Normal eigenvalue: Difference between revisions

Revision as of 01:24, 23 September 2019

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# <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]].

The equivalence of (1) and (35) is proved in LemmaTheorem 4.21 of (Gohberg–Krein 1957, 1960), and then equivalence of (1) with (2) and (4) follows from the continuity of the index.

The equivalence of (1) and (53) is proved in Theorem 24.12 of (Gohberg–Krein 19691957, 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).