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 denotes the identity operator on X, so that 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 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 {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 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 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