Cesà ro-type operators acting on Dirichlet spaces

Abstract

If $(\hat{I}\cdot )=\{\hat{I}{\cdot }_n{\}}_{n=0}^{\infty }$ is a sequence of complex numbers, the Cesà ro-type operator ${\mathcal{C}}_{(\hat{I}\cdot )}$ is formally defined in the space of analytic funtions in the unit disc $\mathbb{D}$ as follows: If $f$ is an analytic function in $\mathbb{D}$, $f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ ($z\in \mathbb{D}$), then ${\mathcal{C}}_{(\hat{I}\cdot )}(f)$ is formally defined by $${\mathcal{C}}_{(\hat{I}\cdot )}(f)(z)={\mathcal{C}}_{\{\hat{I}{\cdot }_n\}}(f)(z)=\sum _{n=0}^{\infty }\hat{I}{\cdot }_n\left(\sum _{k=0}^n{a}_k\right)z^n.$$ The operator ${\mathcal{C}}_{(\hat{I}\cdot )}$ is a natural generalization of the Cesà ro operator. For each $\hat{I}\pm \in \mathbb{R}$ we let ${\mathcal{D}}_{\hat{I}}^2\pm$ be the space of functions $f\in\hol(\mathbb D)$ such that $|a_0{|}^2+{\sum }_{n=1}^{\infty }{n}^{1-\hat{I}\pm }|a_n{|}^2<\infty$ where$f(z)={\sum }_{n=0}^{\infty }{a}_n{z}^n$. In this paper we give a complete characterization of the sequences of complex numbers $(\hat{I}\cdot )$ for which the operator ${\mathcal{C}}_{(\hat{I}\cdot )}$ is bounded (compact) from ${\mathcal{D}}_{\hat{I}}^2\pm$ into ${\mathcal{D}}_{\hat{I}}^2$ for any $\hat{I}\pm ,{\hat{I}}^2\in \mathbb{R}$.