{"paper":{"title":"On mapping theorems for numerical range","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Hubert Klaja, Javad Mashreghi, Thomas Ransford","submitted_at":"2015-10-27T23:41:35Z","abstract_excerpt":"Let $T$ be an operator on a Hilbert space $H$ with numerical radius $w(T)\\le1$. According to a theorem of Berger and Stampfli, if $f$ is a function in the disk algebra such that $f(0)=0$, then $w(f(T))\\le\\|f\\|_\\infty$. We give a new and elementary proof of this result using finite Blaschke products.\n  A well-known result relating numerical radius and norm says $\\|T\\| \\leq 2w(T)$. We obtain a local improvement of this estimate, namely, if $w(T)\\le1$ then \\[ \\|Tx\\|^2\\le 2+2\\sqrt{1-|\\langle Tx,x\\rangle|^2} \\qquad(x\\in H,~\\|x\\|\\le1). \\] Using this refinement, we give a simplified proof of Drury's "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.08132","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"}