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

Abstract

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 ) .