Normal eigenvalue

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 ${\displaystyle A-\lambda I}$ has a bounded inverse.

Root lineal

Let ${\displaystyle {\mathfrak {B}}}$  be a Banach space. The root lineal ${\displaystyle {\mathfrak {L}}_{\lambda }(A)}$  of a linear operator ${\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}$  with ___domain ${\displaystyle {\mathfrak {D}}(A)}$  corresponding to the eigenvalue ${\displaystyle \lambda \in \sigma _{p}(A)}$  is defined as

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

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

Definition

An eigenvalue ${\displaystyle \lambda \in \sigma _{p}(A)}$  of a closed linear operator ${\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}$  in the Banach space ${\displaystyle {\mathfrak {B}}}$  with ___domain ${\displaystyle {\mathfrak {D}}(A)\subset {\mathfrak {B}}}$  is called normal (in the original terminology, ${\displaystyle \lambda }$  corresponds to a normally splitting finite-dimensional root subspace), if the following two conditions are satisfied:

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

That is, the restriction ${\displaystyle A_{2}}$  of ${\displaystyle A}$  onto ${\displaystyle {\mathfrak {N}}_{\lambda }}$  is an operator with ___domain ${\displaystyle {\mathfrak {D}}(A_{2})={\mathfrak {N}}_{\lambda }\cap {\mathfrak {D}}(A)}$  and with the range ${\displaystyle {\mathfrak {R}}(A_{2}-\lambda I)\subset {\mathfrak {N}}_{\lambda }}$  which has a bounded inverse.^[1][2][3]

Equivalent definitions of normal eigenvalues

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

1. ${\displaystyle \lambda \in \sigma (A)}$  is a normal eigenvalue;
2. ${\displaystyle \lambda \in \sigma (A)}$  is an isolated point in ${\displaystyle \sigma (A)}$  and ${\displaystyle A-\lambda I_{\mathfrak {B}}}$  is semi-Fredholm;
3. ${\displaystyle \lambda \in \sigma (A)}$  is an isolated point in ${\displaystyle \sigma (A)}$  and ${\displaystyle A-\lambda I_{\mathfrak {B}}}$  is Fredholm;
4. ${\displaystyle \lambda \in \sigma (A)}$  is an isolated point in ${\displaystyle \sigma (A)}$  and ${\displaystyle A-\lambda I_{\mathfrak {B}}}$  is Fredholm of index zero;
5. ${\displaystyle \lambda \in \sigma (A)}$  is an isolated point in ${\displaystyle \sigma (A)}$  and the rank of the corresponding Riesz projector ${\displaystyle P_{\lambda }}$  is finite;
6. ${\displaystyle \lambda \in \sigma (A)}$  is an isolated point in ${\displaystyle \sigma (A)}$ , its algebraic multiplicity ${\displaystyle \nu =\dim {\mathfrak {L}}_{\lambda }}$  is finite, and the range of ${\displaystyle A-\lambda I_{\mathfrak {B}}}$  is closed. (Gohberg–Krein 1957, 1960, 1969).

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

Decomposition of the spectrum of nonselfadjoint operators

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

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

See also

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

References

1. Gohberg, I. C; Kreĭn, M. G. (1957). "Основные положения о дефектных числах, корневых числах и индексах линейных операторов" [Fundamental aspects of defect numbers, root numbers and indexes of linear operators]. Uspehi Mat. Nauk (N.S.) [Amer. Math. Soc. Transl. (2)]. 12 (2(74)): 43–118.
2. Gohberg, I. C; Kreĭn, M. G. (1960). "Fundamental aspects of defect numbers, root numbers and indexes of linear operators". American Mathematical Society Translations. 13: 185–264.
3. Gohberg, I. C; Kreĭn, M. G. (1969). Introduction to the theory of linear nonselfadjoint operators. American Mathematical Society, Providence, R.I.