The Cesàro operator on Korenblum type spaces of analytic functions

The spectrum of the Cesàro operator C, which is always continuous (but never compact) when acting on the classical Korenblum space and other related weighted Fréchet spaces or (LB)-spaces of analytic functions on the open unit disc, is completely determined. It turns out that such spaces are always Schwartz but, with the exception of the Korenblum space, nevr nuclear. Some consequences concerning the mean ergodicity of C are deduced