Relatively Compact Operators and the Weyl Theorem

Abstract

The notion of a relatively bounded operator is a fruitful one for establishing the self-adjointness of operators that are perturbations of self-adjoint operators. We also want to know about the effect of the perturbation on the spectrum of the original operator. This is the topic of perturbation theory. As with our discussion of spectrum, we will consider the effects of perturbations on both the essential and the discrete spectra. We have already seen two extreme examples of how the spectrum can change under perturbations that preserve self-adjointness, Theorem 10.7 and Theorem 13.9. In Theorem 10.7, we saw that the effect of a perturbation by a positive, increasing potential, although it preserves the self-adjointness, may drastically alter the spectrum of the unperturbed operator, the Laplacian. Such a perturbation is not relatively bounded.