{"paper":{"title":"Concerning $q$-summable Szlenk index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Ryan M. Causey","submitted_at":"2017-12-29T20:50:23Z","abstract_excerpt":"For each ordinal $\\xi$ and each $1\\leqslant q<\\infty$, we define the notion of $\\xi$-$q$-summable Szlenk index. When $\\xi=0$ and $q=1$, this recovers the usual notion of summable Szlenk index. We define for an arbitrary weak$^*$-compact set a transfinite, asymptotic analogue $\\alpha_{\\xi,p}$ of the martingale type norm of an operator. We prove that this quantity is determined by norming sets and determines $\\xi$-Szlenk power type and $\\xi$-$q$-summability of Szlenk index. This fact allows us to prove that the behavior of operators under the $\\alpha_{\\xi,p}$ seminorms passes in the strongest wa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00033","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"}