It is shown that every echelon space λ∞(A), with A an arbitrary Köthe 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 Fréchet 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.
Positivity – Springer Journals
Published: Oct 13, 2006
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