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Toeplitz Operators on Abstract Hardy Spaces Built upon Banach Function Spaces
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Resumo(s)
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.
Descrição
This work was partially supported by the Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matematica e Aplicacoes).
Palavras-chave
REARRANGEMENT-INVARIANT SPACES OSCILLATING WEIGHTS NORM Analysis