Normal eigenvalue explained
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
has a bounded inverse.The set of normal eigenvalues coincides with the
discrete spectrum.
Root lineal
Let
be a
Banach space. The root lineal
of a linear operator
with domain
corresponding to the eigenvalue
is defined as
ak{L}λ(A)=cupk\in\N\{x\inak{D}(A):(A-λIak{B
})^j x\in\mathfrak(A)\,\forall j\in\N,\,j\le k;\, (A-\lambda I_)^k x=0\}\subset\mathfrak,
where
} is the identity operator in
.This set is a
linear manifold but not necessarily a
vector space, since it is not necessarily closed in
. If this set is closed (for example, when it is finite-dimensional), it is called the generalized eigenspace of
corresponding to the eigenvalue
.
Definition of a normal eigenvalue
of a closed linear operator
in the
Banach space
with domain
is called
normal (in the original terminology,
corresponds to a normally splitting finite-dimensional root subspace), if the following two conditions are satisfied:
- The algebraic multiplicity of
is finite:
, where
is the root lineal of
corresponding to the eigenvalue
;
- The space
could be decomposed into a direct sum
, where
is an
invariant subspace of
in which
} has a bounded inverse.
That is, the restriction
of
onto
is an operator with domain
ak{D}(A2)=ak{N}λ\capak{D}(A)
and with the range
ak{R}(A2-λI)\subsetak{N}λ
which has a bounded inverse.
[1] [2] [3] Equivalent characterizations of normal eigenvalues
Let
be a closed linear
densely defined operator in the Banach space
. The following statements are equivalent
[4] (Theorem III.88):
is a normal eigenvalue;
is an isolated point in
and
} is semi-Fredholm;
is an isolated point in
and
} is
Fredholm;
is an isolated point in
and
} is
Fredholm of index zero;
is an isolated point in
and the rank of the corresponding
Riesz projector
is finite;
is an isolated point in
, its algebraic multiplicity
is finite, and the range of
} is
closed.
If
is a normal eigenvalue, then the root lineal
coincides with the range of the Riesz projector,
.
Relation to the discrete spectrum
The above equivalence shows that the set of normal eigenvalues coincides with the discrete spectrum, defined as the set of isolated points of the spectrum with finite rank of the corresponding Riesz projector.[5]
Decomposition of the spectrum of nonselfadjoint operators
The spectrum of a closed operator
in the Banach space
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)=\{normaleigenvaluesof A\}\cup\sigmaess,5(A).
See also
Notes and References
- Gohberg, I. C. Kreĭn, M. G.. Основные положения о дефектных числах, корневых числах и индексах линейных операторов. Fundamental aspects of defect numbers, root numbers and indexes of linear operators. Uspekhi Mat. Nauk . New Series. 12. 2(74). 1957. 43–118. Amer. Math. Soc. Transl. (2).
- Gohberg, I. C. Kreĭn, M. G.. Fundamental aspects of defect numbers, root numbers and indexes of linear operators. American Mathematical Society Translations. 13. 1960. 185–264. 10.1090/trans2/013/08.
- Book: Gohberg, I. C. Kreĭn, M. G.. Introduction to the theory of linear nonselfadjoint operators. 1969. American Mathematical Society, Providence, R.I..
- Book: Boussaid, N.. Comech, A.. Nonlinear Dirac equation. Spectral stability of solitary waves. 2019. American Mathematical Society, Providence, R.I.. 978-1-4704-4395-5.
- Book: Reed, M.. Simon, B.. Methods of modern mathematical physics, vol. IV. Analysis of operators. 1978. Academic Press [Harcourt Brace Jovanovich Publishers], New York.
