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 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 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 {\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 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. (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. Each of the above-defined essential spectra, 1\le k\le 5, 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 is equivalent to Weyl's criterion: is the set of all λ for which there exists a singular sequence. The essential spectrum is invariant under compact perturbations for k = 1,2,3,4, but not for k = 5. The set gives the part of the spectrum that is independent of compact perturbations, that is, 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 where is the discrete spectrum of T.