Generalized Cesàro operator acting on Hilbert spaces of analytic functions

Let $\mathbb{D}$ denote the unit disc in $\mathbb{C}$. We define the generalized Cesàro operator as follows $$C_{\omega }(f)(z)={\int }_0^{1}f(tz)(\frac{1}{z}{\int }_0^z{B}_t^{\omega }(u)du)\omega (t)dt,$$ where $\{B_{\zeta }^{\omega }{\}}_{\zeta \in \mathbb{D}}$ are the reproducing kernels of the Bergman space $A_{\omega }^2$ 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 $A_{\mu }^2$.