Converses for the Dodds-Fremlin and Kalton-Saab theorems. (English) Zbl 0872.47018

Wickstead, A. W.

Math. Proc. Camb. Philos. Soc. 120, No. 1, 175-179 (1996).

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.

Reviewer: W.A.Feldman (Fayetteville)

MSC:

47B65	Positive linear operators and order-bounded operators
46B42	Banach lattices

Keywords:

Banach lattices; positive operator dominated by a compact operator; Dunford-Pettis operator

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