Conjugations of unitary operators, I

Abstract

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 \(CUC = U^{*}\).