Let $\alpha>0$ and let $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_{\mu,\alpha}=(\mu_{n,k,\alpha})_{n,k\ge0}$ with entries
$\mu_{n,k,\alpha}=\int_{[0,1)}^{}\frac{\Gamma(n+\alpha)}{\Gamma(n+1)\Gamma(\alpha)}t^{n+k}{\rm d}\mu(t)$
induces, formally, the generalized-Hilbert operator
$ \mathcal{H}_{\mu,\alpha}\left ( f \right ) \left ( z \right ) =\sum_{n=0}^{\infty} \left (\sum_{k=0}^{\infty} \mu_{n,k,\alpha}a_k \right )z^n,z\in\mathbb{D}, $
where $f(z)={\textstyle \sum_{k=0}^{\infty }} a_kz^k$ is an analytic function in $\mathbb{D}$. This article is devoted to study the measures $\mu$ for which $\mathcal{H}_{\mu,\alpha }$ is a bounded (resp., compact) operator from $H^p(0<p\le1)$ into $H^p(1\le q<\infty)$. We also study the analogous problem in the Hardy spaces $H^p(1\le p\le2)$. Finally, we obtain the essential norm of $\mathcal{H}_{\mu,\alpha}$ from $H^p(0<p\le1)$ into $H^p(1\le q<\infty)$.
