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Weyl–Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle. (English) Zbl 1118.47023

A class of unitary semi-infinite five-diagonal matrices was introduced by M.J.Cantero, L.Moral and L.Velázquez [Linear Algebra Appl.362, 29–56 (2003; Zbl 1022.42013)], called half-lattice CMV operators. Later, B.Simon [“Orthogonal polynomials on the unit circle.Part 1: Classical theory” (Colloquium Publications AMS 54(1), Providence/RI) (2005; Zbl 1082.42020); “Orthogonal polynomials on the unit circle.Part 2: Spectral theory” (Colloquium Publications AMS 51(2), Providence/RI) (2005; Zbl 1082.42021)] introduced the corresponding unitary doubly infinite five-diagonal matrices, the so-called full-lattice CMV operators.
The present paper is essentially in two parts. The first part gives an extensive survey of the Weyl–Titchmarsh theory for half-lattice CMV operators, related systems of orthonormal Laurent polynomials on the unit circle, and spectral representations; although much of this material had already been treated by Simon, the approach here is operator theoretic. The second part of the paper contains corresponding new results on Weyl–Titchmarsh theory for full-lattice CMV operators. These CMV operators are closely related to the trigonometric moment problem and the work of Szegő over eighty years ago on the asymptotic distribution of eigenvalues of finite section Toeplitz forms.

MSC:

47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34B20 Weyl theory and its generalizations for ordinary differential equations
41A30 Approximation by other special function classes
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
47A10 Spectrum, resolvent
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