Conjugations in $L^2(\mathcal{H})$

Abstract: Conjugations commuting with $\mathbf{M}z$ and intertwining $\mathbf{M}_z$ and $\mathbf{M}{\bar z}$ in $L^2(\mathcal{H})$, where $\mathcal{H}$ is a Hilbert space, are characterized. We also investigate which of them leave invariant the whole Hardy space $H^2(\mathcal{H})$ or a model space $K_{\Theta}=H^2(\mathcal{H})\ominus\Theta H^2(\mathcal{H})$, where $\Theta$ is a pure operator valued inner function.