Cesàro-like operators between the Bloch space and Bergman spaces

Let \({\mathbb {D}}\) be the unit disc in the complex plane. Given a positive finite Borel measure \(\mu \) on the radius [0, 1), we denote the n-th moment of \(\mu \) as \(\mu _{n}\), that is, \(\mu _{n}=\int _{[0,1)}t^{n} \textrm{d}\mu (t).\) The Cesàro-like operator \({\mathcal {C}}_{\mu ,s}\) is defined on \(H({\mathbb {D}})\) as follows: If \(f(z)=\sum _{n=0}^{\infty }a_{n}z^{n} \in H({\mathbb {D}} )\) then \({\mathcal {C}}_{\mu ,s}(f)\) is defined by

$$\begin{aligned} {\mathcal {C}}_{\mu ,s}(f)(z)=\sum _{n=0}^{\infty }\left( \mu _{n} \sum _{k=0}^{n}\frac{\Gamma (n-k+s)}{\Gamma (s)(n-k)!}a_{k}\right) z^{n},\ \ z\in {\mathbb {D}}. \end{aligned}$$

In this paper, our focus is on the action of the \(\mathrm Ces\grave{a}ro\)-type operator \({\mathcal {C}}_{\mu ,s}\) on spaces of analytic functions in \({\mathbb {D}}\). We characterize the boundedness (compactness) of the \(\mathrm Ces\grave{a}ro\)-like operator \({\mathcal {C}}_{\mu ,s}\), acting between the Bloch space \({\mathcal {B}}\) and the Bergman space \(A^{p}\).