In this paper we prove that the simplest band representations of unitary
operators on a Hilbert space are five-diagonal. Orthogonal polynomials on the
unit circle play an essential role in the development of this result, and also
provide a parametrization of such five-diagonal representations which shows
specially simple and interesting decomposition and factorization properties. As
an application we get the reduction of the spectral problem of any unitary
Hessenberg matrix to the spectral problem of a five-diagonal one. Two
applications of these results to the study of orthogonal polynomials on the
unit circle are presented: the first one concerns Krein's Theorem; the second
one deals with the movement of mass points of the orthogonality measure under
monoparametric perturbations of the Schur parameters.Comment: 31 page