Singular integrals, rank one perturbations and clark model in general situation

Abstract

We start with considering rank one self-adjoint perturbations Aα = A +α(⋅, φ)φ with cyclic vector φ∈ ℌ on a separable Hilbert space ℌ. The spectral representation of the perturbed operator Aα is realized by a (unitary) operator of a special type: the Hilbert transform in the two-weight setting, the weights being spectral measures of the operators A and Aα. Similar results will be presented for unitary rank one perturbations of unitary operators, leading to singular integral operators on the circle. This motivates the study of abstract singular integral operators, in particular the regularization of such operator in very general settings. Further, starting with contractive rank one perturbations we present the Clark theory for arbitrary spectral measures (i.e. for arbitrary, possibly not inner characteristic functions). We present a description of the Clark operator and its adjoint in the general settings. Singular integral operators, in particular the so-called normalized Cauchy transform again plays a prominent role. Finally, we present a possible way to construct the Clark theory for dissipative rank one perturbations of self-adjoint operators. These lecture notes give an account of the mini-course delivered by the authors, which was centered around (Liaw and Treil, J Funct Anal 257(6):1947–1975, 2009; Rev Mat Iberoam 29(1):53–74, 2013; J Anal Math). Unpublished results are restricted to the last part of this manuscript.