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We will give characterizations of matrix and operator monotonicity by the following generalized Powers-St\\ormer inequality:\n  $$ \\varphi(A + B) - \\varphi(|A - B|) \\leq 2\\varphi(f(A)^1/2g(B)f(A)^1/2), $$ whenever $A, B$ are positive invertible operators in $B(H).$"},"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":"1207.5201","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2012-07-22T06:32:40Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"6fbd7eaab20e0471fa30b2ce03730f8e2a127d647eb365a09fe10aa3de630481","abstract_canon_sha256":"88dffed320d88e5b63e963de70a55a42852d59fa4f0af47b05a0ff90090aa68c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:50:24.164537Z","signature_b64":"+9X5Yi8OocpdEFpTFvYDUIAzMiiSZPn1rERnaPxVVQsEJ9ivPdUfx8BsRzmGqXK4qjF6wyG6mbAOrWbrHHqgCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"182a820ef05a29397305d2bc79ec5aa3735ad944b1a24c9027e76b7bb9006d7f","last_reissued_at":"2026-05-18T03:50:24.163855Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:50:24.163855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Characterization of the monotonicity by the inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Dinh Trung Hoa, Hiroyuki Osaka, Jun Tomiyama","submitted_at":"2012-07-22T06:32:40Z","abstract_excerpt":"Let $\\varphi$ be a normal state on the algebra $B(H)$ of all bounded operators on a Hilbert space $H$, $f$ a strictly positive, continuous function on $(0, \\infty)$, and let $g$ be a function on $(0, \\infty)$ defined by $g(t) = \\frac{t}{f(t)}$. We will give characterizations of matrix and operator monotonicity by the following generalized Powers-St\\ormer inequality:\n  $$ \\varphi(A + B) - \\varphi(|A - B|) \\leq 2\\varphi(f(A)^1/2g(B)f(A)^1/2), $$ whenever $A, B$ are positive invertible operators in $B(H).$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.5201","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":"1207.5201","created_at":"2026-05-18T03:50:24.163969+00:00"},{"alias_kind":"arxiv_version","alias_value":"1207.5201v1","created_at":"2026-05-18T03:50:24.163969+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.5201","created_at":"2026-05-18T03:50:24.163969+00:00"},{"alias_kind":"pith_short_12","alias_value":"DAVIEDXQLIUT","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_16","alias_value":"DAVIEDXQLIUTS4YF","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_8","alias_value":"DAVIEDXQ","created_at":"2026-05-18T12:27:01.376967+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/DAVIEDXQLIUTS4YF2K6HT3C2UN","json":"https://pith.science/pith/DAVIEDXQLIUTS4YF2K6HT3C2UN.json","graph_json":"https://pith.science/api/pith-number/DAVIEDXQLIUTS4YF2K6HT3C2UN/graph.json","events_json":"https://pith.science/api/pith-number/DAVIEDXQLIUTS4YF2K6HT3C2UN/events.json","paper":"https://pith.science/paper/DAVIEDXQ"},"agent_actions":{"view_html":"https://pith.science/pith/DAVIEDXQLIUTS4YF2K6HT3C2UN","download_json":"https://pith.science/pith/DAVIEDXQLIUTS4YF2K6HT3C2UN.json","view_paper":"https://pith.science/paper/DAVIEDXQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1207.5201&json=true","fetch_graph":"https://pith.science/api/pith-number/DAVIEDXQLIUTS4YF2K6HT3C2UN/graph.json","fetch_events":"https://pith.science/api/pith-number/DAVIEDXQLIUTS4YF2K6HT3C2UN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DAVIEDXQLIUTS4YF2K6HT3C2UN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DAVIEDXQLIUTS4YF2K6HT3C2UN/action/storage_attestation","attest_author":"https://pith.science/pith/DAVIEDXQLIUTS4YF2K6HT3C2UN/action/author_attestation","sign_citation":"https://pith.science/pith/DAVIEDXQLIUTS4YF2K6HT3C2UN/action/citation_signature","submit_replication":"https://pith.science/pith/DAVIEDXQLIUTS4YF2K6HT3C2UN/action/replication_record"}},"created_at":"2026-05-18T03:50:24.163969+00:00","updated_at":"2026-05-18T03:50:24.163969+00:00"}