Minimal Rank Decoupling of Full-Lattice CMV Operators with Scalar- and Matrix-Valued Verblunsky Coefficients

TL;DR

This paper investigates how full-lattice CMV operators with scalar and matrix Verblunsky coefficients can be minimally decoupled into two half-lattice operators, providing explicit formulas and analyzing the perturbation rank.

Contribution

It introduces explicit formulas for minimal rank decoupling of full-lattice CMV operators into half-lattice operators, extending prior results and contrasting with the Jacobi case.

Findings

Decoupling full-lattice CMV operators requires twice the minimal rank perturbation compared to Jacobi operators.

Explicit formulas for minimal rank perturbations are derived.

Relations between Weyl--Titchmarsh functions and Green's functions for CMV operators are established.

Abstract

Relations between half- and full-lattice CMV operators with scalar- and matrix-valued Verblunsky coefficients are investigated. In particular, the decoupling of full-lattice CMV operators into a direct sum of two half-lattice CMV operators by a perturbation of minimal rank is studied. Contrary to the Jacobi case, decoupling a full-lattice CMV matrix by changing one of the Verblunsky coefficients results in a perturbation of twice the minimal rank. The explicit form for the minimal rank perturbation and the resulting two half-lattice CMV matrices are obtained. In addition, formulas relating the Weyl--Titchmarsh $m$-functions (resp., matrices) associated with the involved CMV operators and their Green's functions (resp., matrices) are derived.