Equivalent norms with the property $$(\beta )$$ of Rolewicz

Dilworth, S. J.; Kutzarova, Denka; Lancien, Gilles; Randrianarivony, N. Lovasoa

doi:10.1007/s13398-016-0278-2

Cited by 19 publications

(22 citation statements)

References 21 publications

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“…Our next result establishes the duality between strongly AUS and strongly AUC norms by using estimates similar to those in [8]. Recall that, given a continuous function f :…”

Section: On Strongly Auc and Strongly Aus Spacesmentioning

confidence: 71%

“…Following [8], we will consider the set C of finite codimensional σ(X, F )-closed subspaces of X as a directed set with the order given by E F if F ⊂ E. Proposition 2.2. Let F be a norming subspace of X * .…”

Section: On F -Auc and F -Aus Spacesmentioning

confidence: 99%

“…The space X is said to be weak* asymptotically uniformly convex if δ * X (t) > 0 for each t > 0. Let us highlight that it is proved in [8] that a space is AUS if and only if its dual space is weak* AUC. In addition, ρ X is quantitatively related to δ * X by Young's duality.…”

Section: Introduction and Notationmentioning

confidence: 99%

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On strong asymptotic uniform smoothness and convexity

García-Lirola

Raja

2017

Rev Mat Complut

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Abstract. We introduce the notions of strong asymptotic uniform smoothness and convexity. We show that the injective tensor product of strongly asymptotically uniformly smooth spaces is asymptotically uniformly smooth. This applies in particular to uniformly smooth spaces admitting a monotone FDD, extending a result by Dilworth, Kutzarova, Randrianarivony, Revalski and Zhivkov [9]. Our techniques also provide a characterisation of Orlicz functions M, N such that the space of compact operators K(h M , h N ) is asymptotically uniformly smooth. Finally we show that K(X, Y ) is not strictly convex whenever X and Y are at least two-dimensional, which extends a result by Dilworth and Kutzarova [7].

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“…Our next result establishes the duality between strongly AUS and strongly AUC norms by using estimates similar to those in [8]. Recall that, given a continuous function f :…”

Section: On Strongly Auc and Strongly Aus Spacesmentioning

confidence: 71%

Section: On F -Auc and F -Aus Spacesmentioning

confidence: 99%

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On strong asymptotic uniform smoothness and convexity

García-Lirola

Raja

2017

Rev Mat Complut

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“…This metric characterization is only the third of this type in the asymptotic Ribe program after the Baudier-Kalton-Lancien characterization [4] and the Motakis-Schlumprecht characterization [30]. Note that we can omit the separability assumption since it follows from [10], that every (non-separable) reflexive Banach space Y with an unconditional asymptotic structure and Sz(Y * ) > ω has a separable subspace with the same properties. More precisely, the sequence (D ω k ) k∈N is a uniformly characterizing sequence for the class of asymptotically uniformly convexifiable spaces within the class of reflexive Banach spaces with an unconditional asymptotic structure.…”

Section: Applicationsmentioning

confidence: 99%

On the geometry of the countably branching diamond graphs

Baudier

Causey

Dilworth

et al. 2017

Journal of Functional Analysis

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In this article, the bi-Lipschitz embeddability of the sequence of countably branching diamond graphs $(D_k^\omega)_{k\in\mathbb{N}}$ is investigated. In particular it is shown that for every $\varepsilon>0$ and $k\in\mathbb{N}$, $D_k^\omega$ embeds bi-Lipschiztly with distortion at most $6(1+\varepsilon)$ into any reflexive Banach space with an unconditional asymptotic structure that does not admit an equivalent asymptotically uniformly convex norm. On the other hand it is shown that the sequence $(D_k^\omega)_{k\in\mathbb{N}}$ does not admit an equi-bi-Lipschitz embedding into any Banach space that has an equivalent asymptotically midpoint uniformly convex norm. Combining these two results one obtains a metric characterization in terms of graph preclusion of the class of asymptotically uniformly convexifiable spaces, within the class of separable reflexive Banach spaces with an unconditional asymptotic structure. Applications to bi-Lipschitz embeddability into $L_p$-spaces and to some problems in renorming theory are also discussed.Comment: 44 pages, 3 tables, 2 figure

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“…However, property (β) does not imply AUS isometrically [11], but a Banach space with property (β) admits an equivalent norm that is AUS. A complete renorming argument of property (β) can be found in the recent paper by Dilworth, Kutzarova, Lancien and Randrianarivony [6].…”

Section: Introductionmentioning

confidence: 97%

Asymptotic properties of Banach spaces and coarse quotient maps

Zhang

2018

Proc. Amer. Math. Soc.

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We give a quantitative result about asymptotic moduli of Banach spaces under coarse quotient maps. More precisely, we prove that if a Banach space Y is a coarse quotient of a subset of a Banach space X, where the coarse quotient map is coarse Lipschitz, then the (β)-modulus of X is bounded by the modulus of asymptotic uniform smoothness of Y up to some constants. In particular, if the coarse quotient map is a coarse homeomorphism, then the modulus of asymptotic uniform convexity of X is bounded by the modulus of asymptotic uniform smoothness of Y up to some constants.

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Equivalent norms with the property $$(\beta )$$ of Rolewicz

Cited by 19 publications

References 21 publications

On strong asymptotic uniform smoothness and convexity

On strong asymptotic uniform smoothness and convexity

On the geometry of the countably branching diamond graphs

Asymptotic properties of Banach spaces and coarse quotient maps

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