Positivity 11 (2007), 77–93
2006 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010077-17, published online October 13, 2006
Schauder Decompositions and the Grothendieck
and Dunford-Pettis Properties in K¨othe
Echelon Spaces of Inﬁnite Order
Jos´e Bonet and Werner J. Ricker
Abstract. It is shown that every echelon space λ
(A), with A an arbitrary
K¨othe matrix, is a Grothendieck space with the Dunford-Pettis property. Since
(A) is Montel if and only if it coincides with λ
(A), this identiﬁes an
extensive class of non-normable, non-Montel Fr´echet spaces having these two
properties. Even though the canonical unit vectors in λ
(A) fail to form
an unconditional basis whenever λ
(A) = λ
(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. Con-
sequences for spectral measures are also given.
Mathematics Subject Classiﬁcation (2000). Primary 46A45, 46G10, 47B37;
Secondary 46A04, 46A11, 46A35.
Keywords. K¨othe echelon space, Grothendieck space, Dunford-Pettis property,
Schauder decomposition, density condition.
The class of Banach spaces which are Grothendieck spaces with the Dunford-Pettis
property (brieﬂy, GDP-spaces) plays a prominent role in the theory of Banach
spaces and vector measures; see Chapter VI of , especially the Notes and Re-
marks, and , for example. Well known examples of GDP-spaces include L
(D), injective Banach spaces (eg.
) and certain C(K) spaces. A sequence
of continuous projections on a Banach space X is called a (weak) Schau-
der decomposition if:
for all m, n ∈ N,
x} converges (weakly) to x for each x ∈ X,and
for m = n.