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In [[spectral theory]], foran [[eigenvalue]] of a [[Unbounded_operator#Closed_linear_operators|closed linear operatorsoperator]] whichis are not necessarily [[Self-adjoint operators|self-adjoint]], the set ofcalled '''normal eigenvalues''' is defined as a subset of the [[point spectrum]] <math>\sigma_p(A)</math> of <math>A</math> such thatif the space admits a decomposition into a direct sum of a finite-dimensional [[generalized eigenspace]] and an [[invariant subspace]] where <math>A-\lambda I</math> has a bounded inverse.
==Root lineal==
Let <math>\mathbfmathfrak{XB}</math> be a [[Banach space]]. The [[Generalized_eigenvector#Root_lineal_of_a_linear_operator_in_a_Banach_space|root lineal]] <math>\mathfrak{L}_\lambda(A)</math> of a linear operator <math>A:\,\mathbfmathfrak{XB}\to\mathbfmathfrak{XB}</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_{\mathbfmathfrak{XB}})^j x\in\mathfrak{D}(A)\,\forall j\in\N,\,j\le k;\, (A-\lambda I_{\mathbfmathfrak{XB}})^k x=0\}\subset\mathbfmathfrak{B}</math>, where <math>I_{\mathfrak{B}}</math> is the identity operator in <math>\mathfrak{XB}</math>.
This set is a [[linear manifold]] but not necessarily a [[vector space]], since it is not necessarily closed in <math>\mathbfmathfrak{XB}</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:\,\mathbfmathfrak{XB}\to\mathbfmathfrak{XB}</math> in the [[Banach space]] <math>\mathbfmathfrak{XB}</math> with [[Unbounded_operator#Definitions_and_basic_properties|___domain]] <math>\mathfrak{D}(A)\subset\mathbfmathfrak{XB}</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>\mathbfmathfrak{XB}</math> could be decomposed into a direct sum <math>\mathbfmathfrak{XB}=\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_{\mathbfmathfrak{XB}}</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
==Equivalent definitions of normal eigenvalues==
Let <math>A:\,\mathbfmathfrak{XB}\to\mathbfmathfrak{XB}</math> be a closed linear [[densely defined operator]] in the Banach space <math>\mathbfmathfrak{XB}</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_{\mathbfmathfrak{XB}}</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_{\mathbfmathfrak{XB}}</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_{\mathbfmathfrak{XB}}</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_{\mathbfmathfrak{XB}}</math> is [[Closed range theorem|closed]]. (Gohberg–Krein 1957, 1960, 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> (Gohberg–Krein 1969).
==Decomposition of the spectrum of nonselfadjoint operators==
The spectrum of a closed operator <math>A:\,\mathbfmathfrak{XB}\to\mathbfmathfrak{XB}</math> in the Banach space <math>\mathbfmathfrak{XB}</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).
* [[Spectrum of an operator]]
* [[Resolvent formalism]]
* [[Riesz projector]]
* [[Fredholm operator]]
* [[Operator theory]]
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