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In particular, we prove that if $A_{1},A_{2},...,A_{n}\\in {\\mathbb B}({\\mathscr H})$, then \\[|||A_{1}A_{2}^{*}+A_{2}A_{3}^{*}+...+A_{n}A_{1}^{*}|||\\leq|||\\sum_{i=1}^{n}A_{i}A_{i}^{*}|||,\\] for all unitarily invariant norms.  We also show that if $A_{1},A_{2},A_{3},A_{4}$ are projections in ${\\mathbb B}({\\mathscr H})$, then &&|||(\\sum_{i=1}^{4}(-1)^{i+1}A_{i})\\oplus0\\oplus0\\oplus0|||&\\leq&|||(A_{1}+|A_{3}A_{1}|)\\oplus (A_{2}+|A_{4}A_{2}|)\\oplus(A_{3}+|A_{1}A_{3}|)\\oplus(A_{4}+|A_{2}A_{4}|)||| for any unitarily invari"},"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":"1101.3895","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-01-20T13:16:01Z","cross_cats_sorted":[],"title_canon_sha256":"987e44eb152929a32ec52a4ef9ebfb9f2fefe63d4ba517a1aa9140a3868cc147","abstract_canon_sha256":"9a589b73486d444dce86b8d776f99b71a3eeff28c4b8c93ebc77ac309d56c237"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:20.863395Z","signature_b64":"TRFdz2j/Yodr/wEz3ldSN0zDM0SKeXAfLiwHKO66uJzvA4648pNC+Dd2oEy9orzz8v+LfvoAtd9yEikfNlv2Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c76b6dbf74b0b3e83636797fd1b174f83c32faeda81c386fbe98848615ffda7e","last_reissued_at":"2026-05-18T04:31:20.862895Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:20.862895Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unitarily invariant norm inequalities for operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"A. Niknam, M. Erfanian Omidvar, M.S. Moslehian","submitted_at":"2011-01-20T13:16:01Z","abstract_excerpt":"We present several operator and norm inequalities for Hilbert space operators. In particular, we prove that if $A_{1},A_{2},...,A_{n}\\in {\\mathbb B}({\\mathscr H})$, then \\[|||A_{1}A_{2}^{*}+A_{2}A_{3}^{*}+...+A_{n}A_{1}^{*}|||\\leq|||\\sum_{i=1}^{n}A_{i}A_{i}^{*}|||,\\] for all unitarily invariant norms.  We also show that if $A_{1},A_{2},A_{3},A_{4}$ are projections in ${\\mathbb B}({\\mathscr H})$, then &&|||(\\sum_{i=1}^{4}(-1)^{i+1}A_{i})\\oplus0\\oplus0\\oplus0|||&\\leq&|||(A_{1}+|A_{3}A_{1}|)\\oplus (A_{2}+|A_{4}A_{2}|)\\oplus(A_{3}+|A_{1}A_{3}|)\\oplus(A_{4}+|A_{2}A_{4}|)||| for any unitarily invari"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.3895","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":"1101.3895","created_at":"2026-05-18T04:31:20.862966+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.3895v1","created_at":"2026-05-18T04:31:20.862966+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.3895","created_at":"2026-05-18T04:31:20.862966+00:00"},{"alias_kind":"pith_short_12","alias_value":"Y5VW3P3UWCZ6","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_16","alias_value":"Y5VW3P3UWCZ6QNRW","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_8","alias_value":"Y5VW3P3U","created_at":"2026-05-18T12:26:47.523578+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/Y5VW3P3UWCZ6QNRWPF75DMLU7A","json":"https://pith.science/pith/Y5VW3P3UWCZ6QNRWPF75DMLU7A.json","graph_json":"https://pith.science/api/pith-number/Y5VW3P3UWCZ6QNRWPF75DMLU7A/graph.json","events_json":"https://pith.science/api/pith-number/Y5VW3P3UWCZ6QNRWPF75DMLU7A/events.json","paper":"https://pith.science/paper/Y5VW3P3U"},"agent_actions":{"view_html":"https://pith.science/pith/Y5VW3P3UWCZ6QNRWPF75DMLU7A","download_json":"https://pith.science/pith/Y5VW3P3UWCZ6QNRWPF75DMLU7A.json","view_paper":"https://pith.science/paper/Y5VW3P3U","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.3895&json=true","fetch_graph":"https://pith.science/api/pith-number/Y5VW3P3UWCZ6QNRWPF75DMLU7A/graph.json","fetch_events":"https://pith.science/api/pith-number/Y5VW3P3UWCZ6QNRWPF75DMLU7A/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Y5VW3P3UWCZ6QNRWPF75DMLU7A/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Y5VW3P3UWCZ6QNRWPF75DMLU7A/action/storage_attestation","attest_author":"https://pith.science/pith/Y5VW3P3UWCZ6QNRWPF75DMLU7A/action/author_attestation","sign_citation":"https://pith.science/pith/Y5VW3P3UWCZ6QNRWPF75DMLU7A/action/citation_signature","submit_replication":"https://pith.science/pith/Y5VW3P3UWCZ6QNRWPF75DMLU7A/action/replication_record"}},"created_at":"2026-05-18T04:31:20.862966+00:00","updated_at":"2026-05-18T04:31:20.862966+00:00"}