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Integral Equations and Operator Theory

, Volume 83, Issue 2, pp 179–195 | Cite as

The Cesàro Operator and Unconditional Taylor Series in Hardy Spaces

  • Guillermo P. CurberaEmail author
  • Werner J. Ricker
Article

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 Cesàro 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 Cesàro operator acts. The classical majorant property (due to Hardy and Littlewood) provides a means to transfer various results about the Cesàro operator in N p (e.g. continuity, spectrum, etc.) to those for the corresponding Cesàro 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 \({\ell^{p}}\).

Keywords

Cesàro operator Hardy space Taylor series unconditional convergence majorant property 

Mathematics Subject Classification

Primary 30H10 47B37 42A16 Secondary 47A10 46B45 

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© Springer Basel 2015

Authors and Affiliations

  1. 1.Facultad de Matemáticas, IMUSUniversidad de SevillaSevilleSpain
  2. 2.Math.-Geogr. FakultätKatholische Universität Eichstätt-IngolstadtEichstättGermany

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