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. 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 (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 $\mathbf {X}$ 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;
4. $\lambda \in \sigma (A)$ is an isolated point in $\sigma (A)$ and $A-\lambda I_{\mathbf {X} }$ 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 $A-\lambda I_{\mathbf {X} }$ is 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 $\lambda$ is a normal eigenvalue, then ${\mathfrak {L}}_{\lambda }$ coincides with the range of the Riesz projector, ${\mathfrak {R}}(P_{\lambda })$.

Decomposition of the spectrum of nonselfadjoint operators

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
• Fredholm operator
• Operator theory