On strongly lp-summing m-linear operators

Abstract

We introduce and study a new concept of strongly lp-summing m-linear operators in the category of operator spaces. We give some characterizations of this notion such as the Pietsch domination theorem and we show that an m-linear operator is strongly lp-summing if and only if its adjoint is lp-summing. 1. Introduction. The development of the theory of polynomials and multilinear operators can be divided into two periods. The first starts in the thirties of the last century, essentially motivated externally through holomorphic and differential functions on infinite-dimensional spaces. The second begins in the eighties, mainly due to Pietsch (Pie83) where the idea to generalize the theory of ideals to the multilinear setting appears. Motived by the importance of this theory, several authors have developed and studied many concepts relating to summability; see (Ale85, AM89, Dia03, Mat96, Mat03, MT99, Sch91) among so many others. In this note we introduce a new concept concerning summability of multilinear operators. The concept of strongly p-summing linear operators (1 ≤ p < ∞) was introduced by J. S. Cohen (Coh73) in order to obtain a characterization of the conjugates of absolutely p ∗ -summing linear operators. In (AM07), we have generalized this concept to the multilinear case. It is natural to try to develop the same concept in the non-commutative case. In the present work, we introduce a new notion of summability for multilinear operators, which we call strongly lp-summing m-linear oper- ators. Using this notion we prove some properties of multilinear opera- tors in the non-commutative case. Our motivation is that the adjoint of a strongly lp-summing m-linear operator is an lp-summing operator as studied in (Mez02).