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

Abstract

The spectrum of the Ces?ro operator C is determined on the spaces which arises as intersections Ap ?+ (resp. unions Ap ?-) of Bergman spaces Ap? of order 1 < p < 1 induced by standard radial weights (1-|z|)?, for 0 < ? < 1. We treat them as reduced projective limits (resp. inductive limits) of weighted Bergman spaces Ap?, with respect to ?. 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?chet-Schwartz space (resp. a non-nuclear (DFS)-space). We show that C is always continuous, while it fails to be compact or to have bounded inverse on Ap ?+ and Ap ?-.