We obtain a Blaschke-type necessary condition on zeros of analytic functions on the unit disc with different types of exponential growth at the boundary. These conditions are used to prove Lieb-Thirring-type inequalities for the eigenvalues of complex Jacobi matrices.

Abstract. The relation between the Toda lattices and similar nonlinear chains and orthogonal polynomials on the real line has been elaborated immensely for the last decades. We examine another system of differential-difference equations known as the Schur flow, within the framework of the theory of orthogonal polynomials on the unit circle. This system can be displayed in equivalent form as the Lax equation, and the corresponding spectral measure undergoes a simple transformation. The general result is illustrated on the modified Bessel measures on the unit circle and the long time behavior of their Verblunsky coefficients

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Under the mild trace-norm assumptions, we show that the eigenvalues of an arbitrary (non-Hermitian) complex perturbation of a Jacobi matrix sequence (not necessarily real) are still distributed as the real-valued function 2 cos t on [0, π] which characterizes the nonperturbed case. In this way the real interval [- 2, 2] is still a cluster for the asymptotic joint spectrum and, moreover, [- 2, 2] still attracts strongly (with infinite order) the perturbed matrix sequence. The results follow in a straightforward way from more general facts that we prove in an asymptotic linear algebra framework and are plainly generalized to the case of matrix-valued symbols, which arises when dealing with orthogonal polynomials with asymptotically periodic recurrence coefficients

Orthogonal polynomials on the unit circle are completely determined by their reflection coefficients through the Szegő recurrences. We assume that the reflection coefficients converge to some complex number a with 0 < |a| < 1. The polynomials then live essentiallyWe analyze the orthogonal polynomials by comparing them with the orthogonal polynomials with constant reflection coefficients, which were studied earlier by Ya. L. Geronimus and N. I. Akhiezer. In particular, we show that under certain assumptions on the rate of convergence of the reflection coefficients the orthogonality measure will be absolutely continuous on the arc. In addition, we also prove the unit circle analogue of M. G. Krein's characterization of compactly supported nonnegative Borel measures on the real line whose support contains one single limit point in terms of the corresponding system of orthogonal polynomials.1991 Mathematics Subject Classification. 42C05.