Noncompactness of Toeplitz operators between abstract Hardy spaces

In the beginning of 1960s, Brown and Halmos proved that a Toeplitz operator T(a) is compact on the Hardy space H2=H[L2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^2=H[L^2]$$\end{document} over the unit circle T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {T}$$\end{document} if and only if a=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=0$$\end{document} a.e. Recently, Leśnik [13] generalized this result to the setting of Toeplitz operators acting between abstract Hardy spaces H[X] and H[Y] built upon possibly different rearrangement-invariant Banach function spaces X and Y over T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {T}$$\end{document} such that Y has nontrivial Boyd indices. We show that the general principle of noncompactness of nontrivial Toeplitz operators between abstract Hardy spaces H[X] and H[Y] remains true for much more general spaces X and Y. In particular, there are no nontrivial compact Toeplitz operators on the Hardy space H1=H[L1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1=H[L^1]$$\end{document}, although L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^1$$\end{document} has trivial Boyd indices.