Let α > 0 and let μ be a positive Borel measure on the interval [0,1). The Hankel matrix \({\cal{H}}_{\mu,\alpha}=(\mu_{n,k,\alpha})_{n,k\ge0}\) with entries
\(\mu_{n,k,\alpha}=\int_{[0,1)}^{}{{\Gamma (n + \alpha)} \over {\Gamma (n + 1)\Gamma (\alpha)}}t^{n+k}{\rm d}\mu(t)\)
induces, formally, the generalized-Hilbert operator
\({\cal{H}}_{\mu,\alpha}\left (f \right) \left (z \right) =\sum_{n=0}^{\infty} \left (\sum_{k=0}^{\infty} \mu_{n,k,\alpha}a_k \right)z^n,z\in\mathbb{D},\)
where \(f(z)=\sum\nolimits_{k=0}^{\infty} a_kz^k\) is an analytic function in \(\mathbb{D}\) . This article is devoted to study the measures μ for which \({\cal{H}}_{\mu,\alpha}\) is a bounded (resp., compact) operator from Hp(0 < p ≤ 1) into Hp(1 ≤ q < ∞). We also study the analogous problem in the Hardy spaces Hp(1 ≤ p ≤ 2). Finally, we obtain the essential norm of \({\cal{H}}_{\mu,\alpha}\) from Hp(0 < p ≤ 1) into Hp(1 ≤ q < ∞).
