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Journal of the London Mathematical Society 2006 74(3):757-777; doi:10.1112/S0024610706023167
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© The London Mathematical Society

A Borg-Type Theorem Associated with Orthogonal Polynomials on the Unit Circle

Fritz Gesztesy and Maxim Zinchenko

Department of Mathematics, University of Missouri–Columbia Columbia, MO 65211, USA fritz{at}math.missouri.edu
Department of Mathematics, University of Missouri–Columbia Columbia, MO 65211, USA maxim{at}math.missouri.edu

Received 10 July 2005. Revision received 17 January 2006.

We prove a general Borg-type result for reflectionless unitary CMV operators U associated with orthogonal polynomials on the unit circle. The spectrum of U is assumed to be a connected arc on the unit circle. This extends a recent result of Simon in connection with a periodic CMV operator with spectrum the whole unit circle.

In the course of deriving the Borg-type result we also use exponential Herglotz representations of Caratheodory functions to prove an infinite sequence of trace formulas connected with the CMV operator U.


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