Conjugations on the Hardy space $H^{2}$

TL;DR

This paper characterizes all conjugations on the Hardy space $H^{2}$ and identifies complex symmetric Toeplitz operators associated with these conjugations, advancing understanding of symmetries in functional analysis.

Contribution

It provides a complete characterization of conjugations on $H^{2}$ and describes a class of complex symmetric Toeplitz operators linked to these conjugations.

Findings

All conjugations on $H^{2}$ are characterized.

A class of complex symmetric Toeplitz operators is identified.

Connections between conjugations and operator symmetry are established.

Abstract

A conjugation $C$ on a separable complex Hilbert space $\mathcal H$ is an antilinear operator that is isometric and involutive. In this notes, we characterize all conjugations on the Hardy-Hilbert space $H^{2}$ over the disk. In addition, we characterize complex symmetric Toeplitz operators with a special type of these conjugations.