{"paper":{"title":"Crouzeix's conjecture holds for tridiagonal $3\\times 3$ matrices with elliptic numerical range centered at an eigenvalue","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Christer Glader, Mikael Kurula, Mikael Lindstrom","submitted_at":"2017-01-05T16:09:04Z","abstract_excerpt":"M. Crouzeix formulated the following conjecture in (Integral Equations Operator Theory 48, 2004, 461--477): For every square matrix $A$ and every polynomial $p$, $$\n  \\|p(A)\\| \\le 2 \\max_{z\\in W(A)}|p(z)|, $$ where $W(A)$ is the numerical range of $A$. We show that the conjecture holds in its strong, completely bounded form, i.e., where $p$ above is allowed to be any matrix-valued polynomial, for all tridiagonal $3\\times 3$ matrices with constant main diagonal: $$\n  \\left[\\begin{matrix}a&b_1&0\\\\c_1&a&b_2\\\\0&c_2&a\\end{matrix}\\right],\\qquad a,b_k,c_k\\in\\mathbb C, $$ or equivalently, for all comp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.01365","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"}