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:

$$\begin{aligned} C_{\omega }(f)(z)=\int _0^1 f(tz)\left( \frac{1}{z}\int _0^z B^{\omega }_t(u)\,\textrm{d}u\right) \,\omega (t)\textrm{d}t, \end{aligned}$$

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 \(A^{2}_{\mu }\).