LAUSR

Let $$\mu $$ be a positive Borel measure on the interval [0,1). Suppose $${\mathcal {H}}_\mu $$ is the Hankel matrix $$(\mu _{n,k})_{n,k\ge 0}$$ with entries $$\mu _{n,k}=\mu _{n+k}$$ , where $$\mu _n=\int _{[0,1)}t^nd\mu (t)$$ . The matrix formally induces the operator $${\mathcal {H}}_\mu (f)(z)=\sum _{n=0}^{\infty }\big (\sum _{k=0}^{\infty }\mu _{n,k}a_k\big )z^n,$$ which has been widely studied in Bao and Wulan (J Math Anal Appl 409:228–235, 2014), Chatzifountas et al. (J Math Anal Appl 413:154–168, 2014), Galanopoulos and Pelaez (Stud Math 200:201–220, 2010) and Girela and Merchan (Banach J Math Anal 12:374-398, 2018). In this paper, we define the Derivative-Hilbert operator as $$\begin{aligned} \mathcal {DH}_{\mu }(f)(z)=\sum _{n=0}^{\infty } \left( \sum _{k=0}^{\infty } \mu _{n, k} a_{k}\right) (n+1)z^{n}. \end{aligned}$$ We mainly characterize the measures $$\mu $$ for which $$\mathcal {DH}_{\mu }$$ is a bounded (resp., compact) operator on the Bloch space $${\mathscr {B}}$$ . We also characterize those measures $$\mu $$ for which $$\mathcal {DH}_{\mu }$$ is a bounded (resp., compact) operator from the Bloch space $${\mathscr {B}}$$ into the Bergman space $$A^p$$ , $$1\le p<\infty $$ .