Essential Spectrum

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 σ_ess(''T''), is the set of all complex numbers λ such that

:$T-\lambda I_X$

is not a Fredholm operator, where $I_X$ denotes the ''identity operator'' on ''X'', so that $I_X(x)=x$ 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 $T+K$ 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'' for the essential spectrum 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 $\Vert \psi_k\Vert=1$ and

:$\lim_ \left\, T\psi_k - \lambda\psi_k \right\, = 0.$

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
:$\sigma_(T) = \sigma(T) \setminus \sigma_(T).$
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 $\lambda$ 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 space
and let $T:\,X\to X$ be a closed linear operator on ''X'' with dense domain $D(T)$. There are several definitions of the essential spectrum, which are not equivalent.
# The essential spectrum $\sigma_(T)$ is the set of all λ such that $T-\lambda I_X$ 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 $\sigma_(T)$ is the set of all λ such that the range of $T-\lambda I_X$ is not closed or the kernel of $T-\lambda I_X$ is infinite-dimensional.
# The essential spectrum $\sigma_(T)$ is the set of all λ such that $T-\lambda I_X$ 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 $\sigma_(T)$ is the set of all λ such that $T-\lambda I_X$ 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 $\sigma_(T)$ is the union of σ_ess,1(''T'') with all components of $\C\setminus \sigma_(T)$ that do not intersect with the resolvent set $\C \setminus \sigma(T)$.

Each of the above-defined essential spectra $\sigma_(T)$, $1\le k\le 5$, is closed. Furthermore,
:$\sigma_(T) \subset \sigma_(T) \subset \sigma_(T) \subset \sigma_(T) \subset \sigma_(T) \subset \sigma(T) \subset \C,$
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
:$r_(T) = \max \.$
Even though the spectra may be different, the radius is the same for all ''k''.

The definition of the set $\sigma_(T)$ is equivalent to Weyl's criterion: $\sigma_(T)$ is the set of all λ for which there exists a singular sequence.

The essential spectrum $\sigma_(T)$ is invariant under compact perturbations for ''k'' = 1,2,3,4, but not for ''k'' = 5.
The set $\sigma_(T)$ gives the part of the spectrum that is independent of compact perturbations, that is,
:$\sigma_(T) = \bigcap_ \sigma(T+K),$
where $B_0(X)$ 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
:$\sigma(T)=\sigma_(T)\bigsqcup\sigma_(T)$,
where $\sigma_(T)$ is the discrete spectrum of ''T''.

See also

* Spectrum (functional analysis)
* Resolvent formalism
* Decomposition of spectrum (functional analysis)
* Discrete spectrum (mathematics)
* Spectrum of an operator
* Operator theory
* Fredholm theory

References

The self-adjoint case is discussed in
*
*

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.

{{SpectralTheory

Spectral theory