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 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. © 2017 alexei yu. karlovich.