×

zbMATH — the first resource for mathematics

Converses for the Dodds-Fremlin and Kalton-Saab theorems. (English) Zbl 0872.47018
The theorems mentioned in the title provide conditions for Banach lattices \(E\) and \(F\) so that a positive operator dominated by a compact operator in the case of Dodds-Fremlin or by a Dunford-Pettis operator in the case of Kalton-Saab will also have the same property. The author provides a converse to both of these theorems. Specifically, several conditions on the Banach lattice are presented that imply the dominated positive operator inherits the same property and conversely, if each dominated positive operator inherits the property then the Banach lattices \(E\) and \(F\) satisfy one of the conditions. For the compact operator the conditions are
(1) both \(E'\) and \(F\) have order continuous norm,
(2) \(F\) is atomic with order continuous norm or
(3) \(E'\) is atomic with order continuous norm.
For the Dunford-Pettis operator the conditions are
(1) \(F\) has an order continuous norm or
(2) the lattice operators on \(E\) are weakly sequentially continuous.

MSC:
47B65 Positive linear operators and order-bounded operators
46B42 Banach lattices
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Dunford, Linear operators (1976)
[2] DOI: 10.1007/BF02760610 · Zbl 0438.47042 · doi:10.1007/BF02760610
[3] Aliprantis, Positive operators (1985)
[4] Abeamovich, Mat. Zametki 14 pp 723– (1973)
[5] DOI: 10.1007/BF01344465 · Zbl 0273.46006 · doi:10.1007/BF01344465
[6] Wickstead, Bull. Pol. Acad. Sci.
[7] DOI: 10.1007/BF02063211 · Zbl 0153.44001 · doi:10.1007/BF02063211
[8] Meyer-Nieberg, Banach lattices (1991) · doi:10.1007/978-3-642-76724-1
[9] Kalton, Illinois J. Math. 29 pp 382– (1985)
[10] DOI: 10.1093/qmath/32.2.239 · Zbl 0431.47024 · doi:10.1093/qmath/32.2.239
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.