Trace formulas and a Borg‐type theorem for CMV operators with matrix‐valued coefficients

Abstract

AbstractWe prove a general Borg‐type inverse spectral result for a reflectionless unitary CMV operator (CMV for Cantero, Moral, and Velázquez [13]) associated with matrix‐valued Verblunsky coefficients. More precisely, we find an explicit formula for the Verblunsky coefficients of a reflectionless CMV matrix whose spectrum consists of a connected arc on the unit circle. This extends a recent result [39] for CMV operators with scalar‐valued coefficients.In the course of deriving the Borg‐type result we also use exponential Herglotz representations of Caratheodory matrix‐valued functions to prove an infinite sequence of trace formulas connected with CMV operators (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)