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 \(T(a):H[X]\rightarrow H[X]\) $T(a):H[X]\to H[X]$ coincides with \(\Vert a\Vert _{L^\infty }\) ${\Vert a\Vert }_{L^{\infty }}$ for every \(a\in C+H^\infty \) $a\in C+H^{\infty }$ if and only if the essential norm of the backward shift operator \(T(\textbf{e}_{-1}):H[X]\rightarrow H[X]\) $T({\mathbf{e}}_{-1}):H[X]\to H[X]$ is equal to one, where \(\textbf{e}_{-1}(z)=z^{-1}\) ${\mathbf{e}}_{-1}(z)=z^{-1}$ . This result extends an observation by Böttcher, Krupnik, and Silbermann for the case of classical Hardy spaces.