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

TL;DR

This paper extends Borg-type inverse spectral results to matrix-valued CMV operators, providing explicit formulas for Verblunsky coefficients and trace formulas, thus advancing the understanding of reflectionless unitary operators with matrix coefficients.

Contribution

It introduces a general Borg-type inverse spectral theorem for matrix-valued CMV operators, including explicit formulas and trace formulas, extending prior scalar results.

Findings

Explicit formula for Verblunsky coefficients of reflectionless CMV matrices.

Derivation of trace formulas linked to CMV operators.

Extension of scalar results to matrix-valued coefficients.

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.