Positivity 11 (2007), 589–607
2007 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040589-19, published online September 26, 2007
Ideals of Multilinear Forms – a Limit
Daniel Carando, Ver´onica Dimant and Pablo Sevilla-Peris
Abstract. A general theory of limit orders for ideals of multilinear forms is
developed. We relate the limit order of an ideal to those of its maximal hull
and its adjoint ideal. We study the limit orders of the ideals of dominated
and multiple summing multilinear forms. Finally, estimates of the diagonal of
a (non-necessarily diagonal) multilinear form are presented, in terms of the
limit order of the ideals to which it belongs.
Mathematics Subject Classiﬁcation (2000). 46G25, 46A45.
Keywords. Ideals of multilinear mappings, limit orders, sequence spaces.
In 1983, A. Pietsch  presented his “designs of a theory” for ideals of multilinear
forms. His work provided a general framework from which different lines of inves-
tigation developed. Some ideals of multilinear forms appeared as the multilinear
natural extension of ideals of linear operators (e.g., nuclear and integral multilin-
ear 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. Independently of their linear
origin, many ideals of multilinear forms were studied by their own interest and
also in relation to 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, spe-
cially to compare different ideals of linear operators. The concept of limit order
The third author was supported by the MCYT and FEDER Project BFM2002-01423 and grant
GV-GRUPOS04/45. The ﬁrst and second authors were supported by CONICET-PIP 5272. The
ﬁrst author was also supported by UBACyT-X108 and ANPCyT-PICT 0315033.