{"paper":{"title":"A Hilbert-Schmidt analog of Huaxin Lin's Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.SP","authors_text":"Ilya Kachkovskiy, Nikolay Filonov","submitted_at":"2010-08-24T10:09:45Z","abstract_excerpt":"The paper is devoted to the following question: consider two self-adjoint $n\\times n$-matrices $H_1,H_2$, $\\|H_1\\|\\le 1$, $\\|H_2\\|\\le 1$, such that their commutator $[H_1,H_2]$ is small in some sence. Do there exist such self-adjoint commuting matrices $A_1,A_2$, such that $A_i$ is close to $H_i$, $i=1,2$? The answer to this question is positive if the smallness is considered with respect to the operator norm. The following result was established by Huaxin Lin: if $\\|[H_1,H_2]\\|=\\delta$, then we can choose $A_i$ such that $\\|H_i-A_i\\|\\le C(\\delta)$, $i=1,2$, where $C(\\delta)\\to 0$ as $\\delta\\t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.4002","kind":"arxiv","version":2},"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"}