{"paper":{"title":"Spectral sets and distinguished varieties in the symmetrized bidisc","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Orr Shalit, Sourav Pal","submitted_at":"2013-10-10T11:02:14Z","abstract_excerpt":"We show that for every pair of matrices (S,P), having the closed symmetrized bidisc $\\Gamma$ as a spectral set, there is a one dimensional complex algebraic variety $\\Lambda$ in $\\Gamma$ such that for every matrix valued polynomial f, the norm of f(S,P) is less then the sup norm of f on $\\Lambda$.\n  The variety $\\Lambda$ is shown to have a particular determinantal representation, related to the so-called \"fundamental operator\" of the pair (S,P).\n  When (S,P) is a strict $\\Gamma$-contraction, then $\\Lambda$ is a distinguished variety in the symmetrized bidisc, i.e., a one dimensional algebraic "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2769","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"}