Toeplitz Operators on Abstract Hardy Spaces Built upon Banach Function Spaces

Let X be a Banach function space over the unit circle T and let H[X] be the abstract Hardy space built upon X. If the Riesz projection P is bounded on X and a ∈ L^∞, then the Toeplitz operator T_af = P(af) is bounded on H[X]. We extend well-known results by Brown and Halmos for X = L² and show that, under certain assumptions on the space X, the Toeplitz operator T_a is bounded (resp., compact) if and only if a ∈ L^∞ (resp., a = 0). Moreover, ||a||_L ^∞ ≤ ||T_a||_ℬ(H[X]) ≤ ||P||_ℬ(X)||a||_L ^∞. These results are specified to the cases of abstract Hardy spaces built upon Lebesgue spaces with Muckenhoupt weights and Nakano spaces with radial oscillating weights.