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


 * $$T-\lambda I_X$$

is not a Fredholm operator, where $$I_X$$ denotes the identity operator on X, so that $$I_X(x)=x$$ 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 {ψk} in the space X such that $$\Vert \psi_k\Vert=1$$ and


 * $$ \lim_{k\to\infty} \left\| T\psi_k - \lambda\psi_k \right\| = 0. $$

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 $$\{\psi_k\}$$ 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
 * $$ \sigma_{\mathrm{disc}}(T) = \sigma(T) \setminus \sigma_{\mathrm{ess}}(T). $$

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
 * $$ \{ \psi \in X : T\psi = \lambda\psi \} $$

has finite but non-zero dimension and that there is an ε > 0 such that μ ∈ σ(T) and |μ&minus;λ| < ε imply that μ and λ are equal. (For general nonselfadjoint operators in Banach spaces, by definition, a number $$\lambda$$ 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 space and let $$T:\,X\to X$$ 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) The essential spectrum $$\sigma_{\mathrm{ess},1}(T)$$ is the set of all λ such that $$T-\lambda I_X$$ is not semi-Fredholm (an operator is semi-Fredholm if its range is closed and its kernel or its cokernel is finite-dimensional).
 * 2) The essential spectrum $$\sigma_{\mathrm{ess},2}(T)$$ is the set of all λ such that the range of $$T-\lambda I_X$$ is not closed or the kernel of $$T-\lambda I_X$$ is infinite-dimensional.
 * 3) The essential spectrum $$\sigma_{\mathrm{ess},3}(T)$$ is the set of all λ such that $$T-\lambda I_X$$ is not Fredholm (an operator is Fredholm if its range is closed and both its kernel and its cokernel are finite-dimensional).
 * 4) The essential spectrum $$\sigma_{\mathrm{ess},4}(T)$$ is the set of all λ such that $$T-\lambda I_X$$ 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).
 * 5) The essential spectrum $$\sigma_{\mathrm{ess},5}(T)$$ is the union of σess,1(T) with all components of $$\C\setminus \sigma_{\mathrm{ess},1}(T)$$ that do not intersect with the resolvent set $$\C \setminus \sigma(T)$$.

Each of the above-defined essential spectra $$\sigma_{\mathrm{ess},k}(T)$$, $$1\le k\le 5$$, is closed. Furthermore,
 * $$ \sigma_{\mathrm{ess},1}(T) \subset \sigma_{\mathrm{ess},2}(T) \subset \sigma_{\mathrm{ess},3}(T) \subset \sigma_{\mathrm{ess},4}(T) \subset \sigma_{\mathrm{ess},5}(T) \subset \sigma(T) \subset \C,$$

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
 * $$r_{\mathrm{ess},k}(T) = \max \{ |\lambda| : \lambda\in\sigma_{\mathrm{ess},k}(T) \}. $$

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

The definition of the set $$\sigma_{\mathrm{ess},2}(T)$$ is equivalent to Weyl's criterion: $$\sigma_{\mathrm{ess},2}(T)$$ is the set of all λ for which there exists a singular sequence.

The essential spectrum $$\sigma_{\mathrm{ess},k}(T)$$ is invariant under compact perturbations for k = 1,2,3,4, but not for k = 5. The set $$\sigma_{\mathrm{ess},4}(T)$$ gives the part of the spectrum that is independent of compact perturbations, that is,
 * $$ \sigma_{\mathrm{ess},4}(T) = \bigcap_{K \in B_0(X)} \sigma(T+K), $$

where $$B_0(X)$$ 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
 * $$\sigma(T)=\sigma_{\mathrm{ess},5}(T)\bigsqcup\sigma_{\mathrm{d}}(T)$$,

where $$\sigma_{\mathrm{d}}(T)$$ is the discrete spectrum of T.