Normal eigenvalue

In mathematics, specifically in spectral theory, an eigenvalue of a closed linear operator is called normal if 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 ${\mathfrak {B}}$ be a Banach space. The root lineal ${\mathfrak {L}}_{\lambda }(A)$ of a linear operator $A:\,{\mathfrak {B}}\to {\mathfrak {B}}$ with domain ${\mathfrak {D}}(A)$ corresponding to the eigenvalue $\lambda \in \sigma _{p}(A)$ is defined as

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

where $I_{\mathfrak {B}}$ is the identity operator in ${\mathfrak {B}}$. This set is a linear manifold but not necessarily a vector space, since it is not necessarily closed in ${\mathfrak {B}}$. 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 $A:\,{\mathfrak {B}}\to {\mathfrak {B}}$ in the Banach space ${\mathfrak {B}}$ with domain ${\mathfrak {D}}(A)\subset {\mathfrak {B}}$ is called normal (in the original terminology, $\lambda$ corresponds to a normally splitting finite-dimensional root subspace), 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 ${\mathfrak {B}}$ could be decomposed into a direct sum ${\mathfrak {B}}={\mathfrak {L}}_{\lambda }(A)\oplus {\mathfrak {N}}_{\lambda }$, where ${\mathfrak {N}}_{\lambda }$ is an invariant subspace of $A$ in which $A-\lambda I_{\mathfrak {B}}$ 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 $A:\,{\mathfrak {B}}\to {\mathfrak {B}}$ be a closed linear densely defined operator in the Banach space ${\mathfrak {B}}$. 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 $A-\lambda I_{\mathfrak {B}}$ is semi-Fredholm;
3. $\lambda \in \sigma (A)$ is an isolated point in $\sigma (A)$ and $A-\lambda I_{\mathfrak {B}}$ is Fredholm;
4. $\lambda \in \sigma (A)$ is an isolated point in $\sigma (A)$ and $A-\lambda I_{\mathfrak {B}}$ is Fredholm of index zero;
5. $\lambda \in \sigma (A)$ is an isolated point in $\sigma (A)$ and the rank of the corresponding Riesz projector $P_{\lambda }$ is finite;
6. $\lambda \in \sigma (A)$ is an isolated point in $\sigma (A)$, its algebraic multiplicity $\nu =\dim {\mathfrak {L}}_{\lambda }$ is finite, and the range of $A-\lambda I_{\mathfrak {B}}$ is closed. (Gohberg–Krein 1957, 1960, 1969).

If $\lambda$ is a normal eigenvalue, then ${\mathfrak {L}}_{\lambda }$ coincides with the range of the Riesz projector, ${\mathfrak {R}}(P_{\lambda })$ (Gohberg–Krein 1969).

Decomposition of the spectrum of nonselfadjoint operators

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

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

See also

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