On the essential norms of Toeplitz operators on abstract Hardy spaces built upon Banach function spaces

TL;DR

This paper characterizes the essential norms of Toeplitz operators on abstract Hardy spaces built on Banach function spaces, extending classical results to a broader functional analytic setting.

Contribution

It establishes a precise condition linking the essential norm of Toeplitz operators to the backward shift operator on these spaces, generalizing known results from classical Hardy spaces.

Findings

Essential norm of Toeplitz operator equals the supremum norm of the symbol for certain spaces.

The essential norm of the backward shift operator being one is equivalent to the Toeplitz norm characterization.

Extension of classical Hardy space results to abstract Hardy spaces built on Banach function spaces.

Abstract

Let $X$ be a Banach function space over the unit circle such that the Riesz projection $P$ is bounded on $X$ and let $H[X]$ be the abstract Hardy space built upon $X$. We show that the essential norm of the Toeplitz operator $T(a):H[X]\to H[X]$ coincides with $\|a\|_{L^\infty}$ for every $a\in C+H^\infty$ if and only if the essential norm of the backward shift operator $T(\mathbf{e}_{-1}):H[X]\to H[X]$ is equal to one, where $\mathbf{e}_{-1}(z)=z^{-1}$. This result extends an observation by B\"ottcher, Krupnik, and Silbermann for the case of classical Hardy spaces.