Abstract
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, aL∞≤TaB(H[X])≤PB(X)aL∞. 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.
Highlights
The Banach algebra of all bounded linear operators on a Banach space E will be denoted by B(E)
The first aim of this paper is to show that the BrownHalmos theorem remains true for abstract Hardy spaces H[X] built upon reflexive Banach function spaces X if P ∈ B(X)
We show that, under mild assumptions on a Banach function space X, a Toeplitz operator Ta is compact on the abstract Hardy space H[X] built upon X if and only if a = 0
Summary
The Banach algebra of all bounded linear operators on a Banach space E will be denoted by B(E). We show that, under mild assumptions on a Banach function space X, a Toeplitz operator Ta is compact on the abstract Hardy space H[X] built upon X if and only if a = 0 These results are specified to the case of Hardy spaces built upon Lebesgue spaces with Muckenhoupt weights and upon Nakano spaces with certain radial oscillating weights. We prove that if the characteristic functions of all measurable sets E ⊂ T have absolutely continuous norms in X, the sequence {χk}k∈Z+ converges weakly to zero on the abstract Hardy space H[X]. In both cases, it is known that the Riesz projection is bounded
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