Brief description: A geometric approach to rank one perturbations via the theory of dilations
We consider the family of unitary rank one perturbations of a cyclic completely nonunitary contraction with deficiency indices (1,1) on a separable Hilbert space.
We study unitary rank one perturbations from a rather geometric point of view via methods from dilation theory. Using these techniques, we present new proofs of known statements, e.g.~by Livsic and Poltoratski, and hence embed several of the existing approaches into this unifying framework.
Further we derive a certain representation for the characteristic function of the original contraction. An application of this representation then enables us to control the jump behavior of the normalized Cauchy transform of certain measures `across' the unit circle (allowing the case where the family of unitary rank one perturbations has non-empty absolutely continuous spectrum).