Ideals of Multilinear Forms – a Limit Order Approach

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.2000 Mathematics Subject Classification:46G25, 46A45

Unconditionality in Tensor Products and Ideals of Polynomials, Multilinear Forms and Operators

“…Proof. As in the proof of [9,Theorem 3.5] we can see that, if M is a finite dimensional space, then (D n r ) * (M ) = I n r (M ). Let p ∈ {1, 2, ∞}, we have that gl I n r (ℓ m p ) = gl D n r (ℓ m p ) , which we already know by the previous example that this goes to ∞ with m.…”

“…If T is any multilinear form on ℓ p , we denote by D(T ) the multilinear form obtained from T setting to zero all the coefficients outside the diagonal (see [9] for details).…”

Coherent sequences of polynomial ideals on Banach spaces

“…In [5] it is shown that the ideal of r-dominated n-linear forms is maximal and a finitely generated n-tensor norm is presented to which it is dual. Using the same ideas we prove an analogous statement for vector-valued polynomials.…”

Absolutely summing multilinear operators: a panorama

“…Matos' research report [1] are usually quoted as the precursors of the now well-known nonlinear theory of absolutely summing operators. In the last decade this topic of investigation attracted the attention of many authors and various different concepts related to summability of nonlinear operators were introduced; this line of research, besides its intrinsic interests, highlighted abstract questions in the mainstream of the theory of multi-ideals which contributed to the revitalization of the general interest in questions related to ideals of polynomials and multilinear operators (see [9,10,15,27,28]). This paper has a twofold purpose: to summarize/organize some information constructed in the last years concerning the different multilinear generalizations of absolutely summing operators; and to sketch a research project directed to the investigation of the existence of multilinear ideals (related to the ideal of absolutely summing operators) satisfying a list of properties which we consider natural.…”

“…Matos' research report [1] are usually quoted as the precursors of the now well-known nonlinear theory of absolutely summing operators. In the last decade this topic of investigation attracted the attention of many authors and various different concepts related to summability of nonlinear operators were introduced; this line of research, besides its intrinsic interests, highlighted abstract questions in the mainstream of the theory of multi-ideals which contributed to the revitalization of the general interest in questions related to ideals of polynomials and multilinear operators (see [9,10,15,27,28]).…”