Cesàro-like operators between the Bloch space and Bergman spaces

Abstract

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 $A^{p}$.