Discrete spectrum (mathematics) explained

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

λ \in\C in the spectrum \sigma(A) of a closed linear operator A: ak{B}\to ak{B} in the Banach space ak{B} with domain ak{D}(A)\subset ak{B} is said to belong to discrete spectrum \sigma _{disc} (A) of A if the following two conditions are satisfied:^[1] λ is an isolated point in \sigma(A) ;P _{λ= (-1)/(2 \pi i)} \oint _{\Gamma(A-z} I _{ak{B} })^\,dz is finite.

Here

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

Relation to normal eigenvalues

The discrete spectrum

\sigma _{disc} (A) coincides with the set of normal eigenvalues of A :\sigma _{disc} (A) = \{ normaleigenvaluesof A\}. ^{[2] [3] [4]}

Relation to isolated eigenvalues of finite algebraic multiplicity

ak{L} _{λ} of the corresponding eigenvalue, and in particular it is possible to have dim ak{L} _{λ<infty} , rank P _{λ=infty} . So, there is the following inclusion:\sigma _{disc} (A)\subset\{ isolatedpointsofthespectrumof A withfinitealgebraicmultiplicity \}. In particular, for a quasinilpotent operatorQ:l ^{2(\N)\to} l ^{2(\N),} Q:(a _{1,a} _{2,a} _{3,...)\mapsto} ( 0 ,a _{1/2,a} ^{3 , ... ),} ^{3 / 2} one hasak{L} _{λ(Q)=\{0\}} , rank P _{λ=infty} ,\sigma(Q) = \{ 0 \} ,\sigma _{disc} (Q) = \emptyset .

Relation to the point spectrum

The discrete spectrum

\sigma _{disc} (A) of an operator A is not to be confused with the point spectrum \sigma _{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 _{disc} (A)\subset\sigma _{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), L:(a} _{1 ,a} _{2,a} _{3,...)\mapsto} (a _{2,a} _{3,a} _{4,...).} 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 _{p} (L) = D _{1, \sigma(L)=\overline{D} _{1}; \sigma} _{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. Book: Reed, M.. Simon, B.. Methods of modern mathematical physics, vol. IV. Analysis of operators. 1978. Academic Press [Harcourt Brace Jovanovich Publishers], New York.
2. Gohberg, I. C. Kreĭn, M. G.. Fundamental aspects of defect numbers, root numbers and indexes of linear operators. American Mathematical Society Translations. 13. 1960. 185–264.
3. Book: Gohberg, I. C. Kreĭn, M. G.. Introduction to the theory of linear nonselfadjoint operators. 1969. American Mathematical Society, Providence, R.I..
4. Book: Boussaid, N.. Comech, A.. Nonlinear Dirac equation. Spectral stability of solitary waves. 2019. American Mathematical Society, Providence, R.I.. 978-1-4704-4395-5.