Vol. 12, No. 2, 2019

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General Clark model for finite-rank perturbations

Constanze Liaw and Sergei Treil

Vol. 12 (2019), No. 2, 449–492
DOI: 10.2140/apde.2019.12.449
Abstract

All unitary (contractive) perturbations of a given unitary operator $U$ by finite-rank-$d$ operators with fixed range can be parametrized by $d×d$ unitary (contractive) matrices $\Gamma$; 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 $\Gamma$ the resulting perturbed operator ${T}_{\Gamma }$ 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 ${T}_{\Gamma }$ (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.

Keywords
finite-rank perturbations, Clark theory, dilation theory, functional model, normalized Cauchy transform
Mathematical Subject Classification 2010
Primary: 44A15, 47A20, 47A55
Milestones
Received: 20 November 2017
Revised: 17 April 2018
Accepted: 30 May 2018
Published: 14 August 2018
Authors
 Constanze Liaw Department of Mathematical Sciences University of Delaware Newark, DE United States CASPER Baylor University Waco, TX United States Sergei Treil Department of Mathematics Brown University Providence, RI United States