{"paper":{"title":"On powers of operators with spectrum in cantor sets and spectral synthesis","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mohamed Zarrabi (IMB)","submitted_at":"2017-06-09T13:26:03Z","abstract_excerpt":"For $\\xi \\in \\big( 0, \\frac{1}{2} \\big)$, let $E_{\\xi}$ be the perfect symmetric set associated with $\\xi$, that is $$E_{\\xi} = \\Big\\{ \\exp \\Big( 2i \\pi (1-\\xi) \\sum_{n = 1}^{+\\infty} \\epsilon_{n} \\xi^{n-1} \\Big) : \\, \\epsilon_{n} = 0 \\textrm{ or } 1 \\quad (n \\geq 1) \\Big\\}$$ and $$b(\\xi) = \\frac{\\log{\\frac{1}{\\xi}} - \\log{2}}{2\\log{\\frac{1}{\\xi}} - \\log{2}}.$$ Let $q\\geq 3$ be an integer and $s$ be a nonnegative real number. We show that any invertible operator $T$ on a Banach space with spectrum contained in $E_{1/q}$ that satisfies \\begin{eqnarray*} & & \\big\\| T^{n} \\big\\| = O \\big( n^{s} \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.02943","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"}