Generalized Hilbert Operator Acting on Hardy Spaces

Let $\alpha >0$ and $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix ${\mathcal{H}}_{\mu ,\alpha }=({\mu }_{n,k,\alpha }{)}_{n,k\ge 0}$ with entries ${\mu }_{n,k,\alpha }={\int }_{[0,1)}^{}\frac{\mathrm{\Gamma }(n+\alpha )}{\mathrm{\Gamma }(n+1)\mathrm{\Gamma }(\alpha )}{t}^{n+k}d\mu (t)$, induces, formally, the generalized-Hilbert operator as $${\mathcal{H}}_{\mu ,\alpha }\left(f\right)\left(z\right)=\sum _{n=0}^{\mathrm{\infty }}\left(\sum _{k=0}^{\mathrm{\infty }}{\mu }_{n,k,\alpha }{a}_k\right)z^n,z\in \mathbb{D}$$ where $f(z)={\sum }_{k=0}^{\mathrm{\infty }}{a}_k{z}^k$ is an analytic function in $\mathbb{D}$. This article is devoted study the measures $\mu$ for which ${\mathcal{H}}_{\mu ,\alpha }$ is a bounded(resp., compact) operator from $H^p(0<p\le 1)$ into $H^p(1\le q<\mathrm{\infty })$. Then, we also study the analogous problem in the Hardy spaces $H^p(1\le p\le 2)$. Finally, we obtain the essential norm of ${\mathcal{H}}_{\mu ,\alpha }$ from $H^p(0<p\le 1)$ into $H^p(1\le q<\mathrm{\infty })$.