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

Abstract

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 .

Weyl–Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle

“…We conclude this introduction with citing a Borg-type (inverse spectral) result from our paper [20], which motivated us to write the present paper. First we introduce our notation for closed arcs on the unit circle *D,…”

Equality of the Spectral and Dynamical Definitions of Reflection

“…Two-sided CMV matrices were defined first in [40], although related objects appeared earlier in [3,10]. For further study, we mention [4,17,19,39].…”

“…The equivalence is an easy computation using the relations between the Carathéodory and Schur functions (see, e.g., [17]). It is known [17] that (4.8) for one n implies it for all n. It is also known [4] that while (4.8) implies δ n , (C + z)/(C − z)δ n has purely real boundary values a.e. on e, the converse can be false.…”

Right Limits and Reflectionless Measures for CMV Matrices

“…Since Simon analyzed the analogous Jacobi case [35], we feel the following is fitting: Definition 1. 12. We say that a whole-line Jacobi matrix, H, belongs to Simon Class if µ n = µ m for all n, m ∈ Z, where µ j is the spectral measure of H and δ j .…”