The spectrum and spectral inclusions

Abstract

Abstract In this chapter, we investigate more thoroughly the spectrum of a bounded linear operator on a Banach space from the viewpoint of local spectral theory. We already know from Chapter 1 that local spectral properties may be used to identify certain parts of the spectrum. For instance, we have seen in Proposition 1.3.2 that, for every operator with the single-valued extension property, the spectrum coincides with the surjectivity spectrum, and hence also with the union of the local spectra. To obtain further insight, we need certain tools, which will be introduced in the first two sections of this chapter.