{"paper":{"title":"Rates of decay in the classical Katznelson-Tzafriri theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"David Seifert","submitted_at":"2014-10-06T09:25:41Z","abstract_excerpt":"Given a power-bounded operator $T$, the theorem of Katznelson and Tzafriri states that $\\|T^n(I-T)\\|\\to0$ as $n\\to\\infty$ if and only if the spectrum $\\sigma(T)$ of $T$ intersects the unit circle $\\mathbb{T}$ in at most the point 1. This paper investigates the rate at which decay takes place when $\\sigma(T)\\cap\\mathbb{T}=\\{1\\}$. The results obtained lead in particular to both upper and lower bounds on this rate of decay in terms of the growth of the resolvent operator $R(\\mathrm{e}^{\\mathrm{i}\\theta},T)$ as $\\theta\\to0$. In the special case of polynomial resolvent growth, these bounds are then"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.1297","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"}