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

TL;DR

This paper investigates the spectrum of the Cesàro operator on certain weighted Bergman spaces formed as limits, showing it is always continuous but not compact or invertible, and characterizes the spaces as non-nuclear Fréchet or (DFS)-spaces.

Contribution

It determines the spectrum of the Cesàro operator on intersection and union spaces of weighted Bergman spaces and characterizes these spaces as non-nuclear Fréchet or (DFS)-spaces.

Findings

Cesàro operator is always continuous on these spaces.

The operator is not compact and lacks a bounded inverse.

The spaces are non-nuclear Fréchet-Schwartz or (DFS)-spaces.

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 $A^p_{\alpha -}$.