The Cesàro Operator and Unconditional Taylor Series in Hardy Spaces

Abstract

We introduce the spaces \({H^{p}_{uc}}\) consisting of all functions in the Hardy space \({H^p, 1 < p < \infty}\), whose Taylor series are unconditionally convergent and analyze the action of the Cesaro operator in these spaces. There is a related class of Banach sequence spaces \({N^p, 1 < p < \infty}\), arising from harmonic analysis, in which the discrete Cesaro operator acts. The classical majorant property (due to Hardy and Littlewood) provides a means to transfer various results about the Cesaro operator in Np (e.g. continuity, spectrum, etc.) to those for the corresponding Cesaro operator acting in \({H^{p}_{uc}. \,\,{\rm For}\,\, p \neq 2}\), the space \({H^{p}_{uc}}\) is rather different to the classical space \({H^{p}}\). The spaces \({N^{p}}\) also exhibit a remarkable stability property under averaging, akin to that established by Bennett for \({\ell^{p}}\).