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 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 $\Vert \psi _{k}\Vert =1$ and

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

\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

\{\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 $\lambda$ 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.^[1]

The essential spectrum $\sigma _{\mathrm {ess} ,1}(T)$ 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 $\sigma _{\mathrm {ess} ,2}(T)$ 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 $\sigma _{\mathrm {ess} ,3}(T)$ 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 $\sigma _{\mathrm {ess} ,4}(T)$ 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 $\sigma _{\mathrm {ess} ,5}(T)$ is the union of σ_ess,1(T) with all components of $\mathbb {C} \setminus \sigma _{\mathrm {ess} ,1}(T)$ that do not intersect with the resolvent set $\mathbb {C} \setminus \sigma (T)$ .

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

\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

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

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

\sigma _{\mathrm {ess} ,4}(T)=\bigcap _{K\in B_{0}(X)}\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 _{\mathrm {ess} ,5}(T)\bigsqcup \sigma _{\mathrm {d} }(T)

,

where $\sigma _{\mathrm {d} }(T)$ is the discrete spectrum of T.

References

^ Gustafson, Karl (1969). "On the essential spectrum" (PDF). Journal of Mathematical Analysis and Applications. 25 (1): 121–127.

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.

[1] Gustafson, Karl (1969). "On the essential spectrum" (PDF). Journal of Mathematical Analysis and Applications. 25 (1): 121–127.

[1]

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