Conjugations of Unitary operators, I

If U is a unitary operator on a separable complex Hilbert space H, an application of the spectral theorem says there is a conjugation C on H (an antilinear, involutive, isometry on H) for which C U C = U*. In this paper, we fix a unitary operator U and describe all of the conjugations C which satisfy this property. As a consequence of our results, we show that a subspace is hyperinvariant for U if and only if it is invariant for any conjugation C for which CUC = U*.