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#REDIRECT [[Discrete spectrum (mathematics)]]
In [[Spectral theory]], an [[eigenvalue]] <math>\lambda</math> of a [[Unbounded_operator#Closed_linear_operators|closed linear operator]] <math>A:\,\mathbf{X}\to\mathbf{X}</math> in the [[Banach space]] <math>\mathbf{X}</math> with ___domain <math>\mathfrak{D}(A)</math> is called ''normal'' if the following two conditions are satisfied:
# The [[algebraic multiplicity]] of <math>\lambda</math> is finite;
# The space <math>\mathbf{X}</math> could be decomposed into a direct sum <math>\mathbf{X}=\mathfrak{L}_\lambda(A)\oplus \mathfrak{N}_\lambda</math>, where <math>\mathfrak{L}_\lambda(A)</math> is the [[root lineal]] of <math>A</math> corresponding to the eigenvalue <math>\lambda\in\sigma(A)</math> and <math>\mathfrak{N}_\lambda</math> is an [[invariant subspace]] of <math>A</math> in which <math>A-\lambda I_{\mathbf{X}}</math> has a bounded inverse.
{{Rcat shell|
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
{{R from merge}}
|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=Uspehi Mat. Nauk (N.S.)
|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 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.
}}
</ref>
We recall that the [[root lineal]] <math>\mathfrak{L}_\lambda(A)</math> of a linear operator <math>A:\,\mathbf{X}\to\mathbf{X}</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)=\cup_{k\in\N}\{x\in\mathfrak{D}(A):\,(A-\lambda I_{\mathbf{X}})^j x\in\mathfrak{D}(A)\,\forall j\le k,\, (A-\lambda I_{\mathbf{X}})^k x=0\}</math>.
This set is a [[linear manifold]] but not necessarily a [[vector space]], since it is not necessarily closed.
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>.
==See also==
{{Portal|Mathematics}}
* [[Spectrum (functional analysis)]]
* [[Decomposition of spectrum (functional analysis)]]
* [[Essential spectrum]]
* [[Spectrum of an operator]]
* [[Resolvent formalism]]
* [[Operator theory]]
* [[Fredholm theory]]
==References==
<references />
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