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

${\displaystyle T-\lambda I_{X}}$

is not a Fredholm operator, where ${\displaystyle I_{X}}$ denotes the identity operator on X, so that ${\displaystyle I_{X}(x)=x}$ 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 ${\displaystyle T+K}$ 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 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 ${\displaystyle \Vert \psi _{k}\Vert =1}$ and

${\displaystyle \lim _{k\to \infty }\left\|T\psi _{k}-\lambda \psi _{k}\right\|=0.}$

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 ${\displaystyle \{\psi _{k}\}}$ 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

${\displaystyle \sigma _{\mathrm {disc} }(T)=\sigma (T)\setminus \sigma _{\mathrm {ess} }(T).}$

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

${\displaystyle \{\psi \in X:T\psi =\lambda \psi \}}$

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 ${\displaystyle \lambda }$ 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 ${\displaystyle T:\,X\to X}$ be a closed linear operator on X with dense domain ${\displaystyle D(T)}$. There are several definitions of the essential spectrum, which are not equivalent.^[1]

1. The essential spectrum ${\displaystyle \sigma _{\mathrm {ess} ,1}(T)}$ is the set of all λ such that ${\displaystyle T-\lambda I_{X}}$ is not semi-Fredholm (an operator is semi-Fredholm if its range is closed and its kernel or its cokernel is finite-dimensional).
2. The essential spectrum ${\displaystyle \sigma _{\mathrm {ess} ,2}(T)}$ is the set of all λ such that the range of ${\displaystyle T-\lambda I_{X}}$ is not closed or the kernel of ${\displaystyle T-\lambda I_{X}}$ is infinite-dimensional.
3. The essential spectrum ${\displaystyle \sigma _{\mathrm {ess} ,3}(T)}$ is the set of all λ such that ${\displaystyle T-\lambda I_{X}}$ is not Fredholm (an operator is Fredholm if its range is closed and both its kernel and its cokernel are finite-dimensional).
4. The essential spectrum ${\displaystyle \sigma _{\mathrm {ess} ,4}(T)}$ is the set of all λ such that ${\displaystyle T-\lambda I_{X}}$ 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).
5. The essential spectrum ${\displaystyle \sigma _{\mathrm {ess} ,5}(T)}$ is the union of σ_ess,1(T) with all components of ${\displaystyle \mathbb {C} \setminus \sigma _{\mathrm {ess} ,1}(T)}$ that do not intersect with the resolvent set ${\displaystyle \mathbb {C} \setminus \sigma (T)}$.

Each of the above-defined essential spectra ${\displaystyle \sigma _{\mathrm {ess} ,k}(T)}$, ${\displaystyle 1\leq k\leq 5}$, is closed. Furthermore,

${\displaystyle \sigma _{\mathrm {ess} ,1}(T)\subset \sigma _{\mathrm {ess} ,2}(T)\subset \sigma _{\mathrm {ess} ,3}(T)\subset \sigma _{\mathrm {ess} ,4}(T)\subset \sigma _{\mathrm {ess} ,5}(T)\subset \sigma (T)\subset \mathbb {C} ,}$

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

${\displaystyle r_{\mathrm {ess} ,k}(T)=\max\{|\lambda |:\lambda \in \sigma _{\mathrm {ess} ,k}(T)\}.}$

Even though the spectra may be different, the radius is the same for all k.

The definition of the set ${\displaystyle \sigma _{\mathrm {ess} ,2}(T)}$ is equivalent to Weyl's criterion: ${\displaystyle \sigma _{\mathrm {ess} ,2}(T)}$ is the set of all λ for which there exists a singular sequence.

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

${\displaystyle \sigma _{\mathrm {ess} ,4}(T)=\bigcap _{K\in B_{0}(X)}\sigma (T+K),}$

where ${\displaystyle B_{0}(X)}$ 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

${\displaystyle \sigma (T)=\sigma _{\mathrm {ess} ,5}(T)\bigsqcup \sigma _{\mathrm {d} }(T)}$,

where ${\displaystyle \sigma _{\mathrm {d} }(T)}$ 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

• Reed, Michael C.; Simon, Barry (1980), Methods of modern mathematical physics: Functional Analysis, vol. 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.

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.