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".
, is the set of all real numbers
denotes the identity operator on
(An operator is Fredholm if 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 self-adjoint consists only of real numbers.
is self-adjoint, the spectrum is contained on the real axis.
The essential spectrum is invariant under compact perturbations.
is a compact self-adjoint operator on
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.
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
if there exists a sequence satisfying the above condition, which also converges weakly to the zero vector
and its complement is called the discrete spectrum, so If
is self-adjoint, then, by definition, a number
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
(For general, non-self-adjoint operators
on Banach spaces, by definition, a complex number
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.)
There are several definitions of the essential spectrum, which are not equivalent.
[1] Each of the above-defined essential spectra
For self-adjoint operators, all the above definitions of the essential spectrum coincide.
is equivalent to Weyl's criterion:
for which there exists a singular sequence.
is invariant under compact perturbations for
gives the part of the spectrum that is independent of compact perturbations, that is, where
denotes the set of compact operators on
The spectrum of a closed, densely defined operator
can be decomposed into a disjoint union where
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