It is shown that, under some natural additional conditions, a transformation which
intertwines a cyclic singular unitary operator with a one-dimensional perturbation
of another cyclic singular unitary operator is the operator of multiplication by a
multiplier between model spaces. Using this result, it is shown that if T is a one-dimensional
perturbation of a unitary operator and also a quasiaffine transform of a singular
unitary operator, and T is power bounded, then T is similar to a unitary operator.
Moreover,sup(n >= 0) parallel to T-n parallel to <= (2(sup(n >= 0) parallel to T-n
parallel to)(2) + 1) . (sup(n >= 0) parallel to T-n parallel to)(-5).