Conjugations in $L^2$ and their invariants

TL;DR

This paper characterizes conjugations in the space L^2 of the unit circle that commute or intertwine with multiplication by z, and explores their behavior in Hardy and model subspaces.

Contribution

It provides a comprehensive characterization of conjugations in L^2 and analyzes their properties within Hardy and invariant subspaces, extending understanding of symmetries in these spaces.

Findings

Conjugations commuting with multiplication by z are classified.

Conjugations intertwining multiplication by z and orm are described.

Behavior of conjugations in Hardy and model spaces is analyzed.

Abstract

Conjugations in space $L^2$ of the unit circle commuting with multiplication by $z$ or intertwining multiplications by $z$ and $\bar z$ are characterized. We also study their behaviour with respect to the Hardy space, subspaces invariant for the unilateral shift and model spaces.