Normal eigenvalue

In Spectral theory, for closed linear operators which are not necessarily self-adjoint, the set of normal eigenvalues is defined as a subset of the point spectrum $\sigma _{p}(A)$ of $A$ such that the space admits a decomposition into a direct sum of a finite-dimensional generalized eigenspace and an invariant subspace where $A-\lambda I$ has a bounded inverse.

Root lineal

Let $\mathbf {X}$ be a Banach space. We recall that the root lineal ${\mathfrak {L}}_{\lambda }(A)$ of a linear operator $A:\,\mathbf {X} \to \mathbf {X}$ with domain ${\mathfrak {D}}(A)$ corresponding to the eigenvalue $\lambda \in \sigma _{p}(A)$ is defined as

${\mathfrak {L}}_{\lambda }(A)=\cup _{k\in \mathbb {N} }\{x\in {\mathfrak {D}}(A):\,(A-\lambda I_{\mathbf {X} })^{j}x\in {\mathfrak {D}}(A)\,\forall j\in \mathbb {N} ,\,j\leq k;\,(A-\lambda I_{\mathbf {X} })^{k}x=0\}\subset \mathbf {X}$.

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

Definition

An eigenvalue $\lambda \in \sigma _{p}(A)$ of a closed linear operator $A:\,\mathbf {X} \to \mathbf {X}$ in the Banach space $\mathbf {X}$ with domain ${\mathfrak {D}}(A)\subset \mathbf {X}$ is called normal if the following two conditions are satisfied:

1. The algebraic multiplicity of $\lambda$ is finite: $\nu =\dim {\mathfrak {L}}_{\lambda }(A)<\infty$, where ${\mathfrak {L}}_{\lambda }(A)$ is the root lineal of $A$ corresponding to the eigenvalue $\lambda$;
2. The space $\mathbf {X}$ could be decomposed into a direct sum $\mathbf {X} ={\mathfrak {L}}_{\lambda }(A)\oplus {\mathfrak {N}}_{\lambda }$, where ${\mathfrak {N}}_{\lambda }$ is an invariant subspace of $A$ in which $A-\lambda I_{\mathbf {X} }$ has a bounded inverse.

That is, the restriction $A_{2}$ of $A$ onto ${\mathfrak {N}}_{\lambda }$ is an operator with domain ${\mathfrak {D}}(A_{2})={\mathfrak {N}}_{\lambda }\cap {\mathfrak {D}}(A)$ and with the range ${\mathfrak {R}}(A_{2}-\lambda I)\subset {\mathfrak {N}}_{\lambda }$ which has a bounded inverse.

Equivalent definitions of normal eigenvalues

Let $A:\,\mathbf {X} \to \mathbf {X}$ be a closed linear densely defined operator in the Banach space $\mathbf {X}$. The following statements are equivalent:

1. $\lambda \in \sigma (A)$ is a normal eigenvalue;
2. $\lambda \in \sigma (A)$ is an isolated point in $\sigma (A)$ and $A-\lambda I_{\mathbf {X} }$ is semi-Fredholm;
3. $\lambda \in \sigma (A)$ is an isolated point in $\sigma (A)$ and $A-\lambda I_{\mathbf {X} }$ is Fredholm of index zero;
4. $\lambda \in \sigma (A)$ is an isolated point in $\sigma (A)$ and the rank of the corresponding Riesz projector $P_{\lambda }$ is finite;
5. $\lambda \in \sigma (A)$ is an isolated point in $\sigma (A)$, its algebraic multiplicity $\nu =\dim {\mathfrak {L}}_{\lambda }$is finite, and the range of $A-\lambda I_{\mathbf {X} }$ is closed.

Decomposition of the spectrum of nonselfadjoint operators

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

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

See also

• Spectrum (functional analysis)
• Decomposition of spectrum (functional analysis)
• Essential spectrum
• Spectrum of an operator
• Resolvent formalism
• Operator theory
• Fredholm theory