Cesàro-type operators acting on Dirichlet spaces

Abstract

If $(η)=\{ η_n\} _{n=0}^\infty $ is a sequence of complex numbers, the Cesàro-type operator $\mathcal C_{(η)}$ 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 $\mathcal C_{(η)}(f)$ is formally defined by $$\mathcal C_{(η)}(f)(z)=\mathcal C_{\{η_n\}}(f)(z)=\sum_{n=0}^\infty η_n\left (\sum_{k=0}^na_k\right )z^n.$$ The operator $\mathcal C_{(η)}$ is a natural generalization of the Cesàro operator. For each $α\in \mathbb R$ we let $\mathcal D^2_α$ be the space of functions $f\in\hol(\mathbb D)$ such that $|a_0|^2+\sum_{n=1}^\infty n^{1-α} |a_n|^2<\infty$ where$f(z)=\sum_{n=0}^\infty a_nz^n$. In this paper we give a complete characterization of the sequences of complex numbers $(η)$ for which the operator $\mathcal C_{(η)}$ is bounded (compact) from $\mathcal D^2_α$ into $\mathcal D^2_β$ for any $α, β\in \mathbb R$.