A generalized Hilbert matrix acting on Hardy spaces

If $\mu$ is a positive Borel measure on the interval $[0, 1)$, the Hankel matrix $\mathcal H_\mu =(\mu_{n,k})_{n,k\ge 0}$ with entries $\mu_{n,k}=\int_{[0,1)}t^{n+k}\,d\mu(t)$ induces formally the operator $$\mathcal{H}_\mu (f)(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}\mu_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\mathbb{D}$. In this paper we describe those measures $\mu$ for which $\mathcal{H}_\mu$ is a bounded (compact) operator from $H^p$ into $H^q$, $0<p,q<\infty$. We also characterize the measures $\mu$ for which $\mathcal H_\mu$ lies in the Schatten class $S_p(H^2)$, $1<p<\infty$.