On the essential norms of Toeplitz operators with symbols in C + H∞ acting on abstract Hardy spaces built upon translation-invariant Banach function spaces

Let X be a translation-invariant Banach function space on the unit circle and let $H\left[X\right]$ be the abstract Hardy space built upon X. We suppose the Riesz projection P is bounded on X and estimate the essential norms ${\Vert T\left(a\right)\Vert }_{B\left(H\right[X\left]\right),e}$ of Toeplitz operators $T\left(a\right)f:=P\left(af\right)$ with $a\in C+H^{\infty }$. We prove that in this case${\Vert a\Vert }_{L^{\infty }}\le {\Vert T\left(a\right)\Vert }_{B\left(H\right[X\left]\right),e}\le \min \{2,{\Vert P\Vert }_{B\left(X\right)}\}{\Vert a\Vert }_{L^{\infty }},$ extending the results by the second author [27] for classical Hardy spaces $H^p=H\left[L^p\right]$, $1<p<\infty$. In contrast to our previous works [27] and [16], we do not assume that X is reflexive or separable, which complicates the matters, but allows us to include the Hardy-Lorentz spaces $H^{p,q}=H\left[L^{p,q}\right]$ with $1<p<\infty$ and $q=1,\infty$ into consideration.