PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 100, Number 2. lune 1987 POSITIVE p-SUMMING OPERATORS ON LP-SPACES OSCAR BLASCO ABSTRACT. It is shown that for any Banach space B every positive p-sum- ming operator from Lp (ß) in B, 1/p + 1/p' = 1, is also cone absolutely summing. We also prove here that a necessary and sufficient condition that B has the Radon-Nikodym property is that every positive p-summing operator T: Lp (p) —> B is representable by a function / in Lp(p, B). 1. Introduction. In this paper we shall be concerned with a weaker con- cept than a p-absolutely summing operator [5] and stronger than a p-concave one [4]. This concept makes sense when we are dealing with operators T in L{X,B), with A a Banach lattice. An operator which maps positive sequences {xn} with suP||illx*<i ^ K£'2:n)lP < °o in sequences {Txx} such that Y^ ||Tin||p < oo will be called a positive p-summing operator. In case p = 1, such operators are called order summing [2] or cone absolutely summing operators [7] and for 1 < p < oo they have already been considered by the author in [1]. Here we shall investigate the space of positive p-summing operators for spaces C(fi) and Lr{p) (1 < r < oo). We shall find that for any Banach space B the positive p-summing operators from LP {p) in B denoted by AP{LP {p),B), with 1/p + 1/p' = 1, are also cone absolutely summing ones. We shall obtain a necessary and sufficient condition such that B has the Radon- Nikodym property in terms of these operators. This condition can be written as follows: Ap{Lp'{p),B) = L"{p,B) for some p, 1 < p < oo. 2. Definitions and preliminary results. Throughout this paper A will de- note a Banach lattice, B a Banach space, and L{X, B) the space of bounded oper- ators from A into B. We shall write p' for the number such that 1/p + 1/p' = 1. DEFINITION 1. Let 1 < p < oo. An operator T: X —> B is said to be positive p- summing if there exists a constant C > 0 such that for every xi,X2, ■ ■ ■ ,xn, positive elements in A, we have / n \1/P / n \ !/P (1) £lMp <c SUp MTlvC'1')!" • We shall denote by AP(A, B) the space of positive p-summing operators. This space becomes a Banach space with the norm || ||A given by the infimum of the constants verifying (1). Received by the editors March 14, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 47B10, 46B22. Key words and phrases, p-summing operator, cone absolutely summing, Radon-Nikodym prop- erty. ©1987 American Mathematical Society 0002-9939/87 Î1.00 + $.25 per page 275 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use