Abstract
Let μ be a positive Borel measure on the interval [0, 1). The Hankel matrix \({{\cal H}_\mu} = {({\mu _{n,k}})_{n,k \ge 0}}\) with entries μn,k = μn+k, where μn = ∫[0,1)tndμ(t), induces formally the operator as
where \(f(z) = \sum\limits_{n = 0}^\infty {{a_n}{z^n}} \) is an analytic function in \(\mathbb{D}\). We characterize the positive Borel measures on [0,1) such that \({\cal D}{{\cal H}_\mu}(f)(z) = \int_{[0,1)} {{{f(t)} \over {{{(1 - tz)}^2}}}{\rm{d}}\mu (t)} \) for all f in the Hardy spaces Hp(0 < p < ∞), and among these we describe those for which \({\cal D}{{\cal H}_\mu}\) is a bounded (resp., compact) operator from Hp (0 < p < ∞) into Hq (q > p and q ≥ 1). We also study the analogous problem in the Hardy spaces Hp(1 ≤ p ≤ 2).
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The research was supported by the Zhejiang Provincial Natural Science Foundation (LY23A010003) and the National Natural Science Foundation of China (11671357).
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Ye, S., Feng, G. A Derivative-Hilbert operator acting on Hardy spaces. Acta Math Sci 43, 2398–2412 (2023). https://doi.org/10.1007/s10473-023-0605-6
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DOI: https://doi.org/10.1007/s10473-023-0605-6