Noncompactness of Toeplitz operators between abstract Hardy spaces

Abstract

In the beginning of 1960s, Brown and Halmos proved that a Toeplitz operator T(a) is compact on the Hardy space $$H^2=H[L^2]$$ over the unit circle $$\mathbb {T}$$ if and only if $$a=0$$ 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 $$\mathbb {T}$$ 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 $$H^1=H[L^1]$$ , although $$L^1$$ has trivial Boyd indices.