The Ces\`aro operator on weighted Bergman Fr\'echet and (LB)-spaces of analytic functions

Abstract

The spectrum of the Ces\`aro operator $\mathsf{C}$ is determined on the spaces which arises as intersections $A^p_{\alpha +}$ (resp. unions $A^p_{\alpha -}$) 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 $A^p_\alpha$, 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 Fr\'echet-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 $A^p_{\alpha +}$ and $A^p_{\alpha -}$.