Rhaly operators acting on Hardy, Bergman, and Dirichlet spaces

TL;DR

This paper characterizes the boundedness and compactness of Rhaly operators on various analytic function spaces, linking these properties to membership in mean Lipschitz spaces and providing specific conditions and examples.

Contribution

It provides necessary and sufficient conditions for Rhaly operators to be bounded or compact on Hardy, Bergman, and Dirichlet spaces, including new results for derivative-Hardy spaces.

Findings

Boundedness of Rhaly operators on Hardy spaces for certain sequences

Existence of sequences with specific growth where operators are not bounded

Characterization of boundedness and compactness on derivative-Hardy spaces

Abstract

In this article we address the question of characterizing the sequences of complex numbers $(\eta )=\{ \eta_n\}_{n=0}^\infty $ whose associated Rhaly operator $\mathcal R_{(\eta )}$ is bounded or compact on the Hardy spaces $H^p$ ($1\le p<\infty $), on the Bergman spaces $A^p_\alpha $, and on the Dirichlet spaces $\mathcal D^p_\alpha $ ($1\le p<\infty $, $\alpha >-1$). We give a number of conditions which are either necessary or sufficient for the boundedness (compactness) of $\mathcal R_{(\eta )}$ on these spaces. These conditions have to do with the membership in certain mean Lipschitz spaces of analytic functions of the function $F_{(\eta )}$ defined by $F_{(\eta )}(z)=\sum_{n=0}^\infty \eta _nz^n$ ($z\in \mathbb D$). \par We prove that if $2\le p<\infty $ and $\eta_n=\og \left (\frac{1}{n}\right )$, then $\mathcal R_{(\eta )}$ is bounded on $H^p$. However, there exists a sequence $(\eta )$ with $\eta_n=\og \left (\frac{1}{n}\right )$ such that the operator $\mathcal R_{(\eta )}$ 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 $\mathcal R_{(\eta )}$ is a bounded operator from $S^p$ into $S^q$ if and only if it is compact and this happens if and only if $F_{(\eta )}\in S^q$.