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Published 2009 | Submitted
Book Section - Chapter Open

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.

Additional Information

Appeared in Difference Equations and Applications, Proceeding of the 14th International Conference on Difference Equations and Applications, Istanbul, July 21–25, 2008, M. Bohner, Z. Došlá, G. Ladas, M. Ünal, and A. Zafer (eds.), Uğur–Bahçeşehir University Publishing Company, Istanbul, Turkey, 2009, pp. 19–59. Fritz Gesztesy would like to thank all organizers of the 14th International Conference on Difference Equations and Applications (ICDEA 2008), for their kind invitation and the stimulating atmosphere created during the meeting. In addition, he is particularly indebted to Mehmet Ünal for the extraordinary hospitality extended to him during his ten day stay in Istanbul in July of 2008.

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Created:
August 20, 2023
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October 18, 2023