Conjugations of unitary operators, I

Abstract

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