Conjugations of Unitary operators, I

If $U$ is a unitary operator on a separable complex Hilbert space $\mathcal{H}$, an application of the spectral theorem says there is a conjugation $C$ on $\mathcal{H}$ (an antilinear, involutive, isometry on $\mathcal{H}$) 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 $CUC = U^{*}$.