Relation to isolated eigenvalues of finite algebraic multiplicity

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In general, the rank of the Riesz projector can be larger than the dimension of the root lineal L λ {\displaystyle {\mathfrak {L}}_{\lambda }} of the corresponding eigenvalue, and in particular it is possible to have  d i m L λ < ∞ {\displaystyle \mathrm {dim} \,{\mathfrak {L}}_{\lambda }<\infty } ,  r a n k P λ = ∞ {\displaystyle \mathrm {rank} \,P_{\lambda }=\infty } . So, there is the following inclusion:

σ d i s c ( A ) ⊂ { isolated points of the spectrum of A with finite algebraic multiplicity } . {\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

Q : l 2 ( N ) -> l 2 ( N ) , Q : ( a 1 , a 2 , a 3 , … ) ↦ ( 0 , a 1 / 2 , a 2 / 2 2 , a 3 / 2 3 , … ) , {\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  L λ ( Q ) = { 0 } {\displaystyle {\mathfrak {L}}_{\lambda }(Q)=\\{0\\}} ,  r a n k P λ = ∞ {\displaystyle \mathrm {rank} \,P_{\lambda }=\infty } ,  σ ( Q ) = { 0 } {\displaystyle \sigma (Q)=\\{0\\}} ,  σ d i s c ( Q ) = ∅ {\displaystyle \sigma _{\mathrm {disc} }(Q)=\emptyset } .

## Relation to the point spectrum

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

σ d i s c ( A ) ⊂ σ p ( A ) , {\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,  L : l 2 ( N ) -> l 2 ( N ) , L : ( a 1 , a 2 , a 3 , … ) ↦ ( a 2 , a 3 , a 4 , … ) . {\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:

σ p ( L ) = D 1 , σ ( L ) = D 1 ¯ ; σ d i s c ( L ) = ∅ . {\displaystyle \sigma _{\mathrm {p} }(L)=\mathbb {D} _{1},\qquad \sigma (L)={\overline {\mathbb {D} _{1}}};\qquad \sigma _{\mathrm {disc} }(L)=\emptyset .}

## See also

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* 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

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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.

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