  This paper is devoted to the spectral properties of a class of unitary
operators with a matrix representation displaying a band structure. Such band
matrices appear as monodromy operators in the study of certain quantum
dynamical systems. These doubly infinite matrices essentially depend on an
infinite sequence of phases which govern their spectral properties. We prove
the spectrum is purely singular for random phases and purely absolutely
continuous in case they provide the doubly infinite matrix with a periodic
structure in the diagonal direction. We also study some properties of the
singular spectrum of such matrices considered as infinite in one direction
only.
