A Derivative-Hilbert operator Acting on Dirichlet spaces

Abstract

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