{"paper":{"title":"Hypercyclic operators on countably dimensional spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.FA","authors_text":"Andre Schenke, Stanislav Shkarin","submitted_at":"2012-05-02T12:58:44Z","abstract_excerpt":"According to Grivaux, the group $GL(X)$ of invertible linear operators on a separable infinite dimensional Banach space $X$ acts transitively on the set $\\Sigma(X)$ of countable dense linearly independent subsets of $X$. As a consequence, each $A\\in \\Sigma(X)$ is an orbit of a hypercyclic operator on $X$. Furthermore, every countably dimensional normed space supports a hypercyclic operator.\n  We show that for a separable infinite dimensional Fr\\'echet space $X$, $GL(X)$ acts transitively on $\\Sigma(X)$ if and only if $X$ possesses a continuous norm. We also prove that every countably dimension"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.0414","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"}