Cesà ro-like operator acting between Bloch type spaces

Abstract

Let $\hat{I}¼$ be a finite positive Borel measure on the interval $[0,1)$ and $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n\in H(\mathbb{D})$. The Ceà sro-like operator is defined by $${\mathcal{C}}_{\hat{I}}¼(f)(z)=\sum _{n=0}^{\infty }\hat{I}{¼}_n\left(\sum _{k=0}^n{a}_k\right)z^n,z\in \mathbb{D},$$ where, for $n\ge 0$, $\hat{I}{¼}_n$ denotes the $n$-th moment of the measure $\hat{I}¼$, that is, $\hat{I}{¼}_n={\int }_{[0,1)}{t}^{n}d\hat{I}¼(t)$. In this paper, we characterize the measures $\hat{I}¼$ for which ${\mathcal{C}}_{\hat{I}}¼$ is bounded (compact) from one Bloch type space, ${\mathcal{B}}^{\hat{I}}\pm$, into another one, ${\mathcal{B}}^2$.