Trace Formulas and a Borg-type Theorem for CMV Operators with Matrix-valued Coefficients
Mathematical Physics
2008-09-25 v1 math.MP
Spectral Theory
Abstract
We prove a general Borg-type inverse spectral result for a reflectionless unitary CMV operator (CMV for Cantero, Moral, and Vel\'azquez) 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 on 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.
Cite
@article{arxiv.0808.0382,
title = {Trace Formulas and a Borg-type Theorem for CMV Operators with Matrix-valued Coefficients},
author = {Maxim Zinchenko},
journal= {arXiv preprint arXiv:0808.0382},
year = {2008}
}