The Cesàro Operator in Growth Banach Spaces of Analytic Functions

The Ces`aro operator $C$, when acting in the classical growth Banach spaces $A^{-gamma}$ and $A^{-gamma}_0$ , for $gamma>0$, of analytic functions on $D$, 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 $C$ acting in these spaces. In addition, we determine the largest Banach space of analytic functions on $D$ which $C$ maps into $A^{-gamma}$ (resp. into $A^{-gamma}_0$); this optimal domain space always contains $A^{-gamma}$ (resp. $A^{-gamma}_0$ ) as a proper subspace.