Power type ξ-asymptotically uniformly smooth and ξ-asymptotically uniformly flat norms

Concerning $q$-summable Szlenk index

“…Furthermore, Godefroy, Kalton, and Lancien [12] gave a precise renorming theorem for a separable Banach space in terms of the Szlenk power type of the Banach space. This was generalized to non-separable spaces and operators in [5], as well as to higher ordinals in terms of the behavior of special convex combinations of the branches of n-leveled weakly null trees where each level has specified order. Further renorming results were established in [6], analogous to those of Pisier in [18] were investigated in terms of special convex combinations of the branches of ω-leveled weakly null trees where each level has specified order.…”

“…This was generalized to non-separable spaces and operators in [5], as well as to higher ordinals in terms of the behavior of special convex combinations of the branches of n-leveled weakly null trees where each level has specified order. Further renorming results were established in [6], analogous to those of Pisier in [18] were investigated in terms of special convex combinations of the branches of ω-leveled weakly null trees where each level has specified order. The downside of the renorming results from [6] is that they are only produce non-trivial equivalent norms when an operator or a space has some power type behavior of the ε-Szlenk indices.…”

“…Further renorming results were established in [6], analogous to those of Pisier in [18] were investigated in terms of special convex combinations of the branches of ω-leveled weakly null trees where each level has specified order. The downside of the renorming results from [6] is that they are only produce non-trivial equivalent norms when an operator or a space has some power type behavior of the ε-Szlenk indices. That is, for a fixed ξ, the results of [6] produce non-trivial equivalent norms for a Banach space X or an operator A : X → Y if ω ξ k ε Sz(X, ε) ω ξ (k ε + 1) (resp.…”

“…The downside of the renorming results from [6] is that they are only produce non-trivial equivalent norms when an operator or a space has some power type behavior of the ε-Szlenk indices. That is, for a fixed ξ, the results of [6] produce non-trivial equivalent norms for a Banach space X or an operator A : X → Y if ω ξ k ε Sz(X, ε) ω ξ (k ε + 1) (resp. ω ξ k ε Sz(A, ε) ω ξ (k ε + 1)) where k ε ∈ N satisfies lim ε→0 + k ε = ∞ but for some 1 q < ∞, sup ε>0 ε q k ε < ∞.…”

Factorization of Asplund operators

“…Now suppose that A : X → Y is an operator with separable range and Sz(A) ω ζ . Then by [10], A factors through a separable Banach space Z with p ζ (Z) < 2. By Theorem 5.9, there exists a Banach space W with FDD F such that Z is isomorphic to both a subspace and to a quotient of W X ζ,2 ∧ (F).…”