Schauder Decompositions and the Grothendieck and Dunford-Pettis Properties in Köthe Echelon Spaces of Infinite Order

Abstract

It is shown that every echelon space λ∞(A), with A an arbitrary Kothe matrix, is a Grothendieck space with the Dunford-Pettis property. Since λ∞(A) is Montel if and only if it coincides with λ0(A), this identifies an extensive class of non-normable, non-Montel Frechet spaces having these two properties. Even though the canonical unit vectors in λ∞(A) fail to form an unconditional basis whenever λ∞(A) ≠ λ0(A), it is shown, nevertheless, that in this case λ∞(A) still admits unconditional Schauder decompositions (provided it satisfies the density condition). This is in complete contrast to the Banach space setting, where Schauder decompositions never exist. Consequences for spectral measures are also given.