[Paper Review] Three and a half asymptotic properties

Ryan M. Causey|arXiv (Cornell University)|May 7, 2018

Advanced Banach Space Theory5 references3 citations

TL;DR

This paper introduces and analyzes transfinite asymptotic properties—smoothability, type, equal norm type, and intermediate asymptotic smoothability—for operators between Banach spaces, establishing a strict hierarchy of operator ideals indexed by ordinals and exponents. It proves that for any ordinal ξ and 1 < p < ∞, the classes 𝒟ξ ⊊ 𝒯ξ,∞ ⊊ 𝒜ξ,∞ = 𝒩ξ,∞ ⊊ 𝒫ξ,∞ ⊊ 𝒯ξ,p ⊊ 𝒜ξ,p ⊊ 𝒩ξ,p ⊊ 𝒫ξ,p ⊊ 𝒟ξ+1, with strict inclusions when ξ has countable cofinality, and provides a characterization of equal norm type via equivalent norms with controlled asymptotic smoothness modulus.

ABSTRACT

We define and discuss transfinite asymptotic notions of smoothability, type, and equal norm type. We prove distinctness of these notions for a proper class of ordinals and that each class is an ideal. We also extend some results of Godefroy, Kalton, and Lancien to operators and ordinals greater than zero regarding the equivalence of equal norm asymptotic type and uniform renormings with power type smoothness. Finally, we discuss an extension of a non-linear result for quasi-reflexive, asymptotically $p$-smoothable Banach spaces to quasi-reflexive Banach spaces with asymptotic equal norm type $p$.

Motivation & Objective

• To define and analyze transfinite asymptotic analogues of smoothability, type, and equal norm type for operators between Banach spaces.
• To establish a strict, hierarchical inclusion structure among these asymptotic operator classes for all ordinals ξ and exponents p ∈ (1, ∞).
• To extend the characterization of equal norm type p via equivalent norms with power-type asymptotic smoothness to all ordinals and operators.
• To generalize non-linear embedding results for quasi-reflexive Banach spaces with asymptotic equal norm type p.

Proposed method

• Introduces four operator classes: 𝒯ξ,p (asymptotic smoothability), 𝒜ξ,p (asymptotic type), 𝒩ξ,p (equal norm type), and 𝒫ξ,p (intermediate smoothability) for each ordinal ξ and p ∈ (1, ∞).
• Uses the ξ-Szlenk power type pξ(A) as a key invariant to define and analyze the classes, particularly in relation to renorming properties.
• Applies the ξ-modulus of asymptotic uniform smoothness ϱξ(σ; A:X→(Y,|·|)) to characterize membership in 𝒩ξ,p and 𝒩ξ,∞ via equivalent norms with controlled smoothness bounds.
• Employs block sequences, weakly null trees, and basis domination arguments to distinguish the classes, particularly using the formal inclusion I:ℓq → Ti*q and its strict singularity.
• Applies the notion of domination by ℓp-bases and tree estimates to show non-membership in certain classes, especially in the case of quasi-reflexive spaces.
• Extends a result of Lancien and Raja on Lipschitz embeddings of metric spaces Gk into quasi-reflexive Banach spaces by quantifying the distortion in terms of k^{1/p} for spaces with 𝒩0,p structure.

Experimental results

Research questions

• RQ1 Are there distinct transfinite asymptotic analogues of p-smoothability, Haar type, equal norm type, and intermediate smoothability for operators between Banach spaces?
• RQ2 Can the hierarchy of these asymptotic classes be strictly ordered for all ordinals ξ and p ∈ (1, ∞)?
• RQ3 Is there a characterization of equal norm type p in terms of equivalent norms with power-type asymptotic smoothness modulus?
• RQ4 Does the non-linear embedding result of Lancien and Raja extend to quasi-reflexive Banach spaces with asymptotic equal norm type p?
• RQ5 Can the supremum in the renorming theorem for ξ-asymptotically uniformly smooth norms be attained?

Key findings

• For any ordinal ξ and 1 < p < ∞, the classes satisfy the strict inclusion chain: 𝒟ξ ⊊ 𝒯ξ,∞ ⊊ 𝒜ξ,∞ = 𝒩ξ,∞ ⊊ 𝒫ξ,∞ ⊊ 𝒯ξ,p ⊊ 𝒜ξ,p ⊊ 𝒩ξ,p ⊊ 𝒫ξ,p ⊊ 𝒟ξ+1.
• When ξ has countable cofinality, the inclusions 𝒯ξ,p ⊊ 𝒜ξ,p ⊊ 𝒩ξ,p ⊊ 𝒫ξ,p are all strict, confirming the distinctness of the four asymptotic notions.
• An operator A belongs to 𝒩ξ,p if and only if there exist constants C, b ≥ 1 such that for any σ > 0, there is an equivalent norm |·| on Y with b⁻¹B_Y^|·| ⊂ B_Y ⊂ bB_Y^|·| and ϱξ(σ; A:X→(Y,|·|)) ≤ Cσ^p.
• An operator A belongs to 𝒩ξ,∞ if and only if there exist b ≥ 1 and σ > 0 such that for any ε > 0, there is an equivalent norm |·| on Y with b⁻¹B_Y^|·| ⊂ B_Y ⊂ bB_Y^|·| and ϱξ(σ; A:X→(Y,|·|)) ≤ ε.
• For any k ∈ ℕ and Lipschitz map f: G_k → X with X quasi-reflexive and I_X ∈ 𝒩0,p, there exists an infinite subset M ⊂ ℕ such that for any m₁ < n₁ < … < m_k < n_k in M, ||f(m₁,…,m_k) - f(n₁,…,n_k)|| ≤ C·Lip(f)·k^{1/p}.
• No infinite-dimensional quasi-reflexive Banach space can belong to 𝒯0,∞, as such spaces would be isomorphic to a subspace of c₀, contradicting quasi-reflexivity.