Trace Class Perturbations of Spectra of Self-adjoint Operators

Abstract

Chapter 9 deals with trace class perturbations of parts of the spectrum of self-adjoint operators. In the first section, the decomposition of a self-adjoint operator and its spectrum into a pure point part, a singularly continuous part, and an absolutely continuous part is obtained. Let A be a self-adjoint operator, and D a self-adjoint trace class operator acting on the same Hilbert space, and set B=A+D. Krein’s spectral shift associated with such a pair {B,A} is a function on the real line which involves an integral representation formula for the trace of the difference of functions of A and B. The spectral shift and its properties and the proof of Krein’s trace formula are central themes of this chapter. We follow M.G. Krein’s original approach to the spectral shift which was based on perturbation determinants. A short introduction into perturbation determinants is given in a separate section. Another main topic of Chap. 9 is the spectral theory of rank one perturbations B=A+α〈⋅,u〉u of the operator A, where α is a real number, and u is a unit vector. We prove the Aronszajn–Donoghue theorem. It characterizes the various parts of the spectrum of the self-adjoint operator B in terms of boundary values on ℝ  of the holomorphic function 〈R _z (A)u,u〉.