Abstract: Let $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\geq 0}$ with entries $\mu_{n,k}=\mu_{n+k}$, where $\mu_{n}=\int_{[0,1)}t^n{\rm d}\mu(t)$, induces formally the operator as $\mathcal{DH}_\mu(f)(z)=\sum\limits_{n=0}^\infty\Big(\sum\limits_{k=0}^\infty \mu_{n,k}a_k\Big)(n+1)z^n , z\in \mathbb{D},$where $f(z)=\sum\limits_{n=0}^{\infty}a_nz^n$ is an analytic function in $\mathbb{D}$.We characterize the positive Borel measures on $[0,1)$ such that $\mathcal{DH}_\mu(f)(z)= \int_{[0,1)} \frac{f(t)}{{(1-tz)^2}} {\rm d}\mu(t)$ for all $f$ in the Hardy spaces $H^p(0<p<\infty)$, and among these we describe those for which $\mathcal{DH}_\mu$ 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 the Hardy spaces $H^p(1\leq p\leq 2)$.

Key words: Derivative-Hilbert operators, Hardy spaces, Carleson measures