{"paper":{"title":"$\\xi$-asymptotically uniformly smooth, $\\xi$-asymptotically uniformly convex, and $(\\beta)$ operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Ryan M. Causey, Stephen J. Dilworth","submitted_at":"2016-07-05T18:36:32Z","abstract_excerpt":"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 $(\\beta)$ f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.01362","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}