The Essential Spectrum: Weyl’s Criterion

Abstract

We continue to discuss the general properties of the spectrum of a closed operator A. Here we will require that A act on a Hilbert space H. In the general Banach space setting, recall that we decomposed σ(A) into two disjoint parts: σ_d(A) ≡ the “discrete spectrum ” of A, which is the set of all isolated eigenvalues of A with finite algebraic multiplicity; σ_ess(A) ≡ the “essential spectrum” of A, which is simply given by σ(A)\σ_d(A). Properties of σ_d(A) were studied in Chapter 6. The set σ(A)\ σ_d(A) is called the essential spectrum because, unlike other subsets of σ(A), it is stable under relatively compact perturbations of A. This result, called Weyl’s theorem, is discussed in Chapter 14. The main goal of this chapter is to develop a convenient characterization of σ_ess(A). In particular, given λ. ∈ σ(A), we want to know whether λ ∈σ_ess(A). A clue for such a characterization in the self-adjoint case can be found in Theorem 5.10. Elements in σ(A) are approximate eigenvalues and can be characterized by the existence of sequences of approximating eigenfunctions.