Ces\`{a}ro-type operators acting on Dirichlet spaces

If $(\eta )=\{ \eta_n\} _{n=0}^\infty$ is a sequence of complex numbers, the Ces\`aro-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 $\mathbb D$, $f(z)=\sum_{n=0}^\infty a_nz^n$ ($z\in \mathbb D$), then $\mathcal C_{(\eta )}(f)$ is formally defined by $$\mathcal C_{(\eta )}(f)(z)=\mathcal C_{\{\eta_n\}}(f)(z)=\sum_{n=0}^\infty \eta _n\left (\sum_{k=0}^na_k\right )z^n.$$ The operator $\mathcal C_{(\eta )}$ is a natural generalization of the Ces\`{a}ro operator. For each $\alpha\in \mathbb R$ we let $\mathcal D^2_\alpha$ be the space of functions $f\in\hol(\mathbb D)$ such that $|a_0|^2+\sum_{n=1}^\infty n^{1-\alpha} |a_n|^2<\infty$ where$f(z)=\sum_{n=0}^\infty a_nz^n$. 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 $\mathcal D^2_\alpha$ into $\mathcal D^2_\beta$ for any $\alpha , \beta \in \mathbb R$.