Essential Spectrum - The Essential Spectrum of General Bounded Operators

The Essential Spectrum of General Bounded Operators

In the general case, X denotes a Banach space and T is a bounded operator on X. There are several definitions of the essential spectrum in the literature, which are not equivalent.

1. The essential spectrum σ_ess,1(T) is the set of all λ such that λI − T 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 σ_ess,2(T) is the set of all λ such that the range of λI − T is not closed or the kernel of λI − T is infinite-dimensional.
3. The essential spectrum σ_ess,3(T) is the set of all λ such that λI − T 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 σ_ess,4(T) is the set of all λ such that λI − T 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 σ_ess,5(T) is the union of σ_ess,1(T) with all components of C \ σ_ess,1(T) that do not intersect with the resolvent set C \ σ(T).

The essential spectrum of an operator is closed, whatever definition is used. Furthermore,

but any of these inclusions may be strict. However, for self-adjoint operators, all the above definitions for 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 essential spectrum σ_ess,k(T) is invariant under compact perturbations for k = 1,2,3,4, but not for k = 5. The case k = 4 gives the part of the spectrum that is independent of compact perturbations, that is,

where K(X) denotes the set of compact operators on X.

The second definition generalizes Weyl's criterion: σ_ess,2(T) is the set of all λ for which there is no singular sequence.