Normal eigenvalue: Difference between revisions

Revision as of 22:06, 20 September 2019

Revision as of 22:34, 20 September 2019

Line 1:

In [[Spectral theory]], anfor [[eigenvalueUnbounded_operator#Closed_linear_operators|closed linear operators]] <math>\lambda</math>which ofare anot necessarily [[Unbounded_operator#Closed_linear_operatorsSelfadjoint operators|closedselfadjoint]], linearthe set of ''normal eigenvalues'' is defined as a subset of the [[point operatorspectrum]] <math>A:\,\mathbf{X}\to\mathbf{X}sigma_p(A)</math> in the [[Banach space]]of <math>\mathbf{X}A</math> withsuch ___domainthat the space <math>\mathfrakmathbf{DX}(A)</math> isadmits calleda ''normal''decomposition ifinto the following[[eigenspace]] twoand conditions[[invariant are satisfied:space]].

# The [[algebraic multiplicity]] of <math>\lambda</math> is finite;

==Root lineal==

# 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{L}_\lambda(A)</math> is the [[root lineal]] of <math>A</math> corresponding to the eigenvalue <math>\lambda\in\sigma(A)</math> and <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.

We recall that the [[root lineal]] <math>\mathfrak{L}_\lambda(A)</math> of a linear operator <math>A:\,\mathbf{X}\to\mathbf{X}</math> with ___domain <math>\mathfrak{D}(A)</math> corresponding to the eigenvalue <math>\lambda\in\sigma_p(A)</math> is defined as

:<math>\mathfrak{L}_\lambda(A)=\cup_{k\in\N}\{x\in\mathfrak{D}(A):\,(A-\lambda I_{\mathbf{X}})^j x\in\mathfrak{D}(A)\,\forall j\in\N,\,j\le k;\, (A-\lambda I_{\mathbf{X}})^k x=0\}\subset\mathbf{X}</math>.

This set is a [[linear manifold]] but not necessarily a [[vector space]], since it is not necessarily closed in <math>\mathbf{X}</math>. If this set is closed (for example, when it is finite-dimensional), it is called the [[generalized eigenspace]] of <math>A</math> corresponding to the eigenvalue <math>\lambda</math>.

==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'' 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 [[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{L}_\lambda(A)</math> is the [[root lineal]] of <math>A</math> corresponding to the eigenvalue <math>\lambda\in\sigma(A)</math> and <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.

That is, the restriction <math>A_2</math> of <math>A</math> onto <math>\mathfrak{N}_\lambda</math> is an operator with ___domain <math>\mathfrak{D}(A_2)=\mathfrak{N}_\lambda\cap\mathfrak{D}(A)</math> and with the range <math>\mathfrak{R}(A_2-\lambda I)\subset\mathfrak{N}_\lambda</math> which has a bounded inverse.<ref>{{ cite journal

Line 25 ⟶ 33:

</ref>

==Equivalent definitions of normal eigenvalues==

We recall that the [[root lineal]] <math>\mathfrak{L}_\lambda(A)</math> of a linear operator <math>A:\,\mathbf{X}\to\mathbf{X}</math> with ___domain <math>\mathfrak{D}(A)</math> corresponding to the eigenvalue <math>\lambda\in\sigma_p(A)</math> is defined as

:<math>\mathfrak{L}_\lambda(A)=\cup_{k\in\N}\{x\in\mathfrak{D}(A):\,(A-\lambda I_{\mathbf{X}})^j x\in\mathfrak{D}(A)\,\forall j\in\N,\,j\le k;\, (A-\lambda I_{\mathbf{X}})^k x=0\}</math>.

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:

This set is a [[linear manifold]] but not necessarily a [[vector space]], since it is not necessarily closed.

# <math>\lambda\in\sigma(A)</math> is a normal eigenvalue;

If this set is closed (for example, when it is finite-dimensional), it is called the [[generalized eigenspace]] of <math>A</math> corresponding to the eigenvalue <math>\lambda</math>.

# <math>\lambda\in\sigma(A)</math> is an isolated point in <math>\sigma(A)</math> and <math>A-\lambda I_{\mathbf{X}}</math> is [[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;

# <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]].

==Decomposition of the spectrum of nonselfadjoint operators==

One can show that the spectrum of a closed operator <math>A:\,\mathbf{X}\to\mathbf{X}</math> in the Banach space <math>\mathbf{X}</math> can be decomposed into the union of two disjoint sets, the set of normal eigenvalues and the fifth type of the [[essential spectrum]]:

:<math>

\sigma(A)=\{\mathrm{normal\ eigenvalues\ of}\ A\}\cup\sigma_{\mathrm{ess},5}(A).

</math>

==See also==

{{Portal|Mathematics}}

* [[Spectrum (functional analysis)]]

* [[Decomposition of spectrum (functional analysis)]]