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 $\mathcal{H}_{\mu,\gamma}=(\mu_{n,k})_{n,k\geq0}$ 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 $0\leq\alpha<2$, $2\leq\beta<4$, $\gamma\geq1$. we characterize the measure $\mu$ for which $\mathcal{H}_{\mu,\gamma}$ is bounded(resp.,compact)from $\mathcal{D}_{\alpha}$ into $\mathcal{D}_{\beta}$.