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