{"paper":{"title":"Sharp spectral constants for scaled $q$-numerical ranges","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mohamed Amine Aouichaoui, Ryan O'Loughlin","submitted_at":"2026-08-10T17:23:51Z","abstract_excerpt":"For $ n \\geq 2$, $A\\in M_n(\\mathbb C)$ and $0<|q|\\leq 1$, let $\\Omega_q(A)=q^{-1}W_q(A)$ be the scaled $q$-numerical range. We prove that for every $\\gamma \\geq 1$, \\[\n  \\Omega_{\\eta(\\gamma)}(A)\n  =\\bigcup_{\\kappa(S)\\leq\\gamma}W(S^{-1}AS),\n  \\qquad\n  \\eta(\\gamma)=\\frac{2}{\\gamma+\\gamma^{-1}}, \\] where $\\kappa(S)=\\|S\\|\\,\\|S^{-1}\\|$. As a consequence, we prove the sharp inequality \\[\n  \\|p(A)\\|\\leq\n  \\max\\!\\left\\{1,\\frac{2|q|}{1+\\sqrt{1-|q|^2}}\\right\\}\n  \\max_{z\\in\\Omega_q(A)}|p(z)|, \\] for all polynomials $p$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.09866","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.09866/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}