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".
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 
is not a Fredholm operator, where formula_2 denotes the "identity operator" on "X", so that formula_3 for all "x" in "X".
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 formula_4 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 sequence {ψ"k"} in the space "X" such that formula_5 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 formula_7 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
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
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.
The essential spectrum of closed operators in Banach spaces.
Let "X" be a Banach space
and let formula_11 be a closed linear operator on "X" with dense domain formula_12. There are several definitions of the essential spectrum, which are not equivalent.
Each of the above-defined essential spectra formula_25, formula_26, is closed. Furthermore,
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
Even though the spectra may be different, the radius is the same for all "k".
The definition of the set formula_15 is equivalent to Weyl's criterion: formula_15 is the set of all λ for which there exists a singular sequence.
The essential spectrum formula_25 is invariant under compact perturbations for "k" = 1,2,3,4, but not for "k" = 5.
The set formula_20 gives the part of the spectrum that is independent of compact perturbations, that is,
where formula_34 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
where formula_36 is the discrete spectrum of "T".
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