Essential spectrum

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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[edit]

In formal terms, let X be a Hilbert space and let T be a bounded self-adjoint operator on X.

Definition[edit]

The essential spectrum of T, usually denoted σess(T), is the set of all complex numbers λ such that

is not a Fredholm operator.

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[edit]

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 sequencek} in the space X such that ||ψk|| = 1 and

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 is an orthonormal sequence); such a sequence is called a singular sequence.

The discrete spectrum[edit]

The essential spectrum is a subset of the spectrum σ, and its complement is called the discrete spectrum, so

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

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[edit]

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.

  1. 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).
  2. 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.
  3. 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).
  4. 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).
  5. 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,

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

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,

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[edit]

The self-adjoint case is discussed in

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.