Spectral theory 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
−
λ
I
{\displaystyle A-\lambda I}
has a bounded inverse.
The set of normal eigenvalues coincides with the
discrete spectrum .
Root lineal
Let
B
{\displaystyle {\mathfrak {B}}}
be a
Banach space . The
root lineal
L
λ
(
A
)
{\displaystyle {\mathfrak {L}}_{\lambda }(A)}
of a linear operator
A
:
B
→
B
{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}
with domain
D
(
A
)
{\displaystyle {\mathfrak {D}}(A)}
corresponding to the eigenvalue
λ
∈
σ
p
(
A
)
{\displaystyle \lambda \in \sigma _{p}(A)}
is defined as
L
λ
(
A
)
=
⋃
k
∈
N
{
x
∈
D
(
A
)
:
(
A
−
λ
I
B
)
j
x
∈
D
(
A
)
∀
j
∈
N
,
j
≤
k
;
(
A
−
λ
I
B
)
k
x
=
0
}
⊂
B
,
{\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
I
B
{\displaystyle I_{\mathfrak {B}}}
is the identity operator in
B
{\displaystyle {\mathfrak {B}}}
.
This set is a
linear manifold but not necessarily a
vector space , since it is not necessarily closed in
B
{\displaystyle {\mathfrak {B}}}
. If this set is closed (for example, when it is finite-dimensional), it is called the
generalized eigenspace of
A
{\displaystyle A}
corresponding to the eigenvalue
λ
{\displaystyle \lambda }
.
Definition of a normal eigenvalue
An
eigenvalue
λ
∈
σ
p
(
A
)
{\displaystyle \lambda \in \sigma _{p}(A)}
of a
closed linear operator
A
:
B
→
B
{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}
in the
Banach space
B
{\displaystyle {\mathfrak {B}}}
with
domain
D
(
A
)
⊂
B
{\displaystyle {\mathfrak {D}}(A)\subset {\mathfrak {B}}}
is called normal (in the original terminology,
λ
{\displaystyle \lambda }
corresponds to a normally splitting finite-dimensional root subspace ), if the following two conditions are satisfied:
The
algebraic multiplicity of
λ
{\displaystyle \lambda }
is finite:
ν
=
dim
L
λ
(
A
)
<
∞
{\displaystyle \nu =\dim {\mathfrak {L}}_{\lambda }(A)<\infty }
, where
L
λ
(
A
)
{\displaystyle {\mathfrak {L}}_{\lambda }(A)}
is the
root lineal of
A
{\displaystyle A}
corresponding to the eigenvalue
λ
{\displaystyle \lambda }
;
The space
B
{\displaystyle {\mathfrak {B}}}
could be decomposed into a direct sum
B
=
L
λ
(
A
)
⊕
N
λ
{\displaystyle {\mathfrak {B}}={\mathfrak {L}}_{\lambda }(A)\oplus {\mathfrak {N}}_{\lambda }}
, where
N
λ
{\displaystyle {\mathfrak {N}}_{\lambda }}
is an
invariant subspace of
A
{\displaystyle A}
in which
A
−
λ
I
B
{\displaystyle A-\lambda I_{\mathfrak {B}}}
has a bounded inverse.
That is, the restriction
A
2
{\displaystyle A_{2}}
of
A
{\displaystyle A}
onto
N
λ
{\displaystyle {\mathfrak {N}}_{\lambda }}
is an operator with domain
D
(
A
2
)
=
N
λ
∩
D
(
A
)
{\displaystyle {\mathfrak {D}}(A_{2})={\mathfrak {N}}_{\lambda }\cap {\mathfrak {D}}(A)}
and with the range
R
(
A
2
−
λ
I
)
⊂
N
λ
{\displaystyle {\mathfrak {R}}(A_{2}-\lambda I)\subset {\mathfrak {N}}_{\lambda }}
which has a bounded inverse.
[1]
[2]
[3]
Equivalent characterizations of normal eigenvalues
Let
A
:
B
→
B
{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}
be a closed linear
densely defined operator in the Banach space
B
{\displaystyle {\mathfrak {B}}}
. The following statements are equivalent
[4] (Theorem III.88):
λ
∈
σ
(
A
)
{\displaystyle \lambda \in \sigma (A)}
is a normal eigenvalue;
λ
∈
σ
(
A
)
{\displaystyle \lambda \in \sigma (A)}
is an isolated point in
σ
(
A
)
{\displaystyle \sigma (A)}
and
A
−
λ
I
B
{\displaystyle A-\lambda I_{\mathfrak {B}}}
is
semi-Fredholm ;
λ
∈
σ
(
A
)
{\displaystyle \lambda \in \sigma (A)}
is an isolated point in
σ
(
A
)
{\displaystyle \sigma (A)}
and
A
−
λ
I
B
{\displaystyle A-\lambda I_{\mathfrak {B}}}
is
Fredholm ;
λ
∈
σ
(
A
)
{\displaystyle \lambda \in \sigma (A)}
is an isolated point in
σ
(
A
)
{\displaystyle \sigma (A)}
and
A
−
λ
I
B
{\displaystyle A-\lambda I_{\mathfrak {B}}}
is
Fredholm of index zero;
λ
∈
σ
(
A
)
{\displaystyle \lambda \in \sigma (A)}
is an isolated point in
σ
(
A
)
{\displaystyle \sigma (A)}
and the rank of the corresponding
Riesz projector
P
λ
{\displaystyle P_{\lambda }}
is finite;
λ
∈
σ
(
A
)
{\displaystyle \lambda \in \sigma (A)}
is an isolated point in
σ
(
A
)
{\displaystyle \sigma (A)}
, its algebraic multiplicity
ν
=
dim
L
λ
(
A
)
{\displaystyle \nu =\dim {\mathfrak {L}}_{\lambda }(A)}
is finite, and the range of
A
−
λ
I
B
{\displaystyle A-\lambda I_{\mathfrak {B}}}
is
closed .
[1]
[2]
[3]
If
λ
{\displaystyle \lambda }
is a normal eigenvalue, then the root lineal
L
λ
(
A
)
{\displaystyle {\mathfrak {L}}_{\lambda }(A)}
coincides with the range of the Riesz projector,
R
(
P
λ
)
{\displaystyle {\mathfrak {R}}(P_{\lambda })}
.
[3]
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
A
:
B
→
B
{\displaystyle A:\,{\mathfrak {B}}\to {\mathfrak {B}}}
in the Banach space
B
{\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 :
σ
(
A
)
=
{
normal eigenvalues of
A
}
∪
σ
e
s
s
,
5
(
A
)
.
{\displaystyle \sigma (A)=\{{\text{normal eigenvalues of}}\ A\}\cup \sigma _{\mathrm {ess} ,5}(A).}
See also
References
^
a
b 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]. Uspekhi Mat. Nauk [Amer. Math. Soc. Transl. (2) ]. New Series. 12 (2(74)): 43–118.
^
a
b 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.
doi :
10.1090/trans2/013/08 .
^
a
b
c Gohberg, I. C; Kreĭn, M. G. (1969).
Introduction to the theory of linear nonselfadjoint operators . American Mathematical Society, Providence, R.I.
^ Boussaid, N.; Comech, A. (2019).
Nonlinear Dirac equation. Spectral stability of solitary waves . American Mathematical Society, Providence, R.I.
ISBN
978-1-4704-4395-5 .
^ Reed, M.; Simon, B. (1978). Methods of modern mathematical physics, vol. IV. Analysis of operators . Academic Press [Harcourt Brace Jovanovich Publishers], New York.
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