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 ?-.

The Cesaro operator

“…As one can imagine, the Cesàro operator has been studied for many other spaces of analytic functions on D and it is impossible to survey them all. We direct the reader to the papers [18,28,38] which contain useful references to the large literature on this subject.…”

“…There has been renewed interest in the classical Cesàro operator and its generalizations as of late [1,17,28,32,38,44] so perhaps it is a good time to put together an extended survey of what is currently known about this operator. Most of us in analysis know the name Cesàro from his summability method for infinite series and the important role this plays in summing the Fourier series of an integrable function.…”