A Helson-Szeg\"o theorem for subdiagonal subalgebras with applications to Toeplitz operators

TL;DR

This paper extends the classical Helson-Szego theorem to noncommutative subdiagonal subalgebras and applies it to characterize invertible Toeplitz operators on noncommutative Hardy spaces.

Contribution

It formulates a noncommutative Helson-Szego theorem and uses it to characterize symbols of invertible Toeplitz operators in a noncommutative setting.

Findings

Established a noncommutative Helson-Szego theorem.

Provided a characterization of invertible Toeplitz operators.

Connected classical harmonic analysis with noncommutative operator theory.

Abstract

We formulate and establish a noncommutative version of the well known Helson-Szego theorem about the angle between past and future for subdiagonal subalgebras. We then proceed to use this theorem to characterise the symbols of invertible Toeplitz operators on the noncommutative Hardy spaces associated to subdiagonal subalgebras.