Generalized Cesàro operator acting on Hilbert spaces of analytic functions

Abstract

AbstractLet $$\mathbb {D}$$ D denote the unit disc in $$\mathbb {C}$$ C . We define the generalized Cesàro operator as follows: $$\begin{aligned} C_{\omega }(f)(z)=\int _0^1 f(tz)\left( \frac{1}{z}\int _0^z B^{\omega }_t(u)\,\textrm{d}u\right) \,\omega (t)\textrm{d}t, \end{aligned}$$ C ω ( f ) ( z ) = ∫ 0 1 f ( t z ) 1 z ∫ 0 z B t ω ( u ) d u ω ( t ) d t , where $$\{B^{\omega }_\zeta \}_{\zeta \in \mathbb {D}}$$ { B ζ ω } ζ ∈ D are the reproducing kernels of the Bergman space $$A^{2}_{\omega }$$ A ω 2 induced by a radial weight $$\omega $$ ω in the unit disc $$\mathbb {D}$$ D . We study the action of the operator $$C_{\omega }$$ C ω on weighted Hardy spaces of analytic functions $$\mathcal {H}_{\gamma }$$ H γ , $$\gamma >0$$ γ > 0 and on general weighted Bergman spaces $$A^{2}_{\mu }$$ A μ 2 .