Positivity 4: 313–325, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
Positive Compact Operators on Banach Lattices:
Some Loose Ends
Department of Pure Mathematics, The Queen’s University of Belfast, Belfast, BT7 1NN, Northern
Although by now quite a lot is known about positive compact operators on Banach
lattices and their linear span, there remain a few problems that have not been
resolved – not necessarily because of their difﬁculty but because no-one has yet
addressed them. In this note we will tackle two of these.
Several results are known telling us when positive operators dominated by a
compact operator have to be compact. The earliest was the Dodds– Fremlin the-
orem  telling us that if X
and Y both have an order continuous norm then every
positive operator from X into Y which is dominated by a compact operator must
be compact. In  the author proved that the conclusion also holds if either X
Y is atomic with an order continuous norm and in  that these are the only three
cases where the conclusion holds.
In  Aliprantis and Burkinshaw showed that if either X
or Z (or both) has
an order continuous norm then if S
∈ L(X, Y ), S
∈ L(Y, Z),0 S
(i = 1, 2) and T
are compact then the product S
is compact. There
are certainly triples (X,Y,Z) for which this conclusion holds without X
having an order continuous norm (for example if either Y or Y
is atomic with an
order continuous norm) but allowing Y to range over all Banach latices allows us
to prove a converse. The proof of this converse is based on an example in . A
particular consequence of Aliprantis and Burkinshaw’s result is obtained by taking
X = Y .IfS,T ∈ L(X),0 S T and T is compact then S
must be compact
provided either X or X
has an order continuous norm. We show below that if X
is assumed to be Dedekind σ-complete then this characterises Banach lattices X
for which the norm in either X or X
is order continuous and also that if X is not
Dedekind σ-complete then the characterisation fails.
Apart from the linear span of the positive compact operators, the smaller space
consisting of the closure of the ﬁnite rank operators under the regular norm has
been studied. This behaves, in general, much more nicely in many ways than the
larger space so that is of interest to know when the two coincide. It is relatively
simple to show that this is the case if X is atomic with an order continuous norm.