Compactly uniformly convex spaces and property(β)of Rolewicz

On strong asymptotic uniform smoothness and convexity

“…The previous result and the isometry between K(X, Y ) and X * ⊗ ε Y yields the generalisation of Theorem 4.3 in [9]. Theorem 1.2.…”

“…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.…”

“…This result was extended by Besbes in [3], who showed that K(ℓ p , ℓ p ) is AUS whenever 1 < p < ∞. Moreover, in [9] it is proved that K(ℓ p , ℓ q ) is AUS with power type min{p ′ , q} for every 1 < p, q < ∞. On the order hand, Causey recently showed in [5] that the Szlenk index of X ⊗ ε Y is equal to the maximum of the Szlenk indices of X and Y for all Banach spaces X and Y .…”

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

“…It was shown in [13], with a different terminology, that if a norm of a reflexive Banach space is AUS and AUC then it has property (β). We will need the following quantitative version of this result (see Theorem 5.2 in [7]). Theorem 4.1.…”

“…Note that Theorem 5.2 in [7] is stated with indices called b X and d X instead of ρ X and δ X respectively, where b X (t) = sup{lim sup n→∞ x + tx n − 1} and d X (t) = inf{lim inf n→∞ x + tx n − 1}, the above sup and inf being taken over all weakly null sequences in B X . The proof of Theorem 5.2 in [7] can be easily adapted to the moduli ρ X and δ X .…”