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In [[mathematics]], the '''essential spectrum''' of a [[bounded 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 ''X'' be a [[Hilbert space]] and let ''T'' be a [[Operator norm|bounded]] [[self-adjoint operator|self-adjoint]] operator on ''X''.

===Definition===

The '''essential spectrum''' of ''T'', usually denoted σess(''T''), is the set of all [[complex number]]s λ such that

:<math> \lambda\, I - T </math>

is not a [[Fredholm operator]].

Here, an operator is Fredholm if its [[Range (mathematics)|range]] is closed and its [[kernel (algebra)|kernel]] and [[cokernel]] are finite-dimensional. Furthermore, ''I'' denotes the ''identity operator'' on ''X'', so that ''I''(''x'') = ''x'' for all ''x'' in ''X''.

===Properties===

The essential spectrum is always [[closed set|closed]], and it is a subset of the [[spectrum (functional analysis)|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 operator on Hilbert space|compact operator]] on ''X'', then the essential spectra of ''T'' and that of ''T'' + ''K'' coincide. 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'' 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]] {ψ''k''} in the space ''X'' such that ||ψ''k''|| = 1 and

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

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 <math>\{\psi_k\}</math> 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
:<math> \sigma_{\mathrm{discr}}(T) = \sigma(T) \setminus \sigma_{\mathrm{ess}}(T). </math>
A number λ is in the discrete spectrum if it is an isolated eigenvalue of finite multiplicity, meaning that the dimension of the space
:<math> \{ \psi \in X : T\psi = \lambda\psi \} </math>
is finite but non-zero and that there is an ε > 0 such that μ ∈ σ(''T'') and |μ−λ| < ε imply that μ and λ are equal.

==The essential spectrum of general bounded operators==

In the general case, ''X'' denotes a [[Banach space]] and ''T'' is a bounded operator on ''X''. There are several definitions of the essential spectrum in the literature, which are not equivalent.
# The essential spectrum σess,1(''T'') is the set of all λ such that λI − ''T'' 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 σess,2(''T'') is the set of all λ such that the range of λI − ''T'' is not closed or the kernel of λI − ''T'' is infinite-dimensional.
# The essential spectrum σess,3(''T'') is the set of all λ such that λI − ''T'' 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 σess,4(''T'') is the set of all λ such that λI − ''T'' 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 σess,5(''T'') is the union of σess,1(''T'') with all components of '''C''' \ σess,1(''T'') that do not intersect with the resolvent set '''C''' \ σ(''T'').

The essential spectrum of an operator is closed, whatever definition is used. Furthermore,
:<math> \sigma_{\mathrm{ess},1}(T) \subset \sigma_{\mathrm{ess},2}(T) \subset \sigma_{\mathrm{ess},3}(T) \subset \sigma_{\mathrm{ess},4}(T) \subset \sigma_{\mathrm{ess},5}(T) \subset \sigma(T) \subset \mathbf{C}, </math>
but any of these inclusions may be strict. However, for self-adjoint operators, all the above definitions for 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 ''k''.

The essential spectrum σess,''k''(''T'') is invariant under compact perturbations for ''k'' = 1,2,3,4, but not for ''k'' = 5. The case ''k'' = 4 gives the part of the spectrum that is independent of compact perturbations, that is,
:<math> \sigma_{\mathrm{ess},4}(T) = \bigcap_{K \in K(X)} \sigma(T+K), </math>
where ''K''(''X'') denotes the set of compact operators on ''X''.

The second definition generalizes Weyl's criterion: σess,2(''T'') is the set of all λ for which there is no singular sequence.

==References==

The self-adjoint case is discussed in
*{{citation |first1 = Michael C. |last1 = 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=http://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
*H. Weyl (1910), Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, ''Mathematische Annalen'' '''68''', 220–269.

[[Category:Spectral theory]]

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