Conjugations of unitary operators. II (Q6545230)

For a unitary operator \(U\) on a separable complex Hilbert space \(\mathcal{H}\) let \(\mathscr{C}_c(U)\) be the set of all conjugations \(C\) (antilinear, isometric, and involutive maps) on \(\mathcal{H}\) for which \(CUC = U\). Using the multiplicity theory for unitary operators, the authors describe \(\mathscr{C}_c(U)\). Since this set may be empty, it is proved that \(\mathscr{C}_c(U) \neq \varnothing\) if and only if \(U\) is unitarily equivalent to \(U^\ast\).\N\NConcrete descriptions of \(\mathscr{C}_c(U)\) when \(U\) is a unitary matrix, multiplication by an inner function on \(L^2(m)\), the Fourier transform, and the Hilbert transform are included.\N\NFor Part I, see [the authors, Anal. Math. Phys. 14, No. 3, Paper No. 62, 31 p. (2024; Zbl 1546.47031)].