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".

Of self-adjoint operators

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

Definition

The essential spectrum of, usually denoted, is the set of all real numbers such that
is not a Fredholm operator, where denotes the identity operator on, so that, for all.
The definition of essential spectrum will remain unchanged if we allow it to consist of all those complex numbers such that the above condition holds. This is due to the fact that the spectrum of a self-adjoint operator is real.

Properties

The essential spectrum is always closed, and it is a subset of the spectrum. As mentioned above, since is self-adjoint, the spectrum is contained on the real axis.
The spectrum can be partitioned into two parts. One part is the essential spectrum. The other part is the discrete spectrum, which is the set of points such that it is an isolated point, and is a finite dimensional subspace. That is, it is an isolated eigenvalue of finite algebraic multiplicity.
The essential spectrum is invariant under compact perturbations. That is, if is a compact self-adjoint operator on, then the essential spectra of and that of coincide, i.e.. This explains why it is called the essential spectrum: Weyl originally defined the essential spectrum of a certain differential operator to be the spectrum independent of boundary conditions.

The discrete spectrum

The essential spectrum is a subset of the spectrum and its complement is called the discrete spectrum, so
If is self-adjoint, then, by definition, a number is in the discrete spectrum of if it is an isolated eigenvalue of finite multiplicity, meaning that the dimension of the space
has finite but non-zero dimension and that there is an such that and imply that and are equal.

Weyl's criterion

Define the following:

A vector is a unit vector iff it has norm 1.
A sequence of vectors converge to 0 iff. This is written as.
A sequence of vectors converge weakly to 0 iff for any. This is written as.

Under these definitions, we have the following characterization of the spectrum of the operator :

A number is in if and only if there exists a sequence of unit vectors with.

If is on the discrete spectrum, then since is isolated in, any sequence of unit vectors with must converge to, and since is finite-dimensional, must have a convergent subsequence by compactness of the unit sphere of. Therefore,.
Weyl's criterion states that the converse is true as well:

A number is in if and only if there exists a sequence of unit vectors with, and.

Such a sequence is called a singular sequence or Weyl sequence. By sparsifying the sequence and applying Gram–Schmidt process, the sequence can be made orthonormal.

Examples

Let be the multiplication operator defined by. The essential range of is, so the spectrum is. For any, we can explicitly construct a singular sequence as a sequence of increasingly narrow and sharp rectangular functions that are supported on disjoint sets. For example, let, then we can construct to be the rectangular function on of height. They are orthonormal, with. Note that the sequence increasingly resembles the Dirac delta "function" at 0, even though it does not converge.
Let be the momentum operator defined by extending for compactly supported smooth functions. Its essential spectrum is the entire real line. Physicists say that each is an eigenvalue of with eigenfunction. However, this is not technically correct, since has infinite L2-norm. Nevertheless, it is possible to make a similar rigorous statement. While is not in, it can be approached by a Weyl sequence in. The construction is essentially the same, by constructing a sequence approaching the Dirac delta at in momentum space, then performing a Fourier transform to position space.
Let be the Laplace operator, where is the Sobolev space. Its essential spectrum is. For each, and any unit vector, the construction of the Weyl sequence for the "eigenfunction" is similar.

Of densely defined operators

Preliminary concepts

Let be a Banach space, and let be a densely defined operator on. That is, it is of type, where is a dense subspace of. Let the spectrum of be, defined byThe complement of is the resolvent set of.

Definitions

There are several definitions of the essential spectrum of, which are not necessarily the same. Each of these definitions is of the formThere are at least 5 different levels of niceness, increasing in strength. Each increase in strength shrinks the set of nice, thus expands the essential domain.
Let denote an operator of type. Let be its kernel, be its cokernel, be its range. We say that is:

Normally solvable, if is a closed operator, and is a closed set. This can be checked via the closed range theorem.
Semi-Fredholm, if furthermore, is finite-dimensional inclusive-or is finite-dimensional.
Fredholm, if furthermore, is finite-dimensional and is finite-dimensional.
Fredholm with index zero, if furthermore, and has the same dimension.
If furthermore, there exists a deleted neighborhood of zero that is a subset of the resolvent set.
* In other words, zero is not a limit point of.
Has bounded inverse, if there exists a bounded linear operator, such that are inverses of each other.

Now, set. Then conditions 1 to 5 defines 5 essential spectra,, and condition 6 defines the spectrum. It is clear that conditions 1 to 5 increases in strength. One can also show that condition 6 is stronger than condition 5. Thus,Any of these inclusions may be strict.
Different authors defined the essential spectra differently, resulting in different terminologies. For example, Kato used, Wolf used, Schechter used, Browder used. Thus, is also called the Browder essential spectrum, etc.

More definitions

There are even more definitions of the essential spectrum.
The following definition states that the essential spectrum is the part of the spectrum that is stable under compact perturbation:Another definition states that:Given, it is an isolated eigenvalue of with finite multiplicity if and only if has positive finite dimension, and is an isolated point of.

Equalities

Banach space case

If is not closed, then. Because of this, the essential spectrum is uninteresting for these, and we will assume thenceforth that is closed.
If is bounded and either hypernormal or Toeplitz, then.
If is bounded and, then.
for all, where is the transpose operator of.
Define the radius of the essential spectrum by Even though the spectra may be different, the radius is the same for all.
The essential spectrum is invariant under compact perturbations for, but not for. That is, for and any compact operator,. The 4th essential spectrum is in fact the maximal possible that is stable under compact perturbations, in the sense that..
.
, where is the discrete spectrum of.

The definition of the set is equivalent to Weyl's criterion: is the set of all for which there exists a singular sequence.

Hilbert space case

If is a Hilbert space, and is self-adjoint, then all the above definitions of the essential spectrum coincide, except. Concretely, we haveThe issue is that does not include isolated eigenvalues of infinite multiplicity. For example, if and is infinite-dimensional, then is empty, whereas. This is because 1 is an eigenvalue of the identity operator with infinite multiplicity.
If is a Hilbert space, then for all.