{"paper":{"title":"Chaotic Banach algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.FA","authors_text":"Stanislav Shkarin","submitted_at":"2010-08-19T11:00:30Z","abstract_excerpt":"We construct an infinite dimensional non-unital Banach algebra $A$ and $a\\in A$ such that the sets $\\{za^n:z\\in\\C,\\ n\\in\\N\\}$ and $\\{({\\bf 1}+a)^na:n\\in\\N\\}$ are both dense in $A$, where $\\bf 1$ is the unity in the unitalization $A^{\\#}=A\\oplus \\spann\\{{\\bf 1}\\}$ of $A$. As a byproduct, we get a hypercyclic operator $T$ on a Banach space such that $T\\oplus T$ is non-cyclic and $\\sigma(T)=\\{1\\}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.3271","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"}