Essential spectrum

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 σess("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"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 σess,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 σess,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 σess,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 σess,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 σess,5("T") is the union of σess,1("T") with all components of C σess,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 σess,"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) = igcap_{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: σess,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.


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