Ces\`aro operators associated with Borel measures acting on weighted spaces of holomorphic functions with sup-norm

Abstract

Let $\mu$ be a positive finite Borel measure on $[0,1).$ Ces\`aro-type operators $C_{\mu}$ when acting on weighted spaces of holomorphic functions are investigated. In the case of bounded holomorphic functions on the unit disc we prove that $C_\mu$ is continuous if and only if it is compact. In the case of weighted Banach spaces of holomorphic function defined by general weights, we give sufficient and necessary conditions for the continuity and compactness. For standard weights, we characterize the continuity and compactness on classical growth Banach spaces of holomorphic functions. We also study the point spectrum and the spectrum of $C_\mu$ on the space of holomorphic functions on the disc, on the space of bounded holomorphic functions on the disc, and on the classical growth Banach spaces of holomorphic functions. All characterizations are given in terms of the sequence of moments $(\mu_n)_{n\in\N_0}$. The continuity, compactness and spectrum of $C_\mu$ acting on Fr\'echet and (LB) Korenblum type spaces are also considered.