Positive Compact Operators on Banach Lattices: Some Loose Ends

Positive Compact Operators on Banach Lattices: Some Loose Ends 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 [7] 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 [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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Positive Compact Operators on Banach Lattices: Some Loose Ends

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Publisher
Kluwer Academic Publishers
Copyright
Copyright © 2000 by Kluwer Academic Publishers
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009838931430
Publisher site
See Article on Publisher Site

Abstract

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 [7] 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 [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

Journal

PositivitySpringer Journals

Published: Oct 16, 2004

References

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