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 \(F_{(\eta )}(z)=\sum _{n=0}^\infty \eta _nz^n\) ( \(z\in {\mathbb {D}}\) ). We prove that if \(2\le p<\infty \) and \(\eta _n={{\,\textrm{O}\,}}\left( \frac{1}{n}\right) \) , then \({\mathcal {R}}_{(\eta )}\) is bounded on \(H^p\) . However, there exists a sequence \((\eta )\) with \(\eta _n={{\,\textrm{O}\,}}\left( \frac{1}{n}\right) \) such that the operator \({\mathcal {R}}_{(\eta )}\) is not bounded on \(H^p\) for \(1\le p<2\) . 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 \(F_{(\eta )}\in S^q\) .
