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

[math]\displaystyle{ T-\lambda I_X }[/math]

is not a Fredholm operator, where [math]\displaystyle{ I_X }[/math] denotes the identity operator on X, so that [math]\displaystyle{ I_X(x)=x }[/math] 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 [math]\displaystyle{ T+K }[/math] 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 {ψ_k} in the space X such that [math]\displaystyle{ \Vert \psi_k\Vert=1 }[/math] and

[math]\displaystyle{ \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]\displaystyle{ \{\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]\displaystyle{ \sigma_{\mathrm{disc}}(T) = \sigma(T) \setminus \sigma_{\mathrm{ess}}(T). }[/math]

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

[math]\displaystyle{ \{ \psi \in X : T\psi = \lambda\psi \} }[/math]

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 [math]\displaystyle{ \lambda }[/math] 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 [math]\displaystyle{ T:\,X\to X }[/math] be a closed linear operator on X with dense domain [math]\displaystyle{ D(T) }[/math]. There are several definitions of the essential spectrum, which are not equivalent.

1. The essential spectrum [math]\displaystyle{ \sigma_{\mathrm{ess},1}(T) }[/math] is the set of all λ such that [math]\displaystyle{ 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).
2. The essential spectrum [math]\displaystyle{ \sigma_{\mathrm{ess},2}(T) }[/math] is the set of all λ such that the range of [math]\displaystyle{ T-\lambda I_X }[/math] is not closed or the kernel of [math]\displaystyle{ T-\lambda I_X }[/math] is infinite-dimensional.
3. The essential spectrum [math]\displaystyle{ \sigma_{\mathrm{ess},3}(T) }[/math] is the set of all λ such that [math]\displaystyle{ 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).
4. The essential spectrum [math]\displaystyle{ \sigma_{\mathrm{ess},4}(T) }[/math] is the set of all λ such that [math]\displaystyle{ 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).
5. The essential spectrum [math]\displaystyle{ \sigma_{\mathrm{ess},5}(T) }[/math] is the union of σ_ess,1(T) with all components of [math]\displaystyle{ \C\setminus \sigma_{\mathrm{ess},1}(T) }[/math] that do not intersect with the resolvent set [math]\displaystyle{ \C \setminus \sigma(T) }[/math].

Each of the above-defined essential spectra [math]\displaystyle{ \sigma_{\mathrm{ess},k}(T) }[/math], [math]\displaystyle{ 1\le k\le 5 }[/math], is closed. Furthermore,

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

The essential spectrum [math]\displaystyle{ \sigma_{\mathrm{ess},k}(T) }[/math] is invariant under compact perturbations for k = 1,2,3,4, but not for k = 5. The set [math]\displaystyle{ \sigma_{\mathrm{ess},4}(T) }[/math] gives the part of the spectrum that is independent of compact perturbations, that is,

[math]\displaystyle{ \sigma_{\mathrm{ess},4}(T) = \bigcap_{K \in B_0(X)} \sigma(T+K), }[/math]

where [math]\displaystyle{ B_0(X) }[/math] 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

[math]\displaystyle{ \sigma(T)=\sigma_{\mathrm{ess},5}(T)\bigsqcup\sigma_{\mathrm{d}}(T) }[/math],

where [math]\displaystyle{ \sigma_{\mathrm{d}}(T) }[/math] 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

• Reed, Michael C.; Simon, Barry (1980), Methods of modern mathematical physics: Functional Analysis, 1, San Diego: Academic Press, ISBN 0-12-585050-6 
• Teschl, Gerald (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. American Mathematical Society. ISBN 978-0-8218-4660-5. https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/. 

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.