Trace Formulas and a Borg-type Theorem for CMV Operators with Matrix-valued Coefficients

2008/08/04 by Maxim Zinchenko, Zinchenko, Maxim
Mathematics · Physics and Astronomy · #34A55 #47A10 #47B36 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.MP #math.SP #msc:34A55 #msc:47A10 #msc:47B36

paper · pdf · doi:10.48550/arxiv.0808.0382

arxiv created 2008/08/04 · arxiv updated 2009/12/01

Abstract

We prove a general Borg-type inverse spectral result for a reflectionless unitary CMV operator (CMV for Cantero, Moral, and Velázquez) 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.

Related

• A Borg-Type Theorem Associated with Orthogonal Polynomials on the Unit Circle
• Borg-Type Theorems for Matrix-Valued Schrödinger Operators
• A formula related to CMV matrices and Szego cocycles
• Absolutely Continuous Spectrum for CMV Matrices With Small Quasi-Periodic Verblunsky Coefficients
• Generic Spectral Results for CMV Matrices with Dynamically Defined Verblunsky Coefficients