  We start with considering rank one self-adjoint perturbations $A_\alpha =
A+\alpha(\,\cdot\,,\varphi)\varphi$ with cyclic vector $\varphi\in \mathcal{H}$
on a separable Hilbert space $\mathcal H$. The spectral representation of the
perturbed operator $A_\alpha$ 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 $A_\alpha$.
  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 at the Thirteenth New Mexico Analysis Seminar and Afternoon in Honor of
Cora Sadosky. Unpublished results are restricted to the last part of this
manuscript.
