Positive Compact Operators on Banach Lattices: Some Loose Ends

Abstract

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 theorem [7] telling us that if X andY 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 [13] the author proved that the conclusion also holds if either X or Y is atomic with an order continuous norm and in [14] that these are the only three cases where the conclusion holds. In [1] Aliprantis and Burkinshaw showed that if either X or Z (or both) has an order continuous norm then if S1;T12 L.X;Y/,S2, T22 L.Y;Z/ ,0 6 Si 6 Ti.iD 1; 2/ andT1 andT2 are compact then the productS2S1 is compact. There are certainly triples .X;Y;Z/for which this conclusion holds without X or Z 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 [1]. A particular consequence of Aliprantis and Burkinshaw’s result is obtained by taking XD Y .I fS;T2 L.X/ ,0 6 S6 T and T is compact then S 2 must be compact provided either X or X has an order continuous norm. We show below that ifX is assumed to be Dedekind -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 -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.