A Derivative-Hilbert operator acting on Hardy spaces

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 μ_n,k = μ_n+k, where μ_n = ∫_[0,1)tⁿdμ(t), induces formally the operator as

${\cal D}{{\cal H}_\mu}(f)(z) = \sum\limits_{n = 0}^\infty {\left({\sum\limits_{k = 0}^\infty {{\mu _{n,k}}{a_k}}} \right)(n + 1){z^n},z \in \mathbb{D}} $

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 H^p(0 < p < ∞), and among these we describe those for which \({\cal D}{{\cal H}_\mu}\) is a bounded (resp., compact) operator from H^p (0 < p < ∞) into H^q (q > p and q ≥ 1). We also study the analogous problem in the Hardy spaces H^p(1 ≤ p ≤ 2).