{"paper":{"title":"Spectral Sets: Numerical Range and Beyond","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA","math.OA"],"primary_cat":"math.SP","authors_text":"Anne Greenbaum, Michel Crouzeix","submitted_at":"2018-03-29T02:30:56Z","abstract_excerpt":"We extend the proof in [M.~Crouzeix and C.~Palencia, {\\em The numerical range is a $(1 + \\sqrt{2})$-spectral set}, SIAM Jour.~Matrix Anal.~Appl., 38 (2017), pp.~649-655] to show that other regions in the complex plane are $K$-spectral sets. In particular, we show that various annular regions are $(1 + \\sqrt{2} )$-spectral sets and that a more general convex region with a circular hole or cutout is a $(3 + 2 \\sqrt{3} )$-spectral set. We demonstrate how these results can be used to give bounds on the convergence rate of the GMRES algorithm for solving linear systems and on that of rational Krylo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.10904","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"}