{"paper":{"title":"Invertibility threshold for $H^\\infty$ trace algebras, and effective matrix inversions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Nikolai Nikolski, Vasily Vasyunin","submitted_at":"2010-10-28T20:40:58Z","abstract_excerpt":"For a given $\\delta$, $0<\\delta<1$, a Blaschke sequence $\\sigma=\\{\\lambda_j\\}$ is constructed such that every function $f$, $f\\in H^\\infty$, having $\\delta<\\delta_f=\\inf_{\\lambda\\in\\sigma}|f(\\lambda)|\\le\\|f\\|_\\infty\\le1$ is invertible in the trace algebra $H^\\infty|\\sigma$ (with a norm estimate of the inverse depending on $\\delta_f$ only), but there exists $f$ with $\\delta=\\delta_f\\le\\|f\\|_\\infty\\le1$, which does not. As an application, a counterexample to a stronger form of the Bourgain--Tzafriri restricted invertibility conjecture for bounded operators is exhibited, where an ``orthogonal (or"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.6090","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"}