标题： $ξ$-渐近一致光滑，$ξ$-渐近一致凸，和$(β)$算子 显示英文标题

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

摘要： 对于每个序数$\xi$，我们定义了$\xi$-渐近一致光滑和$w^*$-$\xi$-渐近一致凸算子的概念。当$\xi=0$时，这些扩展了渐近一致光滑和$w^*$-渐近一致凸巴拿赫空间的概念。我们根据算子的 Szlenk 指数对这些性质的重新赋范结果进行了完整描述，同时还对这两种性质之间的对偶性进行了完整描述。 我们还定义了具有 Rolewicz 的性质$(\beta)$的算子概念，该概念扩展了巴拿赫空间的性质$(\beta)$的概念。 我们通过算子及其共轭的 Szlenk 指数来表征那些其定义域和值域可以重新赋范使得算子具有性质$(\beta)$的算子。 显示英文摘要

摘要： 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.