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

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.

arXiv comment30 pages

References in corpus (6)

• The Analytic Theory of Matrix Orthogonal Polynomials
• Finite genus solutions for the Ablowitz-Ladik hierarchy
• Local Conservation Laws and the Hamiltonian Formalism for the Ablowitz-Ladik Hierarchy
• Algebro-Geometric Finite-Gap Solutions of the Ablowitz-Ladik Hierarchy
• A connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line
• Conservative discrete time-invariant systems and block operator CMV matrices

Cited by in corpus (2)

• On discrete symplectic systems: Associated maximal and minimal linear relations and nonhomogeneous problems
• Reflectionless CMV matrices and scattering theory