Wikipedia / Essential spectrum
Wikipedia / Essential spectrum
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".
Of self-adjoint operators
be a Hilbert space be a self-adjoint operator.
Definition
The essential spectrum, is the set of all real numbers such that
is not a Fredholm operator denotes the identity operator. (An operator is Fredholm if it is bounded, and its kernel and cokernel are finite-dimensional.)
will remain unchanged if we allow it to consist of all those complex numbers (instead of just real 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 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 such that it is an isolated point is a finite dimensional subspace. That is, it is an isolated eigenvalue of finite algebraic multiplicity (normal eigenvalues).
is a compact. 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.
The discrete spectrum
and its complement is called the discrete spectrum, so
.
is in the discrete spectrum if it is an isolated eigenvalue of finite multiplicity, meaning that the dimension of the space
on Banach spaces 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.)
Weyl's criterion
Define the following:
A vector is a unit vector iff it has norm 1.
.
:..
Weyl's criterion states that the converse is true as well:.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
be the multiplication operator (or the position operator. The essential range. Note that the sequence increasingly resembles the Dirac delta "function" at 0, even though it does not converge.
be the momentum operator in momentum space, then performing a Fourier transform to position space.
be the Laplace operator is the Sobolev space is similar.
Of densely defined operators
Preliminary concepts
be a Banach space be a densely defined operator is the resolvent set.
Definitions
, thus expands the essential domain.
be its kernel be its cokernel is:
Normally solvable is a closed operator is a closed set. This can be checked via the closed range theorem.
is finite-dimensional inclusive-or is finite-dimensional.
Fredholm is finite-dimensional.
has the same dimension.
If furthermore, there exists a deleted neighborhood of zero that is a subset of the resolvent set.
.
are inverses of each other.
Any of these inclusions may be strict.
, Browder is also called the Browder essential spectrum, etc.
More definitions
There are even more definitions of the essential spectrum.
is an isolated point.
Equalities
Banach space case
is closed.
is bounded and either hypernormal or Toeplitz.
.
is the transpose operator.
Define the radius.
. (D.E. Edmunds and W.D. Evans, 1987).
.
is the discrete spectrum.
for which there exists a singular sequence.
Hilbert space case
. Concretely, we have. This is because 1 is an eigenvalue of the identity operator with infinite multiplicity.
.
See also
References
The self-adjoint case is discussed in
A discussion of the spectrum for general operators can be found in
The original definition of the essential spectrum goes back to