Rhaly operators acting on Hardy and Bergman spaces

Abstract: In this article we address the question of characterizing the sequences of complex numbers $(η)={ ηn}{n=0}^\infty $ whose associated Rhaly operator $\mathcal R_{(η)}$ is bounded or compact on the Hardy spaces $H^p$ ($1\le p<\infty $) or on the Bergman spaces $A^p$ ($1\le p<\infty $). Among other results we completely characterize those $(η)$ for which $\mathcal R_{(η)}$ is bounded or compact on $H^p$ ($1<p\le 2$) and on $A^p$ ($1<p<\infty $). We also give conditions on $(η)$ which are either necessary or sufficient for the boundedness (compactness) of $\mathcal R_{(η)}$ on $H^p$ for $p=1$ and $2<p<\infty $. \par In particular, we prove that if $2\le p<\infty $ and $ηn=\og \left (\frac{1}{n}\right )$, then $\mathcal R{(η)}$ is bounded on $H^p$. However, there exists a sequence $(η)$ with $ηn=\og \left (\frac{1}{n}\right )$ such that the operator $\mathcal R{(η)}$ is not bounded on $H^p$ for $1\le p<2$.