Vol. 12, No. 2, 2019

Download this article
Download this article For screen
For printing
Recent Issues

Volume 17
Issue 7, 2247–2618
Issue 6, 1871–2245
Issue 5, 1501–1870
Issue 4, 1127–1500
Issue 3, 757–1126
Issue 2, 379–756
Issue 1, 1–377

Volume 16, 10 issues

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Editors' interests
 
Subscriptions
 
ISSN 1948-206X (online)
ISSN 2157-5045 (print)
 
Author index
To appear
 
Other MSP journals
This article is available for purchase or by subscription. See below.
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 Γ; 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 T .

For a strict contraction Γ the resulting perturbed operator TΓ 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 TΓ (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.

PDF Access Denied

We have not been able to recognize your IP address 3.236.86.184 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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