Essential spectrum explained

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 σess(T), is the set of all complex numbers λ such that

TIX

is not a Fredholm operator, where

IX

denotes the identity operator on X, so that

IX(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 in the space X such that

\Vert\psik\Vert=1

and

\limk\toinfty\left\|T\psik-λ\psik\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

\{\psik\}

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

\sigmadisc(T)=\sigma(T)\setminus\sigmaess(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\inX:T\psi=λ\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

λ

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 spaceand let

T:X\toX

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]
  1. The essential spectrum

\sigmaess,1(T)

is the set of all λ such that

TIX

is not semi-Fredholm (an operator is semi-Fredholm if its range is closed and its kernel or its cokernel is finite-dimensional).
  1. The essential spectrum

\sigmaess,2(T)

is the set of all λ such that the range of

TIX

is not closed or the kernel of

TIX

is infinite-dimensional.
  1. The essential spectrum

\sigmaess,3(T)

is the set of all λ such that

TIX

is not Fredholm (an operator is Fredholm if its range is closed and both its kernel and its cokernel are finite-dimensional).
  1. The essential spectrum

\sigmaess,4(T)

is the set of all λ such that

TIX

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).
  1. The essential spectrum

\sigmaess,5(T)

is the union of σess,1(T) with all components of

\C\setminus\sigmaess,1(T)

that do not intersect with the resolvent set

\C\setminus\sigma(T)

.

Each of the above-defined essential spectra

\sigmaess,k(T)

,

1\lek\le5

, is closed. Furthermore,

\sigmaess,1(T)\subset\sigmaess,2(T)\subset\sigmaess,3(T)\subset\sigmaess,4(T)\subset\sigmaess,5(T)\subset\sigma(T)\subset\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

ress,k(T)=max\{|λ|:λ\in\sigmaess,k(T)\}.

Even though the spectra may be different, the radius is the same for all k.

The definition of the set

\sigmaess,2(T)

is equivalent to Weyl's criterion:

\sigmaess,2(T)

is the set of all λ for which there exists a singular sequence.

The essential spectrum

\sigmaess,k(T)

is invariant under compact perturbations for k = 1,2,3,4, but not for k = 5.The set

\sigmaess,4(T)

gives the part of the spectrum that is independent of compact perturbations, that is,

\sigmaess,4(T)=

cap
K\inB0(X)

\sigma(T+K),

where

B0(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)=\sigmaess,5(T)sqcup\sigmad(T)

,where

\sigmad(T)

is the discrete spectrum of T.

See also

References

The self-adjoint case is discussed in

. Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators . Gerald Teschl . American Mathematical Society . 2009 . 978-0-8218-4660-5 .

A discussion of the spectrum for general operators can be found in

The original definition of the essential spectrum goes back to

Notes and References

  1. Gustafson . Karl. Karl Edwin Gustafson . On the essential spectrum . Journal of Mathematical Analysis and Applications . 1969 . 25 . 1 . 121–127 .