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.