Cesàro-type operators acting on Dirichlet spaces

If $(\eta )=\{{\eta }_n{\}}_{n=0}^{\mathrm{\infty }}$ is a sequence of complex numbers, the Cesàro-type operator ${\mathcal{C}}_{(\eta )}$ is formally defined in the space of analytic funtions in the unit disc $\mathbb{D}$ as follows: If $f$ is an analytic function in $\mathbb{D}$, $f(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ ($z\in \mathbb{D}$), then ${\mathcal{C}}_{(\eta )}(f)$ is formally defined by $${\mathcal{C}}_{(\eta )}(f)(z)={\mathcal{C}}_{\{{\eta }_n\}}(f)(z)=\sum _{n=0}^{\mathrm{\infty }}{\eta }_n\left(\sum _{k=0}^n{a}_k\right)z^n\mathrm{.}$$ The operator ${\mathcal{C}}_{(\eta )}$ is a natural generalization of the Cesàro operator. For each $\alpha \in \mathbb{R}$ we let ${\mathcal{D}}_{\alpha }^2$ be the space of functions $f\in \text{hol}(\mathbb{D})$ such that $\mathrm{\mid }{a}_0{\mathrm{\mid }}^2+{\sum }_{n=1}^{\mathrm{\infty }}{n}^{1-\alpha }\mathrm{\mid }{a}_n{\mathrm{\mid }}^2<\mathrm{\infty }$ where$f(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n$. In this paper we give a complete characterization of the sequences of complex numbers $(\eta )$ for which the operator ${\mathcal{C}}_{(\eta )}$ is bounded (compact) from ${\mathcal{D}}_{\alpha }^2$ into ${\mathcal{D}}_{\beta }^2$ for any $\alpha ,\beta \in \mathbb{R}$.