A tensor norm preserving unconditionality for $\mathcal{L}_p$-spaces

Abstract

Abstract. We show that, for each n ∈ N, there is an n-tensor norm α (in the sense of Grothendieck) with the surprising property that the α-tensor product α (Y 1 , . . . , Y n ) has local unconditional structure for each choice of n arbitrary L p j -spaces Y j . In fact, α is the tensor norm associated to the ideal of multiple 1-summing n-linear forms on Banach spaces.

Almost summing and multiple almost summing multilinear operators on l p spaces

“…In the next result, we use the Maurey-Rosenthal factorization theorem to give a characterisation of almost summing and multiple almost summing operators on a cartesian product of l p spaces, see also [8, page 4] and the proof of Lemma 4.5 in [5].…”

On the Representation of Multi-Ideals by Tensor Norms

“…2 REMARK 2.15. As proved by Defant and Pérez-García [12], the tensor norm that represents the ideal L m( p) of multiple p-summing multilinear mappings is the first example of a tensor norm that preserves unconditionality for L p -spaces. REMARK 2.16.…”

Ideals of Multilinear Forms – a Limit Order Approach

“…Defant and D. Pérez-García used multiple 1-summing in [6] to show that its associated tensor norm preserves unconditionality. Some other properties of the norm were also proved and applied in [6, Section 6] to compute the limit order for bilinear multiple 1-summing operators.…”

“…(see [18,Theorem 4.2], also [6,Theorem 5.1]). On the other hand, by [6,Theorem 5.2], if X has 1-unconditional basis, has cotype 2 and dimX = N , then for S ∈ L( n X) we have…”