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 $H[X]$ be the abstract Hardy space built upon X. We show that the essential norm of the Toeplitz operator T(a):H[X]→ H[X] coincides with \|a\|L^∞ for every a∈ C+H^∞ if and only if the essential norm of the backward shift operator T(e₋₁):H[X]→ H[X] 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.