Positivity 10 (2006), 391–407
© 2006 Birkh
auser Verlag Basel/Switzerland
1385-1292/020391-17, published online April 26, 2006
A Dodds–Fremlin Property for Asplund and
COENRAAD C.A. LABUSCHAGNE
School of Mathematics, University of the Witwatersrand, Johannesburg, Private Bag X3,
2050 WITS, South Africa. E-mail:email@example.com
Received 23 September 2003; accepted 24 December 2004
Abstract. Let E and F be Banach lattices and let S, T : E → F be positive operators
such that 0
T . It is shown that if T is a Radon–Nikod
ym operator, F has order
continuous norm and E and F both have (Schaefer’s) property (P), then S is a Radon–
ym operator; also, if T is an Asplund operator, E
has order continuous norm and
E has property (P), then S is an Asplund operator.
Mathematics Subject Classiﬁcation 2000: Primary 47B65; Secondary 46B22
Key words: Radon–Nikod
ym operator, Asplund operator, Banach lattice
ym property has been studied by many authors (cf. [2, 7,
11, 12, 23, 24]), due to its importance in the geometry of Banach spaces.
Banach spaces for which the continuous dual space has the Radon–Nik-
ym property, are known as an Asplund space (cf. [12 , 23 , 24]).
The notion of a Radon–Nikod
ym operator between Banach spaces can be
traced back to Reinov (cf. ) and Linde (cf. ). The notion of an Aspl-
und operator between Banach spaces can be found in [12 , 23 , 24].
The aim of this paper is to consider order properties of Radon–Nik-
ym and Asplund operators acting between Banach lattices.
Let E and F be Banach lattices such that E
and F both have order
continuous norm and let S, T : E → F be positive operators such that
T . The Dodds–Fremlin result states that, if T is a compact opera-
tor, then S is also a compact operator (cf. ). Similar in spirit, the Kal-
ton-Saab result states that, if T is a Dunford–Pettis operator, then S is also
a Dunford–Pettis operator (cf. ).
Similar results are derived for Radon–Nikod
ym and Asplund operators.