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 Taf = 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 Ta is bounded (resp., compact) if and only if a ∈ L∞ (resp., a = 0). Moreover, ||a||L ∞ ≤ ||Ta||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.
- REARRANGEMENT-INVARIANT SPACES
- OSCILLATING WEIGHTS