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On strong asymptotic uniform smoothness and convexity. (English) Zbl 1403.46016

For a real Banach space \(X\), the moduli of asymptotic uniform convexity and asymptotic uniform smoothness are defined by the formulas \[ \bar{\delta}_X(t)=\inf_{x\in S_X}\sup_{\text{dim} X/Y<\infty}\inf_{y\in S_Y}||x+ty||-1, \;\;\;t>0, \]
\[ \bar{\rho}_X(t)=\sup_{x\in S_X}\inf_{\text{dim} X/Y<\infty}\sup_{y\in S_Y}||x+ty||-1, \;\;\;t>0. \] The space \(X\) is called asymptotically uniformly convex (AUC) if \(\overline{\delta}_X(t)>0\) for any \(t>0\) and asymptotically uniformly smooth (AUS) if \(\lim_{t\to 0}t^{-1}\overline{\rho}_X(t)=0\).
The authors introduce the notion of strongly AUC and strongly AUS Banach spaces, motivated by a reformulation of the moduli recalled above in the case of a Banach space with a monotone shrinking finite dimension decomposition (FDD). For a Banach space \(X\) with an FDD \(E=(E_n)_n\), let \(H_n=\bigoplus_{i=1}^nE_i\) and \(H^n=\bigoplus_{i=n+1}^\infty E_i\), \(n\in\mathbb{N}\). The space \(X\) is called strongly AUC with respect to \(E\) if \(\widehat{\delta}_E(t)>0\) for any \(t>0\), where \[ \widehat{\delta}_E(t)=\inf_{n\in \mathbb{N}}\sup_{m\geq n}\inf\{||x+ty||-1:\;x\in H_m\cap S_X,\, y\in H^m\cap S_X \},\;\;\;t>0. \] The space \(X\) is called strongly AUS with respect to \(E\) if \(\lim_{t\to 0}t^{-1}\widehat{\rho}_E(t)=0\), where \[ \widehat{\rho}_E(t)=\sup_{n\in \mathbb{N}}\inf_{m\geq n}\sup\{||x+ty||-1:\;x\in H_m\cap S_X,\, y\in H^m\cap S_X \},\;\;\;t>0. \] The introduced properties are stronger than AUC and AUS, respectively, and weaker than the uniform convexity and uniform smoothness in the case of a Banach space with a monotone FDD. The duality properties of the new notions and the relation with reflexivity of the space are also discussed.
The main result concerns the injective tensor product of Banach spaces. By the result of R. M. Causey [J. Funct. Anal. 272, No. 8, 3375–3409 (2017; Zbl 1369.46008)], the injective tensor product of Banach spaces that are AUS admits an equivalent AUS norm, however, it is not known if it is AUS in its original norm. The authors prove that the injective tensor product of strongly AUS Banach spaces is AUS, generalizing in particular the result of S. J. Dilworth et al. [J. Math. Anal. Appl. 402, No. 1, 297–307 (2013; Zbl 1276.46006)]. On the other hand, the injective tensor product of strongly AUC spaces need not be AUC. The authors also provide a characterization – in terms of Boyd indices – of Orlicz functions \(M,N\) for which the space of compact operators \(\mathcal{K}(h_M,h_N)\) is AUS and prove that both the space of compact operators \(\mathcal{K}(X,Y)\) and the injective tensor product \(X\widehat{\otimes}_\varepsilon Y\) are not strictly convex, provided that \(X\) and \(Y\) are at least two-dimensional, extending the result of S. J. Dilworth and D. Kutzarova [Lect. Notes Pure Appl. Math. 172, 71–83 (1995; Zbl 0851.46012)].

MSC:

46B20 Geometry and structure of normed linear spaces
46B06 Asymptotic theory of Banach spaces
46B28 Spaces of operators; tensor products; approximation properties
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References:

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