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

TL;DR

This paper characterizes conjugations in $L^2( ext{Hilbert space})$ that commute with multiplication operators and explores their invariance properties on Hardy and model spaces, advancing understanding of operator symmetries.

Contribution

It provides a complete characterization of conjugations commuting with multiplication operators and analyzes their invariance on Hardy and model spaces in a Hilbert space setting.

Findings

Conjugations commuting with $ extbf{M}_z$ are characterized.

Certain conjugations leave Hardy space $H^2( ext{Hilbert space})$ invariant.

Some conjugations preserve model spaces $K_{ heta}$ associated with inner functions.

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.