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In this article we focus our attention on the following questions:\n  $1.$ Under what conditions closedness of range of $T$ will imply the closedness of range of $T+S$?\n  $2.$ What is the relation between $T^{\\dagger}$ and $(T+S)^{\\dagger}$?\n  $3.$ What is the relation between $T^{\\dagger}$ and $S^{\\dagger}$?."},"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":"1510.01534","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-10-06T11:35:22Z","cross_cats_sorted":[],"title_canon_sha256":"2820e251d01bced11801aa1c1581d0658ef6ada061c6b4c5573b63b274f67856","abstract_canon_sha256":"33d38890f595c1fbf595219b256f8735b5d60ef161ebd1f8ca7ae71c9e46b0e9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:50.511332Z","signature_b64":"0egJch80MO+AaWezhRZhSNLDG9BvjHWO//h3PX+9mlRnt+gVd7+V2sd3ASFJcAfGh/C+IQ/heFzwTVtntKeqBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a414a3ac042f6dbac8e034a67f77f36d6cd02388f1f7c664e8dd927024b83995","last_reissued_at":"2026-05-18T01:30:50.510694Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:50.510694Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Perturbation of closed range operators and Moore-Penrose inverse","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"G. 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In this article we focus our attention on the following questions:\n  $1.$ Under what conditions closedness of range of $T$ will imply the closedness of range of $T+S$?\n  $2.$ What is the relation between $T^{\\dagger}$ and $(T+S)^{\\dagger}$?\n  $3.$ What is the relation between $T^{\\dagger}$ and $S^{\\dagger}$?."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.01534","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":"1510.01534","created_at":"2026-05-18T01:30:50.510787+00:00"},{"alias_kind":"arxiv_version","alias_value":"1510.01534v2","created_at":"2026-05-18T01:30:50.510787+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.01534","created_at":"2026-05-18T01:30:50.510787+00:00"},{"alias_kind":"pith_short_12","alias_value":"UQKKHLAEF5W3","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"UQKKHLAEF5W3VSHA","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"UQKKHLAE","created_at":"2026-05-18T12:29:44.643036+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/UQKKHLAEF5W3VSHAGSTH657TNV","json":"https://pith.science/pith/UQKKHLAEF5W3VSHAGSTH657TNV.json","graph_json":"https://pith.science/api/pith-number/UQKKHLAEF5W3VSHAGSTH657TNV/graph.json","events_json":"https://pith.science/api/pith-number/UQKKHLAEF5W3VSHAGSTH657TNV/events.json","paper":"https://pith.science/paper/UQKKHLAE"},"agent_actions":{"view_html":"https://pith.science/pith/UQKKHLAEF5W3VSHAGSTH657TNV","download_json":"https://pith.science/pith/UQKKHLAEF5W3VSHAGSTH657TNV.json","view_paper":"https://pith.science/paper/UQKKHLAE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1510.01534&json=true","fetch_graph":"https://pith.science/api/pith-number/UQKKHLAEF5W3VSHAGSTH657TNV/graph.json","fetch_events":"https://pith.science/api/pith-number/UQKKHLAEF5W3VSHAGSTH657TNV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UQKKHLAEF5W3VSHAGSTH657TNV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UQKKHLAEF5W3VSHAGSTH657TNV/action/storage_attestation","attest_author":"https://pith.science/pith/UQKKHLAEF5W3VSHAGSTH657TNV/action/author_attestation","sign_citation":"https://pith.science/pith/UQKKHLAEF5W3VSHAGSTH657TNV/action/citation_signature","submit_replication":"https://pith.science/pith/UQKKHLAEF5W3VSHAGSTH657TNV/action/replication_record"}},"created_at":"2026-05-18T01:30:50.510787+00:00","updated_at":"2026-05-18T01:30:50.510787+00:00"}