Noncompactness of Toeplitz operators between abstract Hardy spaces

摘要

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 T if and only if a = 0 a.e. Recently, Lesnik [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 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.