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

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