Generalized Cesàro operator acting on Hilbert spaces of analytic functions

Abstract

Let D denote the unit disc in C. We define the generalized Cesàro operator as follows: Cω( f )(z) = 1 0 f (t z) 1 z z 0 Bω t (u) du

ω(t)dt, where {Bω ζ }ζ∈D 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 A2 μ