Conjugations of unitary operators, I

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 ∗ .