The Cesàro Operator in Growth Banach Spaces of Analytic Functions

The Cesàro operator C, when acting in the classical growth Banach spaces A - γ and A 0 - γ , for γ > 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 - γ (resp. into A 0 - γ ); this optimal domain space always contains A - γ (resp. A 0 - γ ) as a proper subspace.