Let \(\textrm{Hol}(\mathbb D)\) be the space of all analytic functions in the unit disc \(\mathbb D\,=\,\{ z\in \mathbb C : \vert z\vert <1\}\) . For each \(\alpha \in \mathbb R\) we let \(\mathcal D^2_\alpha \) be the space of functions \(f\in \textrm{Hol}(\mathbb D)\) such that \(|a_0|^2+\sum _{n=1}^\infty n^{1-\alpha } |a_n|^2<\infty \) where \(f(z)=\sum _{n=0}^\infty a_nz^n\) .
If \((\eta )=\{ \eta _n\}_{n=0}^\infty \) is a sequence of complex numbers and \(f\in \textrm{Hol}(\mathbb D)\) , \(f(z)=\sum _{n=0}^\infty a_nz^n\) ( \(z\in \mathbb D\) ), \(\mathcal C_{(\eta )}(f)=\mathcal C_{(\{\eta _n\})}(f)\) is formally defined by \( \mathcal C_{(\eta )}(f)=\mathcal C_{\{\eta _n\}}(f)(z)=\sum _{n=0}^\infty \eta _n\left( \sum _{k=0}^na_k\right) z^n. \) The operator \(\mathcal C_{(\eta )}\) is a natural generalization of the Cesàro operator. If \(\mu \) is a complex Borel measure on \(\mathbb D\) and, for \(n=0, 1, 2, \dots \) , \(\mu _n=\int _{\mathbb D}w^nd\mu (w)\) , the operator \(\mathcal C_{\{\mu _n\} }\) is denoted by \(\mathcal C_\mu \) .
In a recent paper [J. Funct. Anal. 288 (2025), no. 6, Paper No. 110813], Lin and Xie have studied the question of characterizing the complex Borel measures \(\mu \) on \(\mathbb D\) for which the operator \(\mathcal C_\mu \) is bounded (compact) from \(\mathcal D^2_\alpha \) into \(\mathcal D^2_\beta \) for \(\alpha ,\beta >-1\) , corresponding to the spaces of analytic functions \(f\in \textrm{Hol}(\mathbb D)\) such that \(f'\) belongs to the Bergman spaces \(A^2_\alpha \) and \(A^2_\beta \) respectively, and also from \(\mathcal D^2_{-1}=S^2\) , corresponding to the space of analytic functions \(f\in \textrm{Hol}(\mathbb D)\) such that \(f'\in H^2\) , into itself. They have solved the question for \(\alpha >1\) . For the other values of \(\alpha \) they have given a number of conditions which are either necessary or sufficient. They have also obtained a number of conditions which are either necessary or sufficient for the boundedness (compactness) of \(\mathcal C_\mu \) from \(S^2\) into itself.
In this paper we give a complete characterization of the sequences of complex numbers \((\eta _n )\) for which the operator \(\mathcal C_{(\eta )}\) is bounded (compact) from \(\mathcal D^2_\alpha \) into \(\mathcal D^2_\beta \) for \(\alpha , \beta \in \mathbb R\) .
