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

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