Cesàro-like operator acting between Bloch type spaces

Abstract

Let $μ$ 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 Ceàsro-like operator is defined by $$ \mathcal{C}_μ(f)(z)=\sum^\infty_{n=0}μ_n\left(\sum^n_{k=0}a_k\right)z^n, \ z\in \mathbb{D}, $$ where, for $n\geq 0$, $μ_n$ denotes the $n$-th moment of the measure $μ$, that is, $μ_n=\int_{[0, 1)} t^{n}dμ(t)$. In this paper, we characterize the measures $μ$ for which $\mathcal{C}_μ$ is bounded (compact) from one Bloch type space, $\mathcal {B}^α$, into another one, $\mathcal {B}^β$.