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 $\mathcal{H}_{\mu, \alpha}=(\mu_{n, k, \alpha})_{n, k \geq 0}$ 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}_{\mu, \alpha}(f)(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \mu_{n, k, \alpha} 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 $\mathcal{H}_{\mu, \alpha}(f)(z)=\int_{[0,1)} \frac{f(t)}{(1-t z)^\alpha} \mathrm{d} \mu(t)$. 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 $.