$ξ$-asymptotically uniformly smooth, $ξ$-asymptotically uniformly convex, and $(β)$ operators

For each ordinal $ξ$, we define the notions of $ξ$-asymptotically uniformly smooth and $w^*$-$ξ$-asymptotically uniformly convex operators. When $ξ=0$, these extend the notions of asymptotically uniformly smooth and $w^*$-asymptotically uniformly convex Banach spaces. We give a complete description of renorming results for these properties in terms of the Szlenk index of the operator, as well as a complete description of the duality between these two properties. We also define the notion of an operator with property $(β)$ of Rolewicz which extends the notion of property $(β)$ for a Banach space. We characterize those operators the domain and range of which can be renormed so that the operator has property $(β)$ in terms of the Szlenk index of the operator and its adjoint.

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