A Derivative-Hilbert operator Acting on Dirichlet spaces

Abstract

Let $\mu$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\geq 0}$ with entries $\mu_{n,k}=\mu_{n+k}$, where $\mu_{n}=\int_{[0,1)}t^nd\mu(t)$, induces formally the operator as $$\mathcal{DH}_\mu(f)(z)=\sum_{n=0}^\infty\left(\sum_{k=0}^\infty \mu_{n,k}a_k\right)(n+1)z^n , z\in \mathbb{D},$$ where $f(z)=\sum_{n=0}^{\infty}a_nz^n$ is an analytic function in $\mathbb{D}$. In this paper, we characterize those positive Borel measures on $[0, 1)$ for which $\mathcal{DH}_\mu$ is bounded (resp. compact) from Dirichlet spaces $\mathcal{D}_\alpha ( 0<\alpha\leq2 )$ into $\mathcal{D}_\beta ( 2\leq\beta<4 )$.