Generalize Hilbert operator acting on Dirichlet spaces

Abstract

Let $\mu$ be a positive Borel measure on the interval $[0,1)$. For $\gamma >0$, the Hankel matrix ${\mathcal{H}}_{\mu ,\gamma }=({\mu }_{n,k}{)}_{n,k\ge 0}$ with entries ${\mu }_{n,k}={\mu }_{n+k}$, where ${\mu }_{n+k}={\int }_0^{\infty }{t}^{n+k}d\mu (t)$. formally induces the operator $${\mathcal{H}}_{\mu ,\gamma }=\sum _{n=0}^{\infty }\left(\sum _{k=0}^{\infty }{\mu }_{n,k}{a}_k\right)\frac{\Gamma (n+\gamma )}{n!\Gamma (\gamma )}{z}^n,$$ on the space of all analytic functions $f(z)={\sum }_{k=0}^{\infty }{a}_k{z}^k$ in the unit disc $\mathbb{D}$. Following ideas from \cite{author3} and \cite{author4}, in this paper, for $0\le \alpha <2$, $2\le \beta <4$, $\gamma \ge 1$. we characterize the measure $\mu$ for which ${\mathcal{H}}_{\mu ,\gamma }$ is bounded(resp.,compact)from ${\mathcal{D}}_{\alpha }$ into ${\mathcal{D}}_{\beta }$.

7 pages