Essential spectrum

In mathematics, the essential spectrum of a bounded 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 bounded 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

\lambda \,I-T

is not a Fredholm operator.

Here, an operator is Fredholm if its range is closed and its 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, 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 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

\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 {discr}}}}(T)=\sigma (T)\setminus \sigma _{{{\mathrm {ess}}}}(T).

A number λ is in the discrete spectrum if it is an isolated eigenvalue of finite multiplicity, meaning that the dimension of the space

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

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 σ_ess,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 σ_ess,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 σ_ess,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 σ_ess,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 σ_ess,5(T) is the union of σ_ess,1(T) with all components of C \ σ_ess,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,

\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}},

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

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 essential spectrum σ_ess,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,

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

where K(X) denotes the set of compact operators on X.

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

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.

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.

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