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 Peláez (Stud Math 200:201–220, 2010) and Girela and Merchán (Banach J Math Anal 12:374-398, 2018). In this paper, we define the Derivative-Hilbert operator as

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 \(A^p\), \(1\le p<\infty \).

令\(\mu \)是区间 [0,1) 上的正 Borel 测度。假设\({\mathcal {H}}_\mu \)是 Hankel 矩阵\((\mu _{n,k})_{n,k\ge 0}\)与条目\(\mu _{ n,k}=\mu _{n+k}\)，其中\(\mu _n=\int _{[0,1)}t^nd\mu (t)\)。矩阵形式上归纳了算子\({\mathcal {H}}_\mu (f)(z)=\sum _{n=0}^{\infty }\big (\sum _{k=0}^ {\infty }\mu _{n,k}a_k\big )z^n,\)已在 Bao 和 Wulan (J Math Anal Appl 409:228–235, 2014)、Chatzifuntas 等人中得到广泛研究。(J Math Anal Appl 413:154–168, 2014)、Galanopoulos 和 Peláez (Stud Math 200:201–220, 2010) 以及 Girela 和 Merchán (Banach J Math Anal 12:374-398, 2018)。在本文中，我们将 Derivative-Hilbert 算子定义为

我们主要表征措施\（\亩\）为其\（\ mathcal {DH} _ {\亩} \）是一个有界（相应的，紧凑的）上的布洛赫空间算子\（{\ mathscr {B}} \)。我们还描述了那些度量\(\mu \)，其中\(\mathcal {DH}_{\mu }\)是来自 Bloch 空间\({\mathscr {B}} \)进入伯格曼空间\(A^p\)，\(1\le p<\infty \)。