Riesz projector is finite.) The essential spectrum of closed operators in Banach spaces Let X be a Banach space and let be a closed linear operator on X with dense domain . There are several definitions of the essential spectrum, which are not equivalent. The essential spectrum is the set of all λ such that is not semi-Fredholm (an operator is semi-Fredholm if its range is closed and its kernel or its cokernel is finite-dimensional). The essential spectrum is the set of all λ such that the range of is not closed or the kernel of is infinite-dimensional. The essential spectrum is the set of all λ such that is not Fredholm (an operator is Fredholm if its range is closed and both its kernel and its cokernel are finite-dimensional). The essential spectrum is the set of all λ such that 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). The essential spectrum is the union of σess,1(T) with all components of that do not intersect with the resolvent set . Each of the above-defined essential spectra , , 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 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. See also Spectrum (functional analysis) Resolvent formalism Decomposition of spectrum (functional analysis) Discrete spectrum (mathematics) Spectrum of an operator Operator theory Fredholm theory References The self-adjoint case is discussed in A discussion of the spectrum for general operators can be found in D.E. Edmunds and W.D. Evans (1987), Spectral theory and differential operators, Oxford University Press. . The original definition of