Analysis of the Singular Spectrum for General Perturbations

We investigate the behavior of the singular spectrum of self-adjoint operators under families of Hermitian perturbations, even non-compact ones. Under mild assumptions we have the ``shift'' of the singular spectrum for almost all values of the parameter. Moreover, if we consider trace class perturbations, we observe the ``shift'' outside a countable set of the values of the parameter. While similar results were known for finite rank perturbations, the extensions to trace class perturbations are far from easy, and the proofs require significant new ideas. In addition, some of our results hold (surprisingly enough!) even for all positive and bounded perturbations. This is a consequence of our generalized Aleksandrov disintegration theorem established in this paper. We use some advanced techniques, involving Sz.-Nagy--Foia\c s theory and the operator ${\bf A}_2$ condition.