Toeplitz Operators on Abstract Hardy Spaces Built upon Banach Function Spaces

Let be a Banach function space over the unit circle and let be the abstract Hardy space built upon . If the Riesz projection is bounded on and , then the Toeplitz operator is bounded on . We extend well-known results by Brown and Halmos for and show that, under certain assumptions on the space , the Toeplitz operator is bounded (resp., compact) if and only if (resp., ). Moreover, . 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.

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