Normal eigenvalue: Difference between revisions

Revision as of 05:29, 22 September 2019

Revision as of 01:24, 23 September 2019

Line 7:

==Definition==

An [[eigenvalue]] <math>\lambda\in\sigma_p(A)</math> of a [[Unbounded_operator#Closed_linear_operators|closed linear operator]] <math>A:\,\mathbf{X}\to\mathbf{X}</math> in the [[Banach space]] <math>\mathbf{X}</math> with [[Unbounded_operator#Definitions_and_basic_properties|___domain]] <math>\mathfrak{D}(A)\subset\mathbf{X}</math> is called ''normal'' (in the original terminology, ''<math>\lambda</math> corresponds to a normally splitting finite-dimensional root subspace''), if the following two conditions are satisfied:

# The [[algebraic multiplicity]] of <math>\lambda</math> is finite: <math>\nu=\dim\mathfrak{L}_\lambda(A)<\infty</math>, where <math>\mathfrak{L}_\lambda(A)</math> is the [[Generalized_eigenvector#Root_lineal_of_a_linear_operator_in_a_Banach_space|root lineal]] of <math>A</math> corresponding to the eigenvalue <math>\lambda</math>;

# The space <math>\mathbf{X}</math> could be decomposed into a direct sum <math>\mathbf{X}=\mathfrak{L}_\lambda(A)\oplus \mathfrak{N}_\lambda</math>, where <math>\mathfrak{N}_\lambda</math> is an [[invariant subspace]] of <math>A</math> in which <math>A-\lambda I_{\mathbf{X}}</math> has a bounded inverse.

Line 23:

|url=http://mi.mathnet.ru/umn7581

|trans-journal=Amer. Math. Soc. Transl. (2)

}}</ref><ref>{{ cite bookjournal

}}

|author1=Gohberg, I. C

</ref><ref>{{ cite book

|author2=Kreĭn, M. G.

|title=Fundamental aspects of defect numbers, root numbers and indexes of linear operators

|journal=American Mathematical Society Translations

|volume=13

|year=1960

|pages=185–264

|url=http://mi.mathnet.ru/umn7581

}}</ref><ref>{{ cite book

|author1=Gohberg, I. C

|author2=Kreĭn, M. G.

Line 38 ⟶ 46:

# <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]];

# <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]].

The equivalence of (1) and (3) is proved in Lemma 4.2 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 (5) is proved in Theorem 2.1 of (Gohberg–Krein 1969).

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==