{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:E4UDM4J3J5T3QGFXEV76HOC42K","short_pith_number":"pith:E4UDM4J3","schema_version":"1.0","canonical_sha256":"272836713b4f67b818b7257fe3b85cd2849b39a2c7c0e0ea24b986f99add2171","source":{"kind":"arxiv","id":"1507.01418","version":1},"attestation_state":"computed","paper":{"title":"A semigroup approach to the numerical range of operators on Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Agnes Radl, Martin Adler, Waed Dada","submitted_at":"2015-07-06T12:38:13Z","abstract_excerpt":"We introduce the numerical spectrum $\\sigma_n(A)\\subset \\mathbb{C}$ of an (unbounded) linear operator $A$ on a Banach space $X$ and study its properties. Our definition is closely related to the numerical range $W(A)$ of $A$ and always yields a superset of $W(A)$. In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, $\\sigma_n(A)$ is always closed, convex and contains the spectrum of $A$. In the paper we strongly emphasise the connection of our approach to the theory of $C_0$-semigroups."},"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":"1507.01418","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-07-06T12:38:13Z","cross_cats_sorted":[],"title_canon_sha256":"4d318eeadd727d6c752867bd9bc53259664525771d7b5d73e39be0ee091a3516","abstract_canon_sha256":"ef918f76b625721f15db3a62818d5443803e01a8335d605d7e3c816c9f91db1c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:37:17.006637Z","signature_b64":"G6E4LwXHTZckILXHBc3VNFHsO8aZtSystJimji/wMhUi0BgsUvuKlsS38/PcCIlnYSUKoOFmOEuAGh7xfrt2Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"272836713b4f67b818b7257fe3b85cd2849b39a2c7c0e0ea24b986f99add2171","last_reissued_at":"2026-05-18T01:37:17.006060Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:37:17.006060Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A semigroup approach to the numerical range of operators on Banach spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Agnes Radl, Martin Adler, Waed Dada","submitted_at":"2015-07-06T12:38:13Z","abstract_excerpt":"We introduce the numerical spectrum $\\sigma_n(A)\\subset \\mathbb{C}$ of an (unbounded) linear operator $A$ on a Banach space $X$ and study its properties. Our definition is closely related to the numerical range $W(A)$ of $A$ and always yields a superset of $W(A)$. In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, $\\sigma_n(A)$ is always closed, convex and contains the spectrum of $A$. In the paper we strongly emphasise the connection of our approach to the theory of $C_0$-semigroups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.01418","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":"1507.01418","created_at":"2026-05-18T01:37:17.006149+00:00"},{"alias_kind":"arxiv_version","alias_value":"1507.01418v1","created_at":"2026-05-18T01:37:17.006149+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.01418","created_at":"2026-05-18T01:37:17.006149+00:00"},{"alias_kind":"pith_short_12","alias_value":"E4UDM4J3J5T3","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_16","alias_value":"E4UDM4J3J5T3QGFX","created_at":"2026-05-18T12:29:17.054201+00:00"},{"alias_kind":"pith_short_8","alias_value":"E4UDM4J3","created_at":"2026-05-18T12:29:17.054201+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/E4UDM4J3J5T3QGFXEV76HOC42K","json":"https://pith.science/pith/E4UDM4J3J5T3QGFXEV76HOC42K.json","graph_json":"https://pith.science/api/pith-number/E4UDM4J3J5T3QGFXEV76HOC42K/graph.json","events_json":"https://pith.science/api/pith-number/E4UDM4J3J5T3QGFXEV76HOC42K/events.json","paper":"https://pith.science/paper/E4UDM4J3"},"agent_actions":{"view_html":"https://pith.science/pith/E4UDM4J3J5T3QGFXEV76HOC42K","download_json":"https://pith.science/pith/E4UDM4J3J5T3QGFXEV76HOC42K.json","view_paper":"https://pith.science/paper/E4UDM4J3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1507.01418&json=true","fetch_graph":"https://pith.science/api/pith-number/E4UDM4J3J5T3QGFXEV76HOC42K/graph.json","fetch_events":"https://pith.science/api/pith-number/E4UDM4J3J5T3QGFXEV76HOC42K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/E4UDM4J3J5T3QGFXEV76HOC42K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/E4UDM4J3J5T3QGFXEV76HOC42K/action/storage_attestation","attest_author":"https://pith.science/pith/E4UDM4J3J5T3QGFXEV76HOC42K/action/author_attestation","sign_citation":"https://pith.science/pith/E4UDM4J3J5T3QGFXEV76HOC42K/action/citation_signature","submit_replication":"https://pith.science/pith/E4UDM4J3J5T3QGFXEV76HOC42K/action/replication_record"}},"created_at":"2026-05-18T01:37:17.006149+00:00","updated_at":"2026-05-18T01:37:17.006149+00:00"}