Power type asymptotically uniformly smooth and asymptotically uniformly flat norms

Abstract

We provide a short characterization of p-asymptotic uniform smoothability and asymptotic uniform flatenability of operators and of Banach spaces. We use these characterizations to show that many asymptotic uniform smoothness properties pass to injective tensor products of operators and of Banach spaces. In particular, we prove that the injective tensor product of two asymptotically uniformly smooth Banach spaces is asymptotically uniformly smooth. We prove that for 1 < p < ∞, the class of p-asymptotically uniformly smoothable operators can be endowed with an ideal norm making this class a Banach ideal. We also prove that the class of asymptotically uniformly flattenable operators can be endowed with an ideal norm making this class a Banach ideal.

On Asymptotic Uniform Smoothness and Nonlinear Geometry of Banach Spaces

“…In this subsection, we introduce those notions and prove the necessary results so we can obtain Theorem 1.4. Similar properties have already appeared in the literature, and we refer the interested reader to [11] and [12] for a detailed discussion about transfinite asymptotic properties and their relation with the Szlenk index.…”

Operator ideals and three-space properties of asymptotic ideal seminorms

“…(In the case where A is the identity operator it was proved in [12]). Indeed, if 1 < r < p ′ , then A is r-AUS-able, whence there exists a constant C such that every infinite, weakly null tree in B X has a branch whose images under A are C-dominated by the ℓ r basis [9]. This gives that sup n α r,n (A) C and A has asymptotic basic type r. Conversely, by [7], there exists a universal constant c < 6 such that if Sz(A * B Y * , cε) > n, then there exists a weakly null tree (x t ) t∈D n ⊂ B X such that for every t ∈ D n and every convex combination x of (x s : ∅ < s t) we have Ax ε.…”

“…However, it does not determine whether that supremum is attained. In [9], it was shown that this is determined by the domination of branches of infinite weakly null trees (that is, weakly null trees indexed by D <ω rather than D n ), and that the class T p of p-AUS-able operators can be made into a Banach ideal with a natural norm. One can see that for any 1 < p ∞, the condition sup n α p,n (A) < ∞ does not guarantee that A is p-AUS-able.…”

Prescribed Szlenk index of separable Banach spaces

“…In the terminology introduced in [9] it means that Z satisfies ℓ 2 upper tree estimates. It then follows from Theorem 1.1 in [9] that Z admits an equivalent norm which is 2-AUS.…”