Normal eigenvalue: Difference between revisions

Revision as of 22:37, 20 September 2019

Revision as of 23:06, 20 September 2019

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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]] 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;