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A Derivative-Hilbert operator acting on Hardy spaces

Let $μ$ be a positive Borel measure on the interval [0,1). The Hankel matrix $\mathcal{H}_μ= (μ_{n,k})_{n,k\geq0}$ with entries $μ_{n,k}= μ_{n+k}$, where $μ_n=\int_{ [0,1)}t^ndμ(t)$, induces formally the operator $$\mathcal{DH}_μ(f)(z)=\sum_{n=0}^\infty (\sum_{k=0}^\infty μ_{n,k}a_k)(n+1)z^n$$ on the space of all analytic function $f(z)=\sum_{k=0}^ \infty a_k z^n$ in the unit disc $\mathbb{D}$. We characterize those positive Borel measures on $[0,1)$ such that $\mathcal{DH}_μ(f)(z)= \int_{[0,1)} \frac{f(t)}{(1-tz)^2} dμ(t)$ for all in Hardy spaces $H^p(0<p<\infty)$, and among them we describe those for which $\mathcal{DH}_μ$ is a bounded(resp.,compact) operator from $H^p(0<p <\infty)$ into $H^q(q > p$ and $q\geq 1$). We also study the analogous problem in Hardy spaces $H^p(1\leq p\leq 2)$.