Cesàro-type operators on derivative-type Hilbert spaces of analytic functions: The proof of a conjecture

Abstract

In this paper, we focus on the boundedness and compactness of the Cesàro-type operators$C_{\mu }\left(f\right)\left(z\right):=\sum _{n=0}^{\infty }\left(\underset{D}{\int }{\omega }^{n}d\mu \right(\omega \left)\right)\left(\sum _{k=0}^n{a}_k\right)z^n,z\in D,$ where μ is a complex Borel measure on the unit disc $D$, acting on two derivative-type Hilbert spaces of analytic functions defined in $D$, including the derivative Hardy space $S^2$ and the weighted Dirichlet space $D_{\alpha }^2(-1<\alpha <\infty )$. As a by-product, we not only prove a conjecture (recently posed by Galanopoulos-Girela-Merchán) about the sufficient conditions for the compactness of $C_{\mu }$ acting on weighted Bergman space $A_{\alpha }^2(-1<\alpha <\infty )$, but also give a complete characterization for the boundedness and compactness of $C_{\mu }$ between different weighted Bergman spaces. At last, we collect some unresolved problems and issues for further study.