Normal eigenvalue: Difference between revisions

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# <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]]. (Gohberg–Krein 1957, 1960, 1969).

ifIf <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).

The equivalence of (1) and (5) is proved in Theorem 4.1 of (Gohberg–Krein 1957, 1960).

The equivalence of (1) and (3) is proved in Theorem 4.2 of (Gohberg–Krein 1957, 1960), and then equivalence of (1) with (2) and (4) follows from the [[Fredholm_operator#Properties|stability of the index]].

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==