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arXiv · 1609.00812

The Cesaro operator in growth Banach spaces of analytic functions

Angela A. Albanese·José Bonet·Werner J. Ricker
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Abstract

The Cesaro operator $\mathsf{C}$, when acting in the classical growth Banach spaces $A^{-\gamma}$ and $A_0^{-\gamma}$, for $\gamma > 0 $, of analytic functions on $\mathbb{D}$, is investigated. Based on a detailed knowledge of their spectra (due to A. Aleman and A.-M. Persson) we are able to determine the norms of these operators precisely. It is then possible to characterize the mean ergodic and related properties of $\mathsf{C}$ acting in these spaces. In addition, we determine the largest Banach space of analytic functions on $\mathbb{D}$ which $\mathsf{C}$ maps into $A^{-\gamma}$ (resp. into $A_0^{-\gamma}$); this optimal domain space always contains $A^{-\gamma}$ (resp. $A_0^{-\gamma}$) as a proper subspace.

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Angela A. Albanese, José Bonet, Werner J. Ricker. 2016-09-03. The Cesaro operator in growth Banach spaces of analytic functions. https://doi.org/10.1007/s00020-016-2316-z

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