Editing Essential spectrum

In [[mathematics]], the '''essential spectrum''' of a [[bounded operator]] (or, more generally, of a [[densely defined]] [[Unbounded operator#Closed linear operators|closed linear operator]]) is a certain subset of its [[spectrum (functional analysis)|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 <math>X</math> be a [[Hilbert space]] and let <math>T</math> be a [[self-adjoint operator]] on <math>X</math>.

===Definition===

The '''essential spectrum''' of <math>T</math>, usually denoted <math>\sigma_{\mathrm{ess}}(T)</math>, is the set of all [[real number]]s <math>\lambda \in \R</math> such that

:<math>T-\lambda I_X</math>

is not a [[Fredholm operator]], where <math>I_X</math> denotes the [[identity operator]] on <math>X</math>, so that <math>I_X(x)=x</math>, for all <math>x \in X</math>.
(An operator is Fredholm if its [[kernel (algebra)|kernel]] and [[cokernel]] are finite-dimensional.)

The definition of essential spectrum <math>\sigma_{\mathrm{ess}}(T)</math> will remain unchanged if we allow it to consist of all those [[complex number]]s <math>\lambda \in \C</math> (instead of just real numbers) such that the above condition holds. This is due to the fact that the spectrum of self-adjoint consists only of real numbers.

===Properties===

The essential spectrum is always [[closed set|closed]], and it is a subset of the [[spectrum (functional analysis)|spectrum]] <math>\sigma(T)</math>. As mentioned above, since <math>T</math> is self-adjoint, the spectrum is contained on the real axis.

The essential spectrum is invariant under compact perturbations. That is, if <math>K</math> is a [[Compact operator on Hilbert space|compact]] self-adjoint operator on <math>X</math>, then the essential spectra of <math>T</math> and that of <math>T+K</math> coincide, i.e. <math>\sigma_{\mathrm{ess}}(T)=\sigma_{\mathrm{ess}}(T+K)</math>. This explains why it is called the ''essential spectrum'': [[Hermann Weyl|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 <math>\lambda</math> is in the spectrum <math>\sigma(T)</math> of the operator <math>T</math> if and only if there exists a [[sequence]] <math>\{\psi_k\}_{k\in \N} \subseteq X</math> in the Hilbert space <math>X</math> such that <math>\Vert\psi_k\Vert=1</math> and

:<math> \lim_{k\to\infty} \left\| (T - \lambda)\psi_k \right\| = 0. </math>

Furthermore, <math>\lambda</math> 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 <math>\{\psi_k\}_{k\in \N}</math> is an [[orthonormal]] sequence); such a sequence is called a ''singular sequence''. Equivalently, <math>\lambda</math> is in the essential spectrum <math>\sigma_{\mathrm{ess}}(T)</math> if there exists a sequence satisfying the above condition, which also [[Weak convergence (Hilbert space)|converges weakly]] to the zero vector <math>\mathbf{0}_X</math> in <math>X</math>.

===The discrete spectrum===

The essential spectrum <math>\sigma_{\mathrm{ess}}(T)</math> is a subset of the spectrum <math>\sigma(T)</math> and its complement is called the [[discrete spectrum (mathematics)|discrete spectrum]], so
:<math> \sigma_{\mathrm{disc}}(T) = \sigma(T) \setminus \sigma_{\mathrm{ess}}(T)</math>.
If <math>T</math> is self-adjoint, then, by definition, a number <math>\lambda</math> is in the ''discrete spectrum'' <math>\sigma_{\mathrm{disc}}</math> of <math>T</math> if it is an isolated eigenvalue of finite multiplicity, meaning that the dimension of the space
:<math> \ \mathrm{span} \{ \psi \in X : T\psi = \lambda\psi \} </math>
has finite but non-zero dimension and that there is an <math>\varepsilon>0</math> such that <math>\mu \in \sigma(T)</math> and <math>|\mu-\lambda|<\varepsilon</math> imply that <math>\mu</math> and <math>\lambda</math> are equal.
(For general, non-self-adjoint operators <math>S</math> on [[Banach space]]s, by definition, a complex number <math>\lambda \in \C</math> is in the [[discrete spectrum]] <math>\sigma_{\mathrm{disc}}(S)</math> 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 <math>X</math> be a [[Banach space]]
and let <math>T:\,D(T)\to X</math> be a [[Unbounded operator#Closed linear operators|closed linear operator]] on <math>X</math> with [[Densely defined operator|dense domain]] <math>D(T)</math>. There are several definitions of the essential spectrum, which are not equivalent.<ref>{{cite journal |last1=Gustafson |first1=Karl|author-link=Karl Edwin Gustafson |title=On the essential spectrum |journal=Journal of Mathematical Analysis and Applications |date=1969 |volume=25 |issue=1 |pages=121–127 |url=https://core.ac.uk/download/pdf/82202846.pdf}}</ref>
# The essential spectrum <math>\sigma_{\mathrm{ess},1}(T)</math> is the set of all <math>\lambda</math> such that <math>T-\lambda I_X</math> 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 <math>\sigma_{\mathrm{ess},2}(T)</math> is the set of all <math>\lambda</math> such that the range of <math>T-\lambda I_X</math> is not closed or the kernel of <math>T-\lambda I_X</math> is infinite-dimensional.
# The essential spectrum <math>\sigma_{\mathrm{ess},3}(T)</math> is the set of all <math>\lambda</math> such that <math>T-\lambda I_X</math> 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 <math>\sigma_{\mathrm{ess},4}(T)</math> is the set of all <math>\lambda</math> such that <math>T-\lambda I_X</math> 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 <math>\sigma_{\mathrm{ess},5}(T)</math> is the union of <math>\sigma_{\mathrm{ess},1}(T)</math> with all components of <math>\C\setminus \sigma_{\mathrm{ess},1}(T)</math> that do not intersect with the resolvent set <math>\C \setminus \sigma(T)</math>.

Each of the above-defined essential spectra <math>\sigma_{\mathrm{ess},k}(T)</math>, <math>1\le k\le 5</math>, is closed. Furthermore,
:<math> \sigma_{\mathrm{ess},1}(T) \subseteq \sigma_{\mathrm{ess},2}(T) \subseteq \sigma_{\mathrm{ess},3}(T) \subseteq \sigma_{\mathrm{ess},4}(T) \subseteq \sigma_{\mathrm{ess},5}(T) \subseteq \sigma(T) \subseteq \C,</math>
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
:<math>r_{\mathrm{ess},k}(T) = \max \{ |\lambda| : \lambda\in\sigma_{\mathrm{ess},k}(T) \}. </math>
Even though the spectra may be different, the radius is the same for all <math>k=1,2,3,4,5</math>.

The definition of the set <math>\sigma_{\mathrm{ess},2}(T)</math> is equivalent to Weyl's criterion: <math>\sigma_{\mathrm{ess},2}(T)</math> is the set of all <math>\lambda</math> for which there exists a singular sequence.

The essential spectrum <math>\sigma_{\mathrm{ess},k}(T)</math> is invariant under compact perturbations for <math>k=1,2,3,4</math>, but not for <math>k=5</math>.
The set <math>\sigma_{\mathrm{ess},4}(T)</math> gives the part of the spectrum that is independent of compact perturbations, that is,
:<math> \sigma_{\mathrm{ess},4}(T) = \bigcap_{K \in B_0(X)} \sigma(T+K), </math>
where <math>B_0(X)</math> denotes the set of [[compact operator]]s on <math>X</math> (D.E. Edmunds and W.D. Evans, 1987).

The spectrum of a closed, densely defined operator <math>T</math> can be decomposed into a disjoint union
:<math>\sigma(T)=\sigma_{\mathrm{ess},5}(T)\bigsqcup\sigma_{\mathrm{disc}}(T)</math>,
where <math>\sigma_{\mathrm{disc}}(T)</math> is the [[Discrete spectrum (mathematics)|discrete spectrum]] of <math>T</math>.

==See also==
* [[Spectrum (functional analysis)]]
* [[Resolvent formalism]]
* [[Decomposition of spectrum (functional analysis)]]
* [[Discrete spectrum (mathematics)]]
* [[Spectrum of an operator]]
* [[Operator theory]]
* [[Fredholm theory]]

== References ==
{{Reflist}}
The self-adjoint case is discussed in
* {{citation |first1 = Michael C. |last1 = Reed|author1-link=Michael C. Reed |first2 = Barry |last2 = Simon |author2-link = Barry Simon |title = Methods of modern mathematical physics: Functional Analysis |volume=1 |place = San Diego |publisher = Academic Press |year = 1980 |isbn = 0-12-585050-6}}
* {{Cite book |title=Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators |first= Gerald |last=Teschl |authorlink= Gerald Teschl |publisher= American Mathematical Society |year=2009 |url=https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/ |isbn=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. {{ISBN|0-19-853542-2}}.

The original definition of the essential spectrum goes back to
* [[Hermann Weyl|H. Weyl]] (1910), Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, ''Mathematische Annalen'' '''68''', 220&ndash;269.

{{SpectralTheory}}

[[Category:Spectral theory]]