On strong asymptotic uniform smoothness and convexity

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]. * X by Young's duality. We refer the reader to [16] and the references therein for a detailed study of these properties. The related notions of nearly uniformly convex space (NUC for short) and nearly uniformly smooth (NUS for short) were introduced by Huff [15] and Prus [22]. A space is NUS if and only if it is AUS and reflexive and if and only if its dual is NUC.