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

:λ"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 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:$lim\_\{k\; oinfty\}\; left|\; Tpsi\_k\; -\; lambdapsi\_k\; ight|\; =\; 0.$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:$\{\; psi\; in\; X\; :\; Tpsi\; =\; lambdapsi\; \}$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:$r\_\{mathrm\{ess\},k\}(T)\; =\; max\; \{\; |lambda|\; :\; lambdainsigma\_\{mathrm\{ess\},k\}(T)\; \}.$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,: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
*Michael Reed and Barry Simon (1980), "Functional Analysis," Academic Press, San Diego. ISBN 0-12-585050-6.

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&ndash;269.