Conjugations of unitary operators, II

For a unitary operator U on a separable complex Hilbert space \({\mathcal {H}}\) $\mathcal{H}$ , we describe the set \({\mathscr {C}}_{c}(U)\) ${\mathcal{C}}_c(U)$ of all conjugations C (antilinear, isometric, and involutive maps) on \({\mathcal {H}}\) $\mathcal{H}$ for which \(C U C = U\) $CUC=U$ . As this set might be empty, we also show that \({\mathscr {C}}_{c}(U) \not = \varnothing \) ${\mathcal{C}}_c(U)\ne \varnothing$ if and only if U is unitarily equivalent to \(U^{*}\) $U^{\ast }$ .