Normal eigenvalue: Difference between revisions

Revision as of 03:47, 24 September 2019

Revision as of 03:58, 7 October 2019

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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]].

==Root lineal==

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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 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==

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* [[Spectrum (functional analysis)]]

* [[Decomposition of spectrum (functional analysis)]]

* [[Discrete spectrum (Mathematics)]]

* [[Essential spectrum]]

* [[Spectrum of an operator]]