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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 "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1510.08132","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-10-27T23:41:35Z","cross_cats_sorted":[],"title_canon_sha256":"c27f71eed2a8a18786e3f424354f7e7e45317f85fb6326381242a2798d19d852","abstract_canon_sha256":"18a1635479087944bbfb1b211c6d49f2199612551414a94aa5a1131b7037d49d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:28:37.273461Z","signature_b64":"Nyj7Qf+vtKhtjKeBAnQuKg9+FQ6/HVPWeuBzIPa+G0B9MwvyrNl155KqHTVijhvv6ymipfj8QVDOlOit9+1OCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b86ec881dd3019521aa3a850a00e4b7fe0a3428054cff5791f9bc85549034ed","last_reissued_at":"2026-05-18T01:28:37.272799Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:28:37.272799Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1510.08132","created_at":"2026-05-18T01:28:37.272895+00:00"},{"alias_kind":"arxiv_version","alias_value":"1510.08132v1","created_at":"2026-05-18T01:28:37.272895+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.08132","created_at":"2026-05-18T01:28:37.272895+00:00"},{"alias_kind":"pith_short_12","alias_value":"BODOZCA52MAZ","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_16","alias_value":"BODOZCA52MAZKINK","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_8","alias_value":"BODOZCA5","created_at":"2026-05-18T12:29:14.074870+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BODOZCA52MAZKINKHKCQUAHEW7","json":"https://pith.science/pith/BODOZCA52MAZKINKHKCQUAHEW7.json","graph_json":"https://pith.science/api/pith-number/BODOZCA52MAZKINKHKCQUAHEW7/graph.json","events_json":"https://pith.science/api/pith-number/BODOZCA52MAZKINKHKCQUAHEW7/events.json","paper":"https://pith.science/paper/BODOZCA5"},"agent_actions":{"view_html":"https://pith.science/pith/BODOZCA52MAZKINKHKCQUAHEW7","download_json":"https://pith.science/pith/BODOZCA52MAZKINKHKCQUAHEW7.json","view_paper":"https://pith.science/paper/BODOZCA5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1510.08132&json=true","fetch_graph":"https://pith.science/api/pith-number/BODOZCA52MAZKINKHKCQUAHEW7/graph.json","fetch_events":"https://pith.science/api/pith-number/BODOZCA52MAZKINKHKCQUAHEW7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BODOZCA52MAZKINKHKCQUAHEW7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BODOZCA52MAZKINKHKCQUAHEW7/action/storage_attestation","attest_author":"https://pith.science/pith/BODOZCA52MAZKINKHKCQUAHEW7/action/author_attestation","sign_citation":"https://pith.science/pith/BODOZCA52MAZKINKHKCQUAHEW7/action/citation_signature","submit_replication":"https://pith.science/pith/BODOZCA52MAZKINKHKCQUAHEW7/action/replication_record"}},"created_at":"2026-05-18T01:28:37.272895+00:00","updated_at":"2026-05-18T01:28:37.272895+00:00"}