The Cesàro operator on weighted Bergman Fréchet and (LB)-spaces of analytic functions

Abstract

The spectrum of the Cesàro operator $\mathsf{C}$ is determined on the spaces which arises as intersections $A_{\alpha +}^p$ (resp. unions $A_{\alpha -}^p$) of Bergman spaces $A_{\alpha }^p$ of order $1<p<\infty$ induced by standard radial weights $(1-|z|{)}^{\alpha }$, for $0<\alpha <\infty$. We treat them as reduced projective limits (resp. inductive limits) of weighted Bergman spaces $A_{\alpha }^p$, with respect to $\alpha$. Proving that these spaces admit the monomials as a Schauder basis paves the way for using Grothendieck-Pietsch criterion to deduce that we end up with a non-nuclear Fréchet-Schwartz space (resp. a non-nuclear (DFS)-space). We show that $\mathsf{C}$ is always continuous, while it fails to be compact or to have bounded inverse on $A_{\alpha +}^p$ and $A_{\alpha -}^p$.

arXiv comment15 pages