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 $T(a):H[X]\to H[X]$ coincides with $\mathrm{\parallel }a{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}$ for every $a\in C+H^{\mathrm{\infty }}$ 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 Böttcher, Krupnik, and Silbermann for the case of classical Hardy spaces.