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

Given a complex Borel measure mu on the unit disc D={z. C:|z| < 1}, we consider the Cesaro-type operator C mu defined on the space Hol(D) of all analytic functions in Das follows: If f. Hol(D), f(z) = 8 n=0anzn(z. D), then C mu(f)(z) = 8 n=0 mu n nk= 0ak zn, (z. D), where, for n = 0, mu ndenotes the n-th moment of the measure mu, that is, mu n= Dwnd mu(w). We study the action of the operators C mu on some Hilbert spaces of analytic function in D, namely, the Hardy space H2and the weighted Bergman spaces A2a( a >-1). Among other results, we prove that, if we set F mu(z) = 8 n=0 mu nzn(z. D), then C mu is bounded on H2or on A2aif and only if F mu belongs to the mean Lipschitz space.21/2. We prove also that C mu is a Hilbert-Schmidt operator on H2if and only if F mu belongs to the Dirichlet space D, and that C mu is a Hilbert-Schmidt operator on A2aif and only if F mu belongs to the Dirichlet-type space D2- 1-a. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).