Normal eigenvalue: Difference between revisions

Revision as of 03:58, 7 October 2019

Revision as of 15:13, 13 October 2019

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In mathematics, specifically in [[spectral theory]], an [[eigenvalue]] of a [[Unbounded_operatorUnbounded operator#Closed_linear_operatorsClosed 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]].

==Root lineal==

Let <math>\mathfrak{B}</math> be a [[Banach space]]. The [[Generalized_eigenvectorGeneralized eigenvector#Root_lineal_of_a_linear_operator_in_a_Banach_spaceRoot 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>

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==Definition==

An [[eigenvalue]] <math>\lambda\in\sigma_p(A)</math> of a [[Unbounded_operatorUnbounded operator#Closed_linear_operatorsClosed linear operators|closed linear operator]] <math>A:\,\mathfrak{B}\to\mathfrak{B}</math> in the [[Banach space]] <math>\mathfrak{B}</math> with [[Unbounded_operatorUnbounded operator#Definitions_and_basic_propertiesDefinitions 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_eigenvectorGeneralized eigenvector#Root_lineal_of_a_linear_operator_in_a_Banach_spaceRoot 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.

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|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=UspehiUspekhi Mat. Nauk (N.S.)|series=New Series

|volume=12

|issue=2(74)

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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:

# <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_operatorFredholm operator#semi-Fredholm_operatorsFredholm 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;