{"paper":{"title":"Commutators of small rank and reducibility of operator semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.OA"],"primary_cat":"math.FA","authors_text":"Alexey I. Popov, Ali Jafarian, Heydar Radjavi, Mehdi Radjabalipour","submitted_at":"2013-06-09T01:51:30Z","abstract_excerpt":"It is easy to see that if $\\cG$ is a non-abelian group of unitary matrices, then for no members $A$ and $B$ of $\\cG$ can the rank of $AB-BA$ be one. We examine the consequences of the assumption that this rank is at most two for a general semigroup $\\cS$ of linear operators. Our conclusion is that under obviously necessary, but trivial, size conditions, $\\cS$ is reducible. In the case of a unitary group satisfying the hypothesis, we show that it is contained in the direct sum $\\cG_1\\oplus\\cG_2$ where $\\cG_1$ is at most $3\\times 3$ and $\\cG_2$ is abelian."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.1972","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"}