Rhaly operators acting on Hardy, Bergman, and Dirichlet spaces

Abstract

In this article we address the question of characterizing the sequences of complex numbers $(η)=\{ η_n\}_{n=0}^\infty $ whose associated Rhaly operator $\mathcal R_{(η)}$ is bounded or compact on the Hardy spaces $H^p$ ($1\le p<\infty $), on the Bergman spaces $A^p_α$, and on the Dirichlet spaces $\mathcal D^p_α$ ($1\le p<\infty $, $α>-1$). We give a number of conditions which are either necessary or sufficient for the boundedness (compactness) of $\mathcal R_{(η)}$ on these spaces. These conditions have to do with the membership in certain mean Lipschitz spaces of analytic functions of the function $F_{(η)}$ defined by $F_{(η)}(z)=\sum_{n=0}^\infty η_nz^n$ ($z\in \mathbb D$). \par We prove that if $2\le p<\infty $ and $η_n=\og \left (\frac{1}{n}\right )$, then $\mathcal R_{(η)}$ is bounded on $H^p$. However, there exists a sequence $(η)$ with $η_n=\og \left (\frac{1}{n}\right )$ such that the operator $\mathcal R_{(η)}$ is not bounded on $H^p$ for $1\le p<2$. \par We deal also with the derivative-Hardy spaces. For $p>0$ the derivative-Hardy space $S^p$ consists of those functions $f$, analytic in the unit disc $\mathbb D$, such that $f^\prime \in H^p$. We prove that if $1\le p<\infty $ and $1<q<\infty $ then $\mathcal R_{(η)}$ is a bounded operator from $S^p$ into $S^q$ if and only if it is compact and this happens if and only if $F_{(η)}\in S^q$.