{"paper":{"title":"The star-shapedness of a generalized numerical range","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Nam-Kiu Tsing, Pan-Shun Lau, Tuen-Wai Ng","submitted_at":"2016-08-22T09:21:04Z","abstract_excerpt":"Let $\\mathcal{H}_n$ be the set of all $n\\times n$ Hermitian matrices and $\\mathcal{H}^m_n$ be the set of all $m$-tuples of $n\\times n$ Hermitian matrices. For $A=(A_1,...,A_m)\\in \\mathcal{H}^m_n$ and for any linear map $L:\\mathcal{H}^m_n\\to\\mathbb{R}^\\ell$, we define the $L$-numerical range of $A$ by \\[ W_L(A):=\\{L(U^*A_1U,...,U^*A_mU): U\\in \\mathbb{C}^{n\\times n}, U^*U=I_n\\}. \\] In this paper, we prove that if $\\ell\\leq 3$, $n\\geq \\ell$ and $A_1,...,A_m$ are simultaneously unitarily diagonalizable, then $W_L(A)$ is star-shaped with star center at $L\\left(\\frac{\\mathrm{tr} A_1}{n}I_n,...,\\frac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.06094","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"}