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

Let D denote the unit disc in C. We define the generalized Ces\`aro operator as follows Cω(f)(z)=∫₀¹ f(tz)(1/z∫₀ᶻ Bωₜ(u)\,du)\,ω(t)dt, where ω_ζ\ζ are the reproducing kernels of the Bergman space A²_ω induced by a radial weight ω in the unit disc D. We study the action of the operator Cω on weighted Hardy spaces of analytic functions Hγ, γ >0 and on general weighted Bergman spaces A²μ.