{"paper":{"title":"Commutators close to the identity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Terence Tao","submitted_at":"2018-05-28T19:01:38Z","abstract_excerpt":"Let $D,X \\in B(H)$ be bounded operators on an infinite dimensional Hilbert space $H$. If the commutator $[D,X] = DX-XD$ lies within $\\varepsilon$ in operator norm of the identity operator $1_{B(H)}$, then it was observed by Popa that one has the lower bound $\\| D \\| \\|X\\| \\geq \\frac{1}{2} \\log \\frac{1}{\\varepsilon}$ on the product of the operator norms of $D,X$; this is a quantitative version of the Wintner-Wielandt theorem that $1_{B(H)}$ cannot be expressed as the commutator of bounded operators. On the other hand, it follows easily from the work of Brown and Pearcy that one can construct ex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.11131","kind":"arxiv","version":3},"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"}