Ces\`{a}ro-like operator acting between Bloch type spaces

TL;DR

This paper characterizes measures for which a Cesàro-like operator acts boundedly or compactly between different Bloch type spaces, extending understanding of operator behavior in complex analysis.

Contribution

It provides necessary and sufficient conditions on measures for the Cesàro-like operator to be bounded or compact between Bloch type spaces.

Findings

Characterization of measures for boundedness

Criteria for compactness of the operator

Extension of operator theory in Bloch spaces

Abstract

Let $\mu$ 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 Ce\`{a}sro-like operator is defined by $$ \mathcal{C}_\mu(f)(z)=\sum^\infty_{n=0}\mu_n\left(\sum^n_{k=0}a_k\right)z^n, \ z\in \mathbb{D}, $$ where, for $n\geq 0$, $\mu_n$ denotes the $n$-th moment of the measure $\mu$, that is, $\mu_n=\int_{[0, 1)} t^{n}d\mu(t)$. In this paper, we characterize the measures $\mu$ for which $\mathcal{C}_\mu$ is bounded (compact) from one Bloch type space, $\mathcal {B}^{\alpha}$, into another one, $\mathcal {B}^{\beta}$.