The Brown–Halmos theorem for a pair of abstract Hardy spaces

Let H[X] and H[Y] be abstract Hardy spaces built upon Banach function spaces X and Y over the unit circle T. We prove an analogue of the Brown–Halmos theorem for Toeplitz operators T_a acting from H[X] to H[Y] under the only assumption that the space X is separable and the Riesz projection P is bounded on the space Y. We specify our results to the case of variable Lebesgue spaces X=L^p(⋅) and Y=L^q(⋅) and to the case of Lorentz spaces X=Y=L^p,q(w), 1<p<∞, 1≤q<∞ with Muckenhoupt weights w∈A_p(T).