Generalized Hilbert operators acting from Hardy spaces to weighted Bergman spaces

Abstract

Let $\mu$ be a positive Borel measure on the interval $[0,1)$. For $\alpha >0$, the generalized Hankel matrix ${\mathcal{H}}_{\mu ,\alpha }=({\mu }_{n,k,\alpha }{)}_{n,k\ge 0}$ with entries ${\mu }_{n,k,\alpha }={\int }_{[0,1)}\frac{\Gamma (n+\alpha )}{n!\Gamma (\alpha )}{t}^{n+k}\mathrm{d}\mu (t)$ induces formally the operator \begin{equation*} \mathcal{H}_{μ, α}(f)(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} μ_{n, k, α} a_k\right) z^n \end{equation*} on the space of all analytic function $f(z)={\sum }_{k=0}^{\infty }{a}_k{z}^k$ in the unit disk $\mathbb{D}$. In this paper, we characterize the measures $\mu$ for which ${\mathcal{H}}_{\mu ,\alpha }(f)$ is well defined on the Hardy spaces $H^p(0<p<\infty )$ and satisfies ${\mathcal{H}}_{\mu ,\alpha }(f)(z)={\int }_{[0,1)}\frac{f(t)}{(1-tz{)}^{\alpha }}\mathrm{d}\mu (t)$. Among these measures, we further describe those for which ${\mathcal{H}}_{\mu ,\alpha }(\alpha >1)$ is a bounded (resp., compact) operator from the Hardy spaces $H^p(0<p<\infty )$ into the weighted Bergman spaces $A_{\alpha -2}^q$.