Cesaro-type operators on derivative-type Hilbert spaces of analytic functions: The proof of a conjecture

In this paper, we focus on the boundedness and compactness of the Cesaro-type operators C-mu(f)(z) := & sum;(infinity)(n=0)(integral(D )omega(n)d mu(omega)) (& sum;(n )(k=0)a(k)) z(n), z is an element of D, where mu is a complex Borel measure on the unit disc D, acting on two derivative-type Hilbert spaces of analytic functions defined in D, including the derivative Hardy space S-2 and the weighted Dirichlet space D-alpha(2)(-1 < alpha < infinity). As a by-product, we not only prove a conjecture (recently posed by Galanopoulos-Girela-Merchan) about the sufficient conditions for the compactness of C-mu acting on weighted Bergman space A(alpha)(2)(-1 < alpha < infinity), but also give a complete characterization for the boundedness and compactness of C-mu between different weighted Bergman spaces. At last, we collect some unresolved problems and issues for further study. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.