Conjugations of unitary operators, I

Abstract If U is a unitary operator on a separable complex Hilbert space H H, an application of the spectral theorem says there is a conjugation C on H H (an antilinear, involutive, isometry on H H) for which C U C = U^*. 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^* C U C = U ∗.