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 μ on the unit disc D={z∈C:|z|<1}, we consider the Cesàro-type operator Cμ defined on the space Hol(D) of all analytic functions in D as follows:If f∈Hol(D), f(z)=∑n=0∞anzn (z∈D), then Cμ(f)(z)=∑n=0∞μn(∑k=0nak)zn, (z∈D), where, for n≥0, μn denotes the n-th moment of the measure μ, that is, μn=∫Dwndμ(w).We study the action of the operators Cμ on some Hilbert spaces of analytic function in D, namely, the Hardy space H2 and the weighted Bergman spaces Aα2 (α>−1). Among other results, we prove that, if we set Fμ(z)=∑n=0∞μnzn (z∈D), then Cμ is bounded on H2 or on Aα2 if and only if Fμ belongs to the mean Lipschitz space Λ1/22. We prove also that Cμ is a Hilbert-Schmidt operator on H2 if and only if Fμ belongs to the Dirichlet space D, and that Cμ is a Hilbert-Schmidt operator on Aα2 if and only if Fμ belongs to the Dirichlet-type space D−1−α2.