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A BORG-TYPE THEOREM ASSOCIATED WITH ORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE

Published online by Cambridge University Press:  04 January 2007

FRITZ GESZTESY  and
MAXIM ZINCHENKO
FRITZ GESZTESY
Affiliation:
Department of Mathematics, University of Missouri–Columbia, Columbia, MO 65211, USAfritz@math.missouri.edu
MAXIM ZINCHENKO
Affiliation:
Department of Mathematics, University of Missouri–Columbia, Columbia, MO 65211, USAmaxim@math.missouri.edu
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Abstract

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 $U$.

Keywords

47B3634A5547A10 (primary)34L40 (secondary)
Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 74 , Issue 3 , December 2006 , pp. 757 - 777
Copyright
The London Mathematical Society 2006

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Footnotes

Based on work supported by the US National Science Foundation under Grant No. DMS-0405526.