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The spectrum of the Ces\`aro operator $\mathsf{C}$ is determined on thespaces which arises as intersections $A^p_{\alpha +}$ (resp. unions$A^p_{\alpha -}$) of Bergman spaces $A_\alpha^p$ of order $1<p<\infty$ inducedby standard radial weights $(1-|z|)^\alpha$, for $0<\alpha<\infty$. We treatthem as reduced projective limits (resp. inductive limits) of weighted Bergmanspaces $A^p_\alpha$, with respect to $\alpha$. Proving that these spaces admitthe monomials as a Schauder basis paves the way for using Grothendieck-Pietschcriterion 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 alwayscontinuous, while it fails to be compact or to have bounded inverse on$A^p_{\alpha +}$ and $A^p_{\alpha -}$.

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