Conjugations of unitary operators, I

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