{"paper":{"title":"Some Extensions of the Crouzeix-Palencia Result","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Anne Greenbaum, Kenan Li, Trevor Caldwell","submitted_at":"2017-07-26T18:36:17Z","abstract_excerpt":"In [{\\em The Numerical Range is a $(1 + \\sqrt{2})$-Spectral Set}, SIAM J. Matrix Anal. Appl. 38 (2017), pp.~649-655], Crouzeix and Palencia show that the numerical range of a square matrix or linear operator $A$ is a $(1 + \\sqrt{2})$-spectral set for $A$; that is, for any function $f$ analytic in the interior of the numerical range $W(A)$ and continuous on its boundary, the inequality $\\| f(A) \\| \\leq (1 + \\sqrt{2} ) \\| f \\|_{W(A)}$ holds, where the norm on the left is the operator 2-norm and $\\| f \\|_{W(A)}$ on the right denotes the supremum of $| f(z) |$ over $z \\in W(A)$. In this paper, we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.08603","kind":"arxiv","version":2},"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"}