{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:KRMYFKTRT62WAKM2IS3PQAUTLH","short_pith_number":"pith:KRMYFKTR","canonical_record":{"source":{"id":"2606.08576","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-06-07T11:12:40Z","cross_cats_sorted":[],"title_canon_sha256":"210b441dc1af0164e5ce994ca3fda69b12c0eb4feb3c2d393e34ec8992830d21","abstract_canon_sha256":"f78152d32d72980ec626e8bdee86c48d809cf4cdfcee8dfe87c009caa05fe30a"},"schema_version":"1.0"},"canonical_sha256":"545982aa719fb560299a44b6f8029359c685c4d5425b75f08cd33179cbe1cb6d","source":{"kind":"arxiv","id":"2606.08576","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.08576","created_at":"2026-06-09T01:05:40Z"},{"alias_kind":"arxiv_version","alias_value":"2606.08576v1","created_at":"2026-06-09T01:05:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08576","created_at":"2026-06-09T01:05:40Z"},{"alias_kind":"pith_short_12","alias_value":"KRMYFKTRT62W","created_at":"2026-06-09T01:05:40Z"},{"alias_kind":"pith_short_16","alias_value":"KRMYFKTRT62WAKM2","created_at":"2026-06-09T01:05:40Z"},{"alias_kind":"pith_short_8","alias_value":"KRMYFKTR","created_at":"2026-06-09T01:05:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:KRMYFKTRT62WAKM2IS3PQAUTLH","target":"record","payload":{"canonical_record":{"source":{"id":"2606.08576","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-06-07T11:12:40Z","cross_cats_sorted":[],"title_canon_sha256":"210b441dc1af0164e5ce994ca3fda69b12c0eb4feb3c2d393e34ec8992830d21","abstract_canon_sha256":"f78152d32d72980ec626e8bdee86c48d809cf4cdfcee8dfe87c009caa05fe30a"},"schema_version":"1.0"},"canonical_sha256":"545982aa719fb560299a44b6f8029359c685c4d5425b75f08cd33179cbe1cb6d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T01:05:40.621330Z","signature_b64":"92r2ILv44vO8t/xtVrvAd12yc/gKI0/rwQAWnYOVxSUQHTYETDnABpM1vMDmz9x3mjVTAcTh2Llbqax4Q7L8Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"545982aa719fb560299a44b6f8029359c685c4d5425b75f08cd33179cbe1cb6d","last_reissued_at":"2026-06-09T01:05:40.620908Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T01:05:40.620908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.08576","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-09T01:05:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IHqwqwxb0c3NfRdn9QySCK0gEI02BfZGy0kpab7Au8B6qG4DIxDO3Ka2obLMzoYcP47Qgypw1DsErhUO6T7YAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T21:27:03.294557Z"},"content_sha256":"14dc123f9a9f6edf156a58aefa8232e74f1a364ff84f810147671153116c8dc5","schema_version":"1.0","event_id":"sha256:14dc123f9a9f6edf156a58aefa8232e74f1a364ff84f810147671153116c8dc5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:KRMYFKTRT62WAKM2IS3PQAUTLH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Numerical radius of certain two-by-two block matrices","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Chi-Kwong Li, Hwa-Long Gau, Jia-Huo Hong, Kuo-Zhong Wang","submitted_at":"2026-06-07T11:12:40Z","abstract_excerpt":"We investigate the numerical range $W(T)$ and numerical radius $w(T)$ of operators of the form $T = \\begin{pmatrix} A & B \\\\ 0 & 0 \\end{pmatrix}$. We show that $W(T)$ is the union of the numerical ranges of a family of $2\\times 2$ matrices, $T_x$, leading to several consequences, including improved inequalities for $w(T)$. For cases where $A$ is a self-adjoint involution, we characterize the conditions under which $W(T)$ is an elliptical disk and determine the minimum numerical radius of $T_U = \\begin{pmatrix} U^*AU & B \\\\ 0 & 0 \\end{pmatrix}$ over all unitary operators $U$. Finally, we study "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08576","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.08576/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-09T01:05:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/zvDLXnhk1YVG0mGzhLBg9m4dPjfC6H0q2bsWd2U7mCwzpEF6IH7/OFjsJvhtllvCFxXp7j7D64ffbwWvWPZAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T21:27:03.294932Z"},"content_sha256":"777402d0ff57a4b43e50d2b46f93e4b07e8ceb2f7c37f6a3d689bdc33a80e37d","schema_version":"1.0","event_id":"sha256:777402d0ff57a4b43e50d2b46f93e4b07e8ceb2f7c37f6a3d689bdc33a80e37d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KRMYFKTRT62WAKM2IS3PQAUTLH/bundle.json","state_url":"https://pith.science/pith/KRMYFKTRT62WAKM2IS3PQAUTLH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KRMYFKTRT62WAKM2IS3PQAUTLH/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-20T21:27:03Z","links":{"resolver":"https://pith.science/pith/KRMYFKTRT62WAKM2IS3PQAUTLH","bundle":"https://pith.science/pith/KRMYFKTRT62WAKM2IS3PQAUTLH/bundle.json","state":"https://pith.science/pith/KRMYFKTRT62WAKM2IS3PQAUTLH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KRMYFKTRT62WAKM2IS3PQAUTLH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:KRMYFKTRT62WAKM2IS3PQAUTLH","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f78152d32d72980ec626e8bdee86c48d809cf4cdfcee8dfe87c009caa05fe30a","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-06-07T11:12:40Z","title_canon_sha256":"210b441dc1af0164e5ce994ca3fda69b12c0eb4feb3c2d393e34ec8992830d21"},"schema_version":"1.0","source":{"id":"2606.08576","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.08576","created_at":"2026-06-09T01:05:40Z"},{"alias_kind":"arxiv_version","alias_value":"2606.08576v1","created_at":"2026-06-09T01:05:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.08576","created_at":"2026-06-09T01:05:40Z"},{"alias_kind":"pith_short_12","alias_value":"KRMYFKTRT62W","created_at":"2026-06-09T01:05:40Z"},{"alias_kind":"pith_short_16","alias_value":"KRMYFKTRT62WAKM2","created_at":"2026-06-09T01:05:40Z"},{"alias_kind":"pith_short_8","alias_value":"KRMYFKTR","created_at":"2026-06-09T01:05:40Z"}],"graph_snapshots":[{"event_id":"sha256:777402d0ff57a4b43e50d2b46f93e4b07e8ceb2f7c37f6a3d689bdc33a80e37d","target":"graph","created_at":"2026-06-09T01:05:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.08576/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We investigate the numerical range $W(T)$ and numerical radius $w(T)$ of operators of the form $T = \\begin{pmatrix} A & B \\\\ 0 & 0 \\end{pmatrix}$. We show that $W(T)$ is the union of the numerical ranges of a family of $2\\times 2$ matrices, $T_x$, leading to several consequences, including improved inequalities for $w(T)$. For cases where $A$ is a self-adjoint involution, we characterize the conditions under which $W(T)$ is an elliptical disk and determine the minimum numerical radius of $T_U = \\begin{pmatrix} U^*AU & B \\\\ 0 & 0 \\end{pmatrix}$ over all unitary operators $U$. Finally, we study ","authors_text":"Chi-Kwong Li, Hwa-Long Gau, Jia-Huo Hong, Kuo-Zhong Wang","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-06-07T11:12:40Z","title":"Numerical radius of certain two-by-two block matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.08576","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:14dc123f9a9f6edf156a58aefa8232e74f1a364ff84f810147671153116c8dc5","target":"record","created_at":"2026-06-09T01:05:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f78152d32d72980ec626e8bdee86c48d809cf4cdfcee8dfe87c009caa05fe30a","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.FA","submitted_at":"2026-06-07T11:12:40Z","title_canon_sha256":"210b441dc1af0164e5ce994ca3fda69b12c0eb4feb3c2d393e34ec8992830d21"},"schema_version":"1.0","source":{"id":"2606.08576","kind":"arxiv","version":1}},"canonical_sha256":"545982aa719fb560299a44b6f8029359c685c4d5425b75f08cd33179cbe1cb6d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"545982aa719fb560299a44b6f8029359c685c4d5425b75f08cd33179cbe1cb6d","first_computed_at":"2026-06-09T01:05:40.620908Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-09T01:05:40.620908Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"92r2ILv44vO8t/xtVrvAd12yc/gKI0/rwQAWnYOVxSUQHTYETDnABpM1vMDmz9x3mjVTAcTh2Llbqax4Q7L8Dw==","signature_status":"signed_v1","signed_at":"2026-06-09T01:05:40.621330Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.08576","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:14dc123f9a9f6edf156a58aefa8232e74f1a364ff84f810147671153116c8dc5","sha256:777402d0ff57a4b43e50d2b46f93e4b07e8ceb2f7c37f6a3d689bdc33a80e37d"],"state_sha256":"7115e80f30edf101e7ff7de1d6fcbb0abd05a55aee59effb1023b7af22eaac15"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9B73x+NEzsjnXyzmbS+etSHZkHd2jijFccJhocPC7UDEOE5+tvXfqxW9JN9D9YjxW9S8OEHyyiqlI9bqDXoNDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T21:27:03.297029Z","bundle_sha256":"22d8dfe50ddc697795d94069fa6993284d033ba33cd6c6b241c099192444490e"}}