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#REDIRECT [[Discrete spectrum (mathematics)]]
In [[spectral theory]], for [[Unbounded_operator#Closed_linear_operators|closed linear operators]] which are not necessarily [[Self-adjoint operators|self-adjoint]], the set of '''normal eigenvalues''' is defined as a subset of the [[point spectrum]] <math>\sigma_p(A)</math> of <math>A</math> such that 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.
{{Rcat shell|
==Root lineal==
{{R from merge}}
Let <math>\mathbf{X}</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:\,\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\in\N,\,j\le k;\, (A-\lambda I_{\mathbf{X}})^k x=0\}\subset\mathbf{X}</math>.
This set is a [[linear manifold]] but not necessarily a [[vector space]], since it is not necessarily closed in <math>\mathbf{X}</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:\,\mathbf{X}\to\mathbf{X}</math> in the [[Banach space]] <math>\mathbf{X}</math> with [[Unbounded_operator#Definitions_and_basic_properties|___domain]] <math>\mathfrak{D}(A)\subset\mathbf{X}</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>\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{N}_\lambda</math> is an [[invariant subspace]] of <math>A</math> in which <math>A-\lambda I_{\mathbf{X}}</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=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 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.
}}
</ref>
==Equivalent definitions of normal eigenvalues==
Let <math>A:\,\mathbf{X}\to\mathbf{X}</math> be a closed linear [[densely defined operator]] in the Banach space <math>\mathbf{X}</math>. The following statements are equivalent:
# <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_{\mathbf{X}}</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_{\mathbf{X}}</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_{\mathbf{X}}</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_{\mathbf{X}}</math> is [[Closed range theorem|closed]].
The equivalence of (1) and (3) is proved in Lemma 4.2 of (Gohberg–Krein 1957, 1960), and then equivalence of (1) with (2) and (4) follows from the continuity of the index.
The equivalence of (1) and (5) is proved in Theorem 2.1 of (Gohberg–Krein 1969).
The equivalence of (1) and (6) is stated in (Gohberg–Krein 1969, Chapter 1, §2.1).
By Theorem 2.1 of (Gohberg–Krein 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>.
==Decomposition of the spectrum of nonselfadjoint operators==
The spectrum of a closed operator <math>A:\,\mathbf{X}\to\mathbf{X}</math> in the Banach space <math>\mathbf{X}</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)=\{\mathrm{normal\ eigenvalues\ of}\ A\}\cup\sigma_{\mathrm{ess},5}(A).
</math>
==See also==
* [[Spectrum (functional analysis)]]
* [[Decomposition of spectrum (functional analysis)]]
* [[Essential spectrum]]
* [[Spectrum of an operator]]
* [[Resolvent formalism]]
* [[Fredholm operator]]
* [[Operator theory]]
==References==
<references />
[[Category:Spectral theory]]
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