Golinskii L. Orthogonal polynomials on the unit circle, Szego difference equations and unitary Hessenberg operators

Orthogonal polynomials on the unit circle. To find new patterns in the theory of orthogonal polynomials on circular arcs, to develop spectral theory of the Szego matrix difference equation. Methods of the Hilbert space operator theory, finite-difference equations, function theory in the unit disk. The theory of subordinate solutions and the Simon-Wolff theory are developed; new conditions for absolute continuity of measures in terms of norms of transfer matrices are obtained; new classes of perturbations for Hessenberg and Jacobi matrices are introduced and the properties of their spectra are studied; direct and inverse theorems for the boundary values of bounded analytic functions in the unit disk and their Schur parameters are proved. The results are important for the theory of random stationary sequences and theory of integrable systems.