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 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.