Functional model for finite rank perturbations

The unitary perturbations of a given unitary operator by finite rank d operators can be parametrized by dxd unitary matrices; this generalizes the rank one setting, where the Clark family is parametrized by the scalars on the unit circle. For finite rank perturbations we investigate the functional model of a related class of contractions, as well as a (unitary) Clark operator that realizes such a model representation. We express the adjoint of the Clark operator through a matrix-valued Cauchy integral operator. We determine the matrix-valued characteristic functions of the model (for contractions). In the case of inner characteristic functions results suggest a generalization of the normalized Cauchy transform to the finite rank setting. This presentation is based on joint work with Sergei Treil

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