Dilation to the unilateral shifts

The classical result of Foias says that an operator power dilates to a unilateral shift if and only if it is aC ₀ contraction. In this paper, we consider the corresponding question of dilating to a unilateral shift. We show tht for contractions with at least one defect index finite, dilation and power dilation to some unilateral shift amount to the same thing. The only difference is on the minimum multiplicity of the unilateral shift to which the contraction can be (power) dilated. We also obtain a characterization of contractions which are finite-rank perturbations of a unilateral shift, generalizing the rank-one perturbation result of Nakamura.