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

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

2005-01-14

arxiv.org/abs/math/0501212v1

Mathematics

Spectral Theory

28 pages

Scientific paper

We prove a general Borg-type result for reflectionless unitary Cantero-Moral-Velazquez (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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