Ideals of Multilinear Forms - a Limit Order Approach

In 1983, A. Pietsch [20] presented his “designs of a theory” for ideals of multilinear forms. This work provided a general framework from which different lines of investigation developed. Some ideals of multilinear forms appeared as the multilinear natural extension of ideals of linear operators (e.g., nuclear and integral multilinear forms). However, it is not always clear what the multilinear analogous of a linear operators ideal should be. For example, the ideal of absolutely r-summing operators lead to the development of many ideals of multilinear forms: absolutely r-summing, r-dominated, multiple r-summing, etc. Also, some ideals of multilinear forms do not share this linear origin, but appeared by their own interest (sometimes, in relation with ideals of polynomials and holomorphy types). In any case, the theory of ideals of multilinear forms allows to deal with all the different situations in a unified way. In the linear theory, the calculus of limit orders proved a useful tool, specially to compare different ideals of linear operators. In [5], we introduced the concept of limit order for multilinear forms. As an application, we could show that some properties of bilinear ideals were no longer valid for the n-linear case (n ≥ 3). In this article, we aim to develop a bit further the general theory of limit orders for multilinear forms. Even though some of our proofs are simple adaptations of the linear analogous results, we chose to present them here. Our motivation was to give a self-contained treatment of the subject, since the linear versions of these results come from different sources and with different notations. In the first section, we recall the definitions of limit order and present some general properties. The second section deals with maximal and adjoint ideals of multilinear forms and their corresponding limit orders.