The Cesaro operator in growth Banach spaces of analytic functions

The Cesaro operator $\mathsf{C}$, when acting in the classical growth Banach spaces $A^{-γ}$ and $A_0^{-γ}$, for $γ> 0$, 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 $A^{-γ}$ (resp. into $A_0^{-γ}$); this optimal domain space always contains $A^{-γ}$ (resp. $A_0^{-γ}$) as a proper subspace.

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