Cesàro-type operators associated with Borel measures on the unit disc acting on some Hilbert spaces of analytic functions

Abstract

Given a complex Borel measure μon the unit disc D={z∈C:|z| <1}, we consider the Cesàro-type operator Cμdefined on the space Hol(D)of all analytic functions in Das follows: If f∈Hol(D), f(z) = ∞n=0anzn(z∈D), then Cμ(f)(z) = ∞n=0μn nk=0ak zn, (z∈D), where, for n ≥0, μndenotes the n-th moment of the measure μ, that is, μn= Dwndμ(w). We study the action of the operators Cμon some Hilbert spaces of analytic function in D, namely, the Hardy space H2and the weighted Bergman spaces A2α(α >−1). Among other results, we prove that, if we set Fμ(z) = ∞n=0μnzn(z∈D), then Cμis bounded on H2or on A2αif and only if Fμbelongs to the mean Lipschitz space Λ21/2. We prove also that Cμis a Hilbert-Schmidt operator on H2if and only if Fμbelongs to the Dirichlet space D, and that Cμis a Hilbert-Schmidt operator on A2αif and only if Fμbelongs to the Dirichlet-type space D2−1−α.