Szegő asymptotics for matrix-valued measures with countably many bound states

Abstract

Let μ be a matrix-valued measure with the essential spectrum a single interval and countably many point masses outside of it. Under the assumption that the absolutely continuous part of μ satisfies Szegő’s condition and the point masses satisfy a Blaschke-type condition, we obtain the asymptotic behavior of the orthonormal polynomials on and off the support of the measure. The result generalizes the scalar analogue of Peherstorfer–Yuditskii (2001) [12] and the matrix-valued result of Aptekarev–Nikishin (1983) [1], which handles only a finite number of mass points.