Essential spectrum explained
In mathematics, the essential spectrum of a bounded operator (or, more generally, of a densely defined closed linear operator) is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, "fails badly to be invertible".
The essential spectrum of self-adjoint operators
In formal terms, let X be a Hilbert space and let T be a self-adjoint operator on X.
Definition
The essential spectrum of T, usually denoted σess(T), is the set of all complex numbers λ such that
is not a Fredholm operator, where
denotes the
identity operator on
X, so that
for all
x in
X.(An operator is Fredholm if its
kernel and
cokernel are finite-dimensional.)
Properties
The essential spectrum is always closed, and it is a subset of the spectrum. Since T is self-adjoint, the spectrum is contained on the real axis.
The essential spectrum is invariant under compact perturbations. That is, if K is a compact self-adjoint operator on X, then the essential spectra of T and that of
coincide. This explains why it is called the
essential spectrum:
Weyl (1910) originally defined the essential spectrum of a certain differential operator to be the spectrum independent of boundary conditions.
Weyl's criterion is as follows. First, a number λ is in the spectrum of T if and only if there exists a sequence in the space X such that
and
\limk\toinfty\left\|T\psik-λ\psik\right\|=0.
Furthermore, λ is in the essential spectrum if there is a sequence satisfying this condition, but such that it contains no convergent subsequence (this is the case if, for example
is an
orthonormal sequence); such a sequence is called a
singular sequence.
The discrete spectrum
The essential spectrum is a subset of the spectrum σ, and its complement is called the discrete spectrum, so
\sigmadisc(T)=\sigma(T)\setminus\sigmaess(T).
If
T is self-adjoint, then, by definition, a number λ is in the
discrete spectrum of
T if it is an isolated eigenvalue of finite multiplicity, meaning that the dimension of the space
has finite but non-zero dimension and that there is an ε > 0 such that μ ∈ σ(
T) and |μ-λ| < ε imply that μ and λ are equal.(For general nonselfadjoint operators in
Banach spaces, by definition, a number
is in the discrete spectrum if it is a
normal eigenvalue; or, equivalently, if it is an isolated point of the spectrum and the rank of the corresponding
Riesz projector is finite.)
The essential spectrum of closed operators in Banach spaces
Let X be a Banach spaceand let
be a closed linear operator on
X with
dense domain
. There are several definitions of the essential spectrum, which are not equivalent.
[1] - The essential spectrum
is the set of all λ such that
is not semi-Fredholm (an operator is semi-Fredholm if its range is closed and its kernel or its cokernel is finite-dimensional).
- The essential spectrum
is the set of all λ such that the range of
is not closed or the kernel of
is infinite-dimensional.
- The essential spectrum
is the set of all λ such that
is not Fredholm (an operator is Fredholm if its range is closed and both its kernel and its cokernel are finite-dimensional).
- The essential spectrum
is the set of all λ such that
is not Fredholm with index zero (the index of a Fredholm operator is the difference between the dimension of the kernel and the dimension of the cokernel).
- The essential spectrum
is the union of σ
ess,1(
T) with all components of
\C\setminus\sigmaess,1(T)
that do not intersect with the resolvent set
.
Each of the above-defined essential spectra
,
, is closed. Furthermore,
\sigmaess,1(T)\subset\sigmaess,2(T)\subset\sigmaess,3(T)\subset\sigmaess,4(T)\subset\sigmaess,5(T)\subset\sigma(T)\subset\C,
and any of these inclusions may be strict. For self-adjoint operators, all the above definitions of the essential spectrum coincide.
Define the radius of the essential spectrum by
ress,k(T)=max\{|λ|:λ\in\sigmaess,k(T)\}.
Even though the spectra may be different, the radius is the same for all
k.
The definition of the set
is equivalent to Weyl's criterion:
is the set of all λ for which there exists a singular sequence.
The essential spectrum
is invariant under compact perturbations for
k = 1,2,3,4, but not for
k = 5.The set
gives the part of the spectrum that is independent of compact perturbations, that is,
\sigmaess,4(T)=
\sigma(T+K),
where
denotes the set of
compact operators on
X (D.E. Edmunds and W.D. Evans, 1987).
The spectrum of a closed densely defined operator T can be decomposed into a disjoint union
\sigma(T)=\sigmaess,5(T)sqcup\sigmad(T)
,where
is the
discrete spectrum of
T.
See also
References
The self-adjoint case is discussed in
- Book: Teschl, Gerald . Gerald Teschl
. Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators . Gerald Teschl . American Mathematical Society . 2009 . 978-0-8218-4660-5 .
A discussion of the spectrum for general operators can be found in
- D.E. Edmunds and W.D. Evans (1987), Spectral theory and differential operators, Oxford University Press. .
The original definition of the essential spectrum goes back to
- H. Weyl (1910), Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Mathematische Annalen 68, 220 - 269.
Notes and References
- Gustafson . Karl. Karl Edwin Gustafson . On the essential spectrum . Journal of Mathematical Analysis and Applications . 1969 . 25 . 1 . 121–127 .