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.