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We show that if $e_1, ..., e_m$ are vectors in a Hilbert module ${\\mathfrak X}$ over a $C^*$-algebra ${\\mathfrak A}$ with unit 1 such that $<e_i,e_j>=0 (1\\leq i\\neq j \\leq m)$ and $\\|e_i\\|=1 (1\\leq i\\leq m)$, and also $r_k,\\rho_k\\in\\Bbb{R} (1\\leq k\\leq m)$ and $x_1, ..., x_n\\in {\\mathfrak X}$ satisfy $$0\\leq r_k^2 \\|x_j\\|\\leq {Re}< r_ke_k,x_j> ,\\quad0\\leq \\rho_k^2 \\|x_j\\| \\leq {Im}< \\rho_ke_k,x_j> ,$$ then [\\sum_{k=1}^m(r_k^2+\\rho_k^2)]^{{1/2}}\\sum_{j=1}^n \\|x_j\\|\\leq\\|\\sum_{j=1}^nx_j\\|, and the equality holds i"},"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":"0911.2751","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-11-14T07:57:58Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"f1dee0ed0bf3858078eda76139cf9468a501de3d973af03280000d7df6e9c427","abstract_canon_sha256":"6d346a35986fb60a0c9e23eab71379765e4f20950b01588fee4347136e5fe31a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:59:38.606889Z","signature_b64":"DqkyC3XBzsvGkvegYooKDP5c5LPjK6Dxat/fqyQIUtT596a9hTADS/6oQJMi4oqvH1AgrMX3PeB8rmuE56tBCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"615697f3c52af0c6dd0b100072899064a8180b4fbea8cd89771f3f25bcb0a3c7","last_reissued_at":"2026-05-18T03:59:38.606176Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:59:38.606176Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reverse triangle inequality in Hilbert $C^*$-modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"H. Mahyar, M. Khosravi, M.S. Moslehian","submitted_at":"2009-11-14T07:57:58Z","abstract_excerpt":"We prove several versions of reverse triangle inequality in Hilbert $C^*$-modules. We show that if $e_1, ..., e_m$ are vectors in a Hilbert module ${\\mathfrak X}$ over a $C^*$-algebra ${\\mathfrak A}$ with unit 1 such that $<e_i,e_j>=0 (1\\leq i\\neq j \\leq m)$ and $\\|e_i\\|=1 (1\\leq i\\leq m)$, and also $r_k,\\rho_k\\in\\Bbb{R} (1\\leq k\\leq m)$ and $x_1, ..., x_n\\in {\\mathfrak X}$ satisfy $$0\\leq r_k^2 \\|x_j\\|\\leq {Re}< r_ke_k,x_j> ,\\quad0\\leq \\rho_k^2 \\|x_j\\| \\leq {Im}< \\rho_ke_k,x_j> ,$$ then [\\sum_{k=1}^m(r_k^2+\\rho_k^2)]^{{1/2}}\\sum_{j=1}^n \\|x_j\\|\\leq\\|\\sum_{j=1}^nx_j\\|, and the equality holds i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.2751","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":"0911.2751","created_at":"2026-05-18T03:59:38.606282+00:00"},{"alias_kind":"arxiv_version","alias_value":"0911.2751v1","created_at":"2026-05-18T03:59:38.606282+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0911.2751","created_at":"2026-05-18T03:59:38.606282+00:00"},{"alias_kind":"pith_short_12","alias_value":"MFLJP46FFLYM","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_16","alias_value":"MFLJP46FFLYMNXIL","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_8","alias_value":"MFLJP46F","created_at":"2026-05-18T12:26:00.592388+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/MFLJP46FFLYMNXILCAAHFCMQMS","json":"https://pith.science/pith/MFLJP46FFLYMNXILCAAHFCMQMS.json","graph_json":"https://pith.science/api/pith-number/MFLJP46FFLYMNXILCAAHFCMQMS/graph.json","events_json":"https://pith.science/api/pith-number/MFLJP46FFLYMNXILCAAHFCMQMS/events.json","paper":"https://pith.science/paper/MFLJP46F"},"agent_actions":{"view_html":"https://pith.science/pith/MFLJP46FFLYMNXILCAAHFCMQMS","download_json":"https://pith.science/pith/MFLJP46FFLYMNXILCAAHFCMQMS.json","view_paper":"https://pith.science/paper/MFLJP46F","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0911.2751&json=true","fetch_graph":"https://pith.science/api/pith-number/MFLJP46FFLYMNXILCAAHFCMQMS/graph.json","fetch_events":"https://pith.science/api/pith-number/MFLJP46FFLYMNXILCAAHFCMQMS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MFLJP46FFLYMNXILCAAHFCMQMS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MFLJP46FFLYMNXILCAAHFCMQMS/action/storage_attestation","attest_author":"https://pith.science/pith/MFLJP46FFLYMNXILCAAHFCMQMS/action/author_attestation","sign_citation":"https://pith.science/pith/MFLJP46FFLYMNXILCAAHFCMQMS/action/citation_signature","submit_replication":"https://pith.science/pith/MFLJP46FFLYMNXILCAAHFCMQMS/action/replication_record"}},"created_at":"2026-05-18T03:59:38.606282+00:00","updated_at":"2026-05-18T03:59:38.606282+00:00"}