Positivity 2: 171–191, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
Some Applications of Rademacher Sequences in
and A.W. WICKSTEAD
Department of Applied Mathematics, Southwest Jiaotong University, Chengdu Sichuan 610031,
Department of Pure Mathematics, The Queen’s University of Belfast, Belfast, BT7 1NN,
Northern Ireland, UK
(Received: 29 July 1997; accepted in revised form: 18 March 1998)
Abstract. We give several applications of Rademacher sequences in abstract Banach lattices. We
characterise those Banach lattices with an atomic dual in terms of weak
We give an alternative treatment of results of Rosenthal, generalising a classical result of Pitt, on the
compactness of operators from L
. Finally we generalise earlier work of ours by showing
that, amongst Banach lattices F with an order continuous norm, those having the property that the
linear span of the positive compact operators from E into F is complete under the regular norm for
all Banach lattices E are precisely the atomic lattices.
Mathematics Subject Classiﬁcations (1991): 46B42, 47B65
Key words: Banach lattices, Rademacher sequences, atomicity, compact operators
It is well-known that the standard Rademacher functions on [0, 1] have many
important properties and applications. In particular, they play a key role in the
study of L
[0, 1]. In a general Banach lattice setting their use dates back at least to
[A] where a number of applications to the study of weak compactness in Banach
lattices are given. In this note we give several more applications of their use in a
Banach lattice setting.
In §3 we give a characterisation of those Banach lattices with an atomic dual
in terms of weak
sequential convergence. Section four contains an alternative
treatment of results of Rosenthal, which in turn generalised a classical result of
Pitt, on the compactness of operators from L
The ﬁnal section is devoted to the topic of when K
(E, F ), the linear span of
the positive compact operators from E into F, is complete for the regular norm.
The main result of  was that K
[0, 1]) is not complete under the
regular norm. Here we extend the results of  by showing that, amongst Banach
lattices with an order continuous norm, those having the property that K
(E, F ) is
always complete under the regular norm are precisely the atomic lattices. We also
deduce the dual result and some other related results.
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