Normal eigenvalue

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

1. The algebraic multiplicity of $\lambda$ is finite;
2. The space $\mathbf {X}$ could be decomposed into a direct sum $\mathbf {X} ={\mathfrak {L}}_{\lambda }(A)\oplus {\mathfrak {N}}_{\lambda }$, where ${\mathfrak {L}}_{\lambda }(A)$ is the root lineal of $A$ corresponding to the eigenvalue $\lambda \in \sigma (A)$ and ${\mathfrak {N}}_{\lambda }$ 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.

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

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

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

See also

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