*#REDIRECT [[Discrete spectrum ( Mathematicsmathematics)]] ▼In mathematics, specifically in [[spectral theory]], an [[eigenvalue]] of a [[Unbounded operator#Closed linear operators|closed linear operator]] is called '''normal''' if the space admits a decomposition into a direct sum of a finite-dimensional [[generalized eigenspace]] and an [[invariant subspace]] where <math>A-\lambda I</math> has a bounded inverse.
The set of normal eigenvalues coincides with the [[discrete spectrum (mathematics)|discrete spectrum]].
{{Rcat shell|
==Root lineal==
{{R from merge}}
Let <math>\mathfrak{B}</math> be a [[Banach space]]. The [[Generalized eigenvector#Root lineal of a linear operator in a Banach space|root lineal]] <math>\mathfrak{L}_\lambda(A)</math> of a linear operator <math>A:\,\mathfrak{B}\to\mathfrak{B}</math> with ___domain <math>\mathfrak{D}(A)</math> corresponding to the eigenvalue <math>\lambda\in\sigma_p(A)</math> is defined as
: <math>\mathfrak{L}_\lambda(A)=\bigcup_{k\in\N}\{x\in\mathfrak{D}(A):\,(A-\lambda I_{\mathfrak{B}})^j x\in\mathfrak{D}(A)\,\forall j\in\N,\,j\le k;\, (A-\lambda I_{\mathfrak{B}})^k x=0\}\subset\mathfrak{B}, </math>
where <math>I_{\mathfrak{B}}</math> is the identity operator in <math>\mathfrak{B}</math>.
This set is a [[linear manifold]] but not necessarily a [[vector space]], since it is not necessarily closed in <math>\mathfrak{B}</math>. If this set is closed (for example, when it is finite-dimensional), it is called the [[generalized eigenspace]] of <math>A</math> corresponding to the eigenvalue <math>\lambda</math>.
==Definition==
An [[eigenvalue]] <math>\lambda\in\sigma_p(A)</math> of a [[Unbounded operator#Closed linear operators|closed linear operator]] <math>A:\,\mathfrak{B}\to\mathfrak{B}</math> in the [[Banach space]] <math>\mathfrak{B}</math> with [[Unbounded operator#Definitions and basic properties|___domain]] <math>\mathfrak{D}(A)\subset\mathfrak{B}</math> is called ''normal'' (in the original terminology, ''<math>\lambda</math> corresponds to a normally splitting finite-dimensional root subspace''), if the following two conditions are satisfied:
# The [[algebraic multiplicity]] of <math>\lambda</math> is finite: <math>\nu=\dim\mathfrak{L}_\lambda(A)<\infty</math>, where <math>\mathfrak{L}_\lambda(A)</math> is the [[Generalized eigenvector#Root lineal of a linear operator in a Banach space|root lineal]] of <math>A</math> corresponding to the eigenvalue <math>\lambda</math>;
# The space <math>\mathfrak{B}</math> could be decomposed into a direct sum <math>\mathfrak{B}=\mathfrak{L}_\lambda(A)\oplus \mathfrak{N}_\lambda</math>, where <math>\mathfrak{N}_\lambda</math> is an [[invariant subspace]] of <math>A</math> in which <math>A-\lambda I_{\mathfrak{B}}</math> has a bounded inverse.
That is, the restriction <math>A_2</math> of <math>A</math> onto <math>\mathfrak{N}_\lambda</math> is an operator with ___domain <math>\mathfrak{D}(A_2)=\mathfrak{N}_\lambda\cap\mathfrak{D}(A)</math> and with the range <math>\mathfrak{R}(A_2-\lambda I)\subset\mathfrak{N}_\lambda</math> which has a bounded inverse.<ref>{{ cite journal
|author1=Gohberg, I. C
|author2=Kreĭn, M. G.
|title=Основные положения о дефектных числах, корневых числах и индексах линейных операторов
|trans-title=Fundamental aspects of defect numbers, root numbers and indexes of linear operators
|journal=Uspekhi Mat. Nauk |series=New Series
|volume=12
|issue=2(74)
|year=1957
|pages=43–118
|url=http://mi.mathnet.ru/umn7581
|trans-journal=Amer. Math. Soc. Transl. (2)
}}</ref><ref>{{ 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
}}</ref><ref>{{ 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
}}
</ref>
==Equivalent definitions of normal eigenvalues==
Let <math>A:\,\mathfrak{B}\to\mathfrak{B}</math> be a closed linear [[densely defined operator]] in the Banach space <math>\mathfrak{B}</math>. The following statements are equivalent<ref>{{ 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
}}</ref>(Theorem III.88):
# <math>\lambda\in\sigma(A)</math> is a normal eigenvalue;
# <math>\lambda\in\sigma(A)</math> is an isolated point in <math>\sigma(A)</math> and <math>A-\lambda I_{\mathfrak{B}}</math> is [[Fredholm operator#semi-Fredholm operators|semi-Fredholm]];
# <math>\lambda\in\sigma(A)</math> is an isolated point in <math>\sigma(A)</math> and <math>A-\lambda I_{\mathfrak{B}}</math> is [[Fredholm operator|Fredholm]];
# <math>\lambda\in\sigma(A)</math> is an isolated point in <math>\sigma(A)</math> and <math>A-\lambda I_{\mathfrak{B}}</math> is [[Fredholm operator|Fredholm]] of index zero;
# <math>\lambda\in\sigma(A)</math> is an isolated point in <math>\sigma(A)</math> and the rank of the corresponding [[Riesz projector]] <math>P_\lambda</math> is finite;
# <math>\lambda\in\sigma(A)</math> is an isolated point in <math>\sigma(A)</math>, its algebraic multiplicity <math>\nu=\dim\mathfrak{L}_\lambda</math> is finite, and the range of <math>A-\lambda I_{\mathfrak{B}}</math> is [[Closed range theorem|closed]]. (Gohberg–Krein 1957, 1960, 1969).
If <math>\lambda</math> is a normal eigenvalue, then <math>\mathfrak{L}_\lambda</math> coincides with the range of the Riesz projector, <math>\mathfrak{R}(P_\lambda)</math> (Gohberg–Krein 1969).
==Relation to the discrete spectrum==
The above equivalence shows that the set of normal eigenvalues coincides with the [[Discrete spectrum (Mathematics)|discrete spectrum]], defined as the set of isolated points of the spectrum with finite rank of the corresponding Riesz projector.<ref>{{ 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
}}
</ref>
==Decomposition of the spectrum of nonselfadjoint operators==
The spectrum of a closed operator <math>A:\,\mathfrak{B}\to\mathfrak{B}</math> in the Banach space <math>\mathfrak{B}</math> can be decomposed into the union of two disjoint sets, the set of normal eigenvalues and the fifth type of the [[essential spectrum]]:
:<math>
\sigma(A)=\{\text{normal eigenvalues of}\ A\}\cup\sigma_{\mathrm{ess},5}(A).
</math>
==See also==
* [[Spectrum (functional analysis)]]
* [[Decomposition of spectrum (functional analysis)]]
▲* [[Discrete spectrum (Mathematics)]]
* [[Essential spectrum]]
* [[Spectrum of an operator]]
* [[Resolvent formalism]]
* [[Riesz projector]]
* [[Fredholm operator]]
* [[Operator theory]]
==References==
<references />
[[Category:Spectral theory]]
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