arXiv2026
Let \(η=(η_n)_{n\ge0}\) be a complex sequence such that \(F_η(z)=\sum_{n\ge0}η_nz^n\in\Hol(\D)\), and let \(\mathcal R_{(η)}\) be the associated Rhaly operator. We give a complete characterization of boundedness and compactness of \(\mathcal R_{(η)}:\mathcal D^p_α\to\mathcal D^q_β\) for \(1 -1\). The characterization is formulated in terms of the \(H^q\)-norms of the dyadic blocks of \(F_η\). For \(-1<α p-2\), they are expressed in analytic Besov spaces. When boundedness and compactness do not coincide, compactness is characterized by a little analytic truncated Besov space or a little Besov space. We also obtain operator norm estimates and, in these cases, essential norm estimates in terms of the distance of \(F_η\) to these little spaces. As applications, we obtain boundedness and compactness characterizations for Rhaly operators between weighted Bergman spaces and for Cesàro-type operators induced by positive measures. In particular, we characterize \(C_μ:B^p\to B^p\), answering a question left open by Sun, Ye and Zhou and later noted by Tang, and obtain the corresponding essential norm estimate. For each \(p>2\), we also construct a symbol \(F_η\in H^\infty\capλ^p_{1/p}\) for which \(\mathcal R_{(η)}\) is not bounded on \(H^p\).