Normal eigenvalue: Difference between revisions

Revision as of 15:13, 13 October 2019

Revision as of 00:02, 13 January 2020

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==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]];