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Suppose that V is a bounded self-adjoint operator on H such that ||V||<d/2 and let L=A+V. Denote by P the spectral projection of A associated with the spectral set {\\sigma} and let Q be the spectral projection of L corresponding to the closed ||V||-neighborhood of {\\sigma}. We prove a bound of the form arcsin(||P-Q||)\\leq M(||V||/d), M: [0,1/2)-->R^+, that is essentially stronger than the previously known es"},"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":"1112.0149","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2011-12-01T11:59:52Z","cross_cats_sorted":["math-ph","math.FA","math.MP"],"title_canon_sha256":"c4d149bfd7bf169fd0036a7aba18a91bcb0ec078d4cb7ced8a6095bba7d8a441","abstract_canon_sha256":"43fbbf299ef62d41234fc06e90cc1f1b49d08e192aab1e6bfd4515782950a9e9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:17:57.538073Z","signature_b64":"AHv1ROOyaoXo8jlKbtemYmkgQBcNcASjseVQdHJo1P7hQRn66wbOcUnqueBzEL44XIBR7eq7A5CwEUBTp4nrDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0744aa6c52dbaa2c790a90b6247630d9702e45502a508f008fc5c931f3be08a8","last_reissued_at":"2026-05-18T03:17:57.537418Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:17:57.537418Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharpening the norm bound in the subspace perturbation theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.SP","authors_text":"Alexander K. Motovilov, Sergio Albeverio","submitted_at":"2011-12-01T11:59:52Z","abstract_excerpt":"Let A be a self-adjoint operator on a Hilbert space H. Assume that {\\sigma} is an isolated component of the spectrum of A, i.e. dist({\\sigma},{\\Sigma})=d>0 where {\\Sigma}=spec(A)\\{\\sigma}. Suppose that V is a bounded self-adjoint operator on H such that ||V||<d/2 and let L=A+V. Denote by P the spectral projection of A associated with the spectral set {\\sigma} and let Q be the spectral projection of L corresponding to the closed ||V||-neighborhood of {\\sigma}. We prove a bound of the form arcsin(||P-Q||)\\leq M(||V||/d), M: [0,1/2)-->R^+, that is essentially stronger than the previously known es"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.0149","kind":"arxiv","version":3},"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":"1112.0149","created_at":"2026-05-18T03:17:57.537518+00:00"},{"alias_kind":"arxiv_version","alias_value":"1112.0149v3","created_at":"2026-05-18T03:17:57.537518+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.0149","created_at":"2026-05-18T03:17:57.537518+00:00"},{"alias_kind":"pith_short_12","alias_value":"A5CKU3CS3OVC","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_16","alias_value":"A5CKU3CS3OVCY6IK","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_8","alias_value":"A5CKU3CS","created_at":"2026-05-18T12:26:24.575870+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/A5CKU3CS3OVCY6IKSC3CI5RQ3F","json":"https://pith.science/pith/A5CKU3CS3OVCY6IKSC3CI5RQ3F.json","graph_json":"https://pith.science/api/pith-number/A5CKU3CS3OVCY6IKSC3CI5RQ3F/graph.json","events_json":"https://pith.science/api/pith-number/A5CKU3CS3OVCY6IKSC3CI5RQ3F/events.json","paper":"https://pith.science/paper/A5CKU3CS"},"agent_actions":{"view_html":"https://pith.science/pith/A5CKU3CS3OVCY6IKSC3CI5RQ3F","download_json":"https://pith.science/pith/A5CKU3CS3OVCY6IKSC3CI5RQ3F.json","view_paper":"https://pith.science/paper/A5CKU3CS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1112.0149&json=true","fetch_graph":"https://pith.science/api/pith-number/A5CKU3CS3OVCY6IKSC3CI5RQ3F/graph.json","fetch_events":"https://pith.science/api/pith-number/A5CKU3CS3OVCY6IKSC3CI5RQ3F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A5CKU3CS3OVCY6IKSC3CI5RQ3F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A5CKU3CS3OVCY6IKSC3CI5RQ3F/action/storage_attestation","attest_author":"https://pith.science/pith/A5CKU3CS3OVCY6IKSC3CI5RQ3F/action/author_attestation","sign_citation":"https://pith.science/pith/A5CKU3CS3OVCY6IKSC3CI5RQ3F/action/citation_signature","submit_replication":"https://pith.science/pith/A5CKU3CS3OVCY6IKSC3CI5RQ3F/action/replication_record"}},"created_at":"2026-05-18T03:17:57.537518+00:00","updated_at":"2026-05-18T03:17:57.537518+00:00"}