Orthogonal polynomials on the unit circle: New results

Abstract

We announce numerous new results in the theory of orthogonal polynomials on the unit circle, most of which involve the connection between a measure on the unit circle in the complex plane and the coefficients in the recursion relations for the polynomials known as Verblunsky coefficients. Included are several applications of the recently discovered matrix realization of Cantero, Moral, and Velázquez. In analogy with the spectral theory of Jacobi matrices, several classes of exotic Verblunsky coefficients are studied. A version of Rahkmanov's theorem is proven with a single gap with eigenvalues allowed in the gap. Analogs of Borg's theorem and the Birman-Schwinger principle are found.