Generalized Ces\`aro operator acting on Hilbert spaces of analytic functions

Let $\mathbb{D}$ denote the unit disc in $\mathbb{C}$. We define the generalized Ces\`aro operator as follows $$C_{\omega}(f)(z)=\int_0^1 f(tz)\left(\frac{1}{z}\int_0^z B^{\omega}_t(u)\,du\right)\,\omega(t)dt,$$ where $\{B^{\omega}_\zeta\}_{\zeta\in\mathbb{D}}$ are the reproducing kernels of the Bergman space $A^2_\omega$ induced by a radial weight $\omega$ in the unit disc $\mathbb{D}$. We study the action of the operator $C_{\omega}$ on weighted Hardy spaces of analytic functions $\mathcal{H}_{\gamma}$, $\gamma >0$ and on general weighted Bergman spaces $A^2_{\mu}$.