The Cesaro operator in growth Banach spaces of analytic functions

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.

17 pages