This is an announcement for the paper “$\xi$-asymptotically uniformly smooth, $\xi$--asymptotically uniformly convex, and $(\beta)$-operators” by Ryan M. Causey<http://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1>, Stephen J. Dilworth<http://arxiv.org/find/math/1/au:+Dilworth_S/0/1/0/all/0/1>. Abstract: For each ordinal $\xi$, we define the notions of $\xi$-asymptotically uniformly smooth and $w^*$-$\xi$-asymptotically uniformly convex operators. When $\xi=0$, 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 $(\beta)$ of Rolewicz which extends the notion of property $(\beta)$ for a Banach space. We characterize those operators the domain and range of which can be renormed so that the operator has property $(\beta)$ in terms of the Szlenk index of the operator and its adjoint. The paper may be downloaded from the archive by web browser from URL http://arxiv.org/abs/1607.01362