Generalized Cesà ro operator acting on Hilbert spaces of analytic functions

Abstract

Let $\mathbb{D}$ denote the unit disc in $\mathbb{C}$. We define the generalized Cesà ro operator as follows $$C_{\ddot{I}}‰(f)(z)={\int }_0^{1}f(tz)\left(\frac{1}{z}{\int }_0^z{B}^{\ddot{I}}{‰}_t(u)du\right)\ddot{I}‰(t)dt,$$ where $\{B^{\ddot{I}}{‰}_{\hat{I}}\P {\}}_{\hat{I}\P \in \mathbb{D}}$ are the reproducing kernels of the Bergman space $A_{\ddot{I}}^2‰$ induced by a radial weight $\ddot{I}‰$ in the unit disc $\mathbb{D}$. We study the action of the operator $C_{\ddot{I}}‰$ on weighted Hardy spaces of analytic functions ${\mathcal{H}}_{\hat{I}}^3$, ${\hat{I}}^3>0$ and on general weighted Bergman spaces $A_{\hat{I}}^2¼$.