General Clark model for finite-rank perturbations

Abstract

All unitary (contractive) perturbations of a given unitary operator $U$ by finite-rank-$d$ operators with fixed range can be parametrized by $d\times d$ unitary (contractive) matrices $\mathrm{\Gamma }$; this generalizes unitary rank-one ($d=1$) perturbations, where the Aleksandrov–Clark family of unitary perturbations is parametrized by the scalars on the unit circle $\mathbb{T}\subset ℂ$.

For a strict contraction $\mathrm{\Gamma }$ the resulting perturbed operator $T_{\mathrm{\Gamma }}$ is (under the natural assumption about star cyclicity of the range) a completely nonunitary contraction, so it admits the functional model.

We investigate the Clark operator, i.e., a unitary operator that intertwines $T_{\mathrm{\Gamma }}$ (written in the spectral representation of the nonperturbed operator $U$) and its model. We make no assumptions on the spectral type of the unitary operator $U$; an absolutely continuous spectrum may be present.

We first find a universal representation of the adjoint Clark operator in the coordinate-free Nikolski–Vasyunin functional model; the word “universal” means that it is valid in any transcription of the model. This representation can be considered to be a special version of the vector-valued Cauchy integral operator.

Combining the theory of singular integral operators with the theory of functional models, we derive from this abstract representation a concrete formula for the adjoint of the Clark operator in the Sz.-Nagy–Foiaş transcription. As in the scalar case, the adjoint Clark operator is given by a sum of two terms: one is given by the boundary values of the vector-valued Cauchy transform (postmultiplied by a matrix-valued function) and the second one is just the multiplication operator by a matrix-valued function.

Finally, we present formulas for the direct Clark operator in the Sz.-Nagy–Foiaş transcription.