Positivity (2005) 9:249–257 © Springer 2005
On Positive Operators on Some Ordered Banach
4, Rue Antoine Chantin; 75014 Paris, France. E-mail: firstname.lastname@example.org
(Received 28 January 2003; accepted 19 November 2003)
Abstract. In this paper we study to what extent some classical results concerning oper-
ators T ,fromaC(K)-space to a Banach space, or from a Banach space to a L
can be precised, when the Banach spaces involved are ordered (by a normal cone in the
ﬁrst case, by a closed generating proper convex cone in the second case) and when the
operators T are positive.
Mathematics Subject Classiﬁcations. 46B40, 46E30, 47A68, 47B10, 57B65
Key words: C(K) and L
spaces, summing operators, ordered Banach spaces
In this paper we examine to what extent some classical Theorems concerning
operators between Banach spaces can be precised when the Banach spaces
involved are ordered by convex proper cones, and when the operators are
Recall the following Theorem of B. Maurey ( Proposition 3 or 
Proposition 1.4 or  11.14, b) (most of the deﬁnitions will be recalled in
Section 2, in order to keep the paper self-contained):
THEOREM 1. Let B be a Banach space having cotype q 2; for any com-
pact Hausdorff space K, any continuous map T : C(K) → B and any r>q,
then T is r-summing.
In Section 3 we shall establish an analogous result (Theorem 7), assum-
ing that B is ordered by a normal cone and that T is positive; we shall give
applications to conical measures.
Indeed, the proof of our Theorem 7 is quite different of those of The-
orem 1 given in () or in () or in (), but we shall give, in our
Remark 8, a sketch proof of Theorem 1, analogous in some sense to the
one of our Theorem 7.
In Section 4 we shall consider the case of positive operators from
a Banach space B, ordered by a closed generating proper convex cone