{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2007:SLU7CJ7N4G2O3KT6WEVGYDDKQR","short_pith_number":"pith:SLU7CJ7N","schema_version":"1.0","canonical_sha256":"92e9f127ede1b4edaa7eb12a6c0c6a8459d848a9263332a220850db40eced716","source":{"kind":"arxiv","id":"math/0701129","version":2},"attestation_state":"computed","paper":{"title":"On the Araki-Lieb-Thirring inequality","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Koenraad M.R. Audenaert","submitted_at":"2007-01-04T08:55:15Z","abstract_excerpt":"We prove an inequality that complements the famous Araki-Lieb-Thirring (ALT) inequality for positive matrices $A$ and $B$, by giving a lower bound on the quantity $\\trace[A^r B^r A^r]^q$ in terms of $\\trace[ABA]^{rq}$ for $0\\le r\\le 1$ and $q\\ge0$, whereas the ALT inequality gives an upper bound. The bound contains certain norms of $A$ and $B$ as additional ingredients and is therefore of a different nature than the Kantorovich type inequality obtained by Bourin (\\textit{Math. Inequal. Appl.} \\textbf{8}(2005) pp. 373--378) and others. Secondly, we also prove a generalisation of the ALT inequal"},"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":"math/0701129","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"2007-01-04T08:55:15Z","cross_cats_sorted":[],"title_canon_sha256":"d98c2567634ea905fb74166701aabac05ba42552683a2d87f90571ae95c49f12","abstract_canon_sha256":"fa00975a2fa6addf75d765178e09f27a5325c70bad7d90a41d9a7e4ce98ed55f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:19:03.439757Z","signature_b64":"ZJbkHId55S7JrbMndLedY0k/LicnLUPZ0tSdk3cNS3tDIYbblIkQX3diBY4gFQE73sroGY8MtVasUsUf7e73BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"92e9f127ede1b4edaa7eb12a6c0c6a8459d848a9263332a220850db40eced716","last_reissued_at":"2026-05-18T04:19:03.439259Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:19:03.439259Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Araki-Lieb-Thirring inequality","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Koenraad M.R. Audenaert","submitted_at":"2007-01-04T08:55:15Z","abstract_excerpt":"We prove an inequality that complements the famous Araki-Lieb-Thirring (ALT) inequality for positive matrices $A$ and $B$, by giving a lower bound on the quantity $\\trace[A^r B^r A^r]^q$ in terms of $\\trace[ABA]^{rq}$ for $0\\le r\\le 1$ and $q\\ge0$, whereas the ALT inequality gives an upper bound. The bound contains certain norms of $A$ and $B$ as additional ingredients and is therefore of a different nature than the Kantorovich type inequality obtained by Bourin (\\textit{Math. Inequal. Appl.} \\textbf{8}(2005) pp. 373--378) and others. Secondly, we also prove a generalisation of the ALT inequal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0701129","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0701129","created_at":"2026-05-18T04:19:03.439346+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0701129v2","created_at":"2026-05-18T04:19:03.439346+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0701129","created_at":"2026-05-18T04:19:03.439346+00:00"},{"alias_kind":"pith_short_12","alias_value":"SLU7CJ7N4G2O","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_16","alias_value":"SLU7CJ7N4G2O3KT6","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_8","alias_value":"SLU7CJ7N","created_at":"2026-05-18T12:25:56.245647+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.07257","citing_title":"Quantifying and detecting quantum-state texture","ref_index":24,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SLU7CJ7N4G2O3KT6WEVGYDDKQR","json":"https://pith.science/pith/SLU7CJ7N4G2O3KT6WEVGYDDKQR.json","graph_json":"https://pith.science/api/pith-number/SLU7CJ7N4G2O3KT6WEVGYDDKQR/graph.json","events_json":"https://pith.science/api/pith-number/SLU7CJ7N4G2O3KT6WEVGYDDKQR/events.json","paper":"https://pith.science/paper/SLU7CJ7N"},"agent_actions":{"view_html":"https://pith.science/pith/SLU7CJ7N4G2O3KT6WEVGYDDKQR","download_json":"https://pith.science/pith/SLU7CJ7N4G2O3KT6WEVGYDDKQR.json","view_paper":"https://pith.science/paper/SLU7CJ7N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0701129&json=true","fetch_graph":"https://pith.science/api/pith-number/SLU7CJ7N4G2O3KT6WEVGYDDKQR/graph.json","fetch_events":"https://pith.science/api/pith-number/SLU7CJ7N4G2O3KT6WEVGYDDKQR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SLU7CJ7N4G2O3KT6WEVGYDDKQR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SLU7CJ7N4G2O3KT6WEVGYDDKQR/action/storage_attestation","attest_author":"https://pith.science/pith/SLU7CJ7N4G2O3KT6WEVGYDDKQR/action/author_attestation","sign_citation":"https://pith.science/pith/SLU7CJ7N4G2O3KT6WEVGYDDKQR/action/citation_signature","submit_replication":"https://pith.science/pith/SLU7CJ7N4G2O3KT6WEVGYDDKQR/action/replication_record"}},"created_at":"2026-05-18T04:19:03.439346+00:00","updated_at":"2026-05-18T04:19:03.439346+00:00"}