Positivity 4: 313–325, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands. Positive Compact Operators on Banach Lattices: Some Loose Ends A.W. WICKSTEAD Department of Pure Mathematics, The Queen’s University of Belfast, Belfast, BT7 1NN, Northern Ireland 1. Introduction 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 or 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
Positivity – Springer Journals
Published: Oct 16, 2004
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