{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:ITVWHWXWZFNZWTP6BL4CA5LJVA","short_pith_number":"pith:ITVWHWXW","canonical_record":{"source":{"id":"1310.2036","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-10-08T08:25:11Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"a246a5d026573df5b10afbc1b530317de3097c80c5b916bab6f71325286167b3","abstract_canon_sha256":"1839de08584df16a3c0c54b864f15c959742170665783b6d804f18069a40a257"},"schema_version":"1.0"},"canonical_sha256":"44eb63daf6c95b9b4dfe0af8207569a802e8c5bca4b242d98c5bce66a49e3079","source":{"kind":"arxiv","id":"1310.2036","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.2036","created_at":"2026-05-18T02:46:32Z"},{"alias_kind":"arxiv_version","alias_value":"1310.2036v1","created_at":"2026-05-18T02:46:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.2036","created_at":"2026-05-18T02:46:32Z"},{"alias_kind":"pith_short_12","alias_value":"ITVWHWXWZFNZ","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"ITVWHWXWZFNZWTP6","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"ITVWHWXW","created_at":"2026-05-18T12:27:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:ITVWHWXWZFNZWTP6BL4CA5LJVA","target":"record","payload":{"canonical_record":{"source":{"id":"1310.2036","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-10-08T08:25:11Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"a246a5d026573df5b10afbc1b530317de3097c80c5b916bab6f71325286167b3","abstract_canon_sha256":"1839de08584df16a3c0c54b864f15c959742170665783b6d804f18069a40a257"},"schema_version":"1.0"},"canonical_sha256":"44eb63daf6c95b9b4dfe0af8207569a802e8c5bca4b242d98c5bce66a49e3079","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:46:32.822081Z","signature_b64":"+GxxqDl6m1QS5SsWBho5BBuyyFsMAsgTlLN60JqnzTqZOlfe2iviVKWLEBMW0jQNFZHze/gDJegtLVa66HHcAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"44eb63daf6c95b9b4dfe0af8207569a802e8c5bca4b242d98c5bce66a49e3079","last_reissued_at":"2026-05-18T02:46:32.821698Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:46:32.821698Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1310.2036","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:46:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Weh5u36cKf7LScLK2QfcWfCK4xxehPIDImpevGjIENw2ptvYH/4BysVxoDkqF7dwHKtzghzLCR092gUahuUVDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-19T23:41:52.289605Z"},"content_sha256":"6e7c333bff350e9689575eb5a1cb49b31ed88bf1cb72496a6e67d8124881f841","schema_version":"1.0","event_id":"sha256:6e7c333bff350e9689575eb5a1cb49b31ed88bf1cb72496a6e67d8124881f841"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:ITVWHWXWZFNZWTP6BL4CA5LJVA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Notes on the sin2\\Theta\\ theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.SP","authors_text":"Albrecht Seelmann","submitted_at":"2013-10-08T08:25:11Z","abstract_excerpt":"An analogue of the Davis-Kahan sin2\\Theta\\ theorem from [SIAM J. Numer. Anal. 7 (1970), 1--46] is proved under a general spectral separation condition. This extends the generic sin2\\theta\\ estimates recently shown by Albeverio and Motovilov in [Complex Anal. Oper. Theory 7 (2013), 1389--1416]. The result is applied to the subspace perturbation problem to obtain a bound on the arcsine of the norm of the difference of the spectral projections associated with isolated components of the spectrum of the unperturbed and perturbed operators, respectively."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2036","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:46:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Fv2NvmzIvkgzFrdG0VD9n9fV9d5Gy+bYvljoIuq9X+nr1S18p4jJW+81fc98kFXNIaov7Iu+4Dlk3WTGQ4tvBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-19T23:41:52.289946Z"},"content_sha256":"522e1f2743cd780b897b492246c5901fd90c75feab50b2deb53c90733c1c50f3","schema_version":"1.0","event_id":"sha256:522e1f2743cd780b897b492246c5901fd90c75feab50b2deb53c90733c1c50f3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ITVWHWXWZFNZWTP6BL4CA5LJVA/bundle.json","state_url":"https://pith.science/pith/ITVWHWXWZFNZWTP6BL4CA5LJVA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ITVWHWXWZFNZWTP6BL4CA5LJVA/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-19T23:41:52Z","links":{"resolver":"https://pith.science/pith/ITVWHWXWZFNZWTP6BL4CA5LJVA","bundle":"https://pith.science/pith/ITVWHWXWZFNZWTP6BL4CA5LJVA/bundle.json","state":"https://pith.science/pith/ITVWHWXWZFNZWTP6BL4CA5LJVA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ITVWHWXWZFNZWTP6BL4CA5LJVA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:ITVWHWXWZFNZWTP6BL4CA5LJVA","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1839de08584df16a3c0c54b864f15c959742170665783b6d804f18069a40a257","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-10-08T08:25:11Z","title_canon_sha256":"a246a5d026573df5b10afbc1b530317de3097c80c5b916bab6f71325286167b3"},"schema_version":"1.0","source":{"id":"1310.2036","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.2036","created_at":"2026-05-18T02:46:32Z"},{"alias_kind":"arxiv_version","alias_value":"1310.2036v1","created_at":"2026-05-18T02:46:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.2036","created_at":"2026-05-18T02:46:32Z"},{"alias_kind":"pith_short_12","alias_value":"ITVWHWXWZFNZ","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_16","alias_value":"ITVWHWXWZFNZWTP6","created_at":"2026-05-18T12:27:49Z"},{"alias_kind":"pith_short_8","alias_value":"ITVWHWXW","created_at":"2026-05-18T12:27:49Z"}],"graph_snapshots":[{"event_id":"sha256:522e1f2743cd780b897b492246c5901fd90c75feab50b2deb53c90733c1c50f3","target":"graph","created_at":"2026-05-18T02:46:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"An analogue of the Davis-Kahan sin2\\Theta\\ theorem from [SIAM J. Numer. Anal. 7 (1970), 1--46] is proved under a general spectral separation condition. This extends the generic sin2\\theta\\ estimates recently shown by Albeverio and Motovilov in [Complex Anal. Oper. Theory 7 (2013), 1389--1416]. The result is applied to the subspace perturbation problem to obtain a bound on the arcsine of the norm of the difference of the spectral projections associated with isolated components of the spectrum of the unperturbed and perturbed operators, respectively.","authors_text":"Albrecht Seelmann","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-10-08T08:25:11Z","title":"Notes on the sin2\\Theta\\ theorem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2036","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6e7c333bff350e9689575eb5a1cb49b31ed88bf1cb72496a6e67d8124881f841","target":"record","created_at":"2026-05-18T02:46:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1839de08584df16a3c0c54b864f15c959742170665783b6d804f18069a40a257","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2013-10-08T08:25:11Z","title_canon_sha256":"a246a5d026573df5b10afbc1b530317de3097c80c5b916bab6f71325286167b3"},"schema_version":"1.0","source":{"id":"1310.2036","kind":"arxiv","version":1}},"canonical_sha256":"44eb63daf6c95b9b4dfe0af8207569a802e8c5bca4b242d98c5bce66a49e3079","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"44eb63daf6c95b9b4dfe0af8207569a802e8c5bca4b242d98c5bce66a49e3079","first_computed_at":"2026-05-18T02:46:32.821698Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:46:32.821698Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+GxxqDl6m1QS5SsWBho5BBuyyFsMAsgTlLN60JqnzTqZOlfe2iviVKWLEBMW0jQNFZHze/gDJegtLVa66HHcAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:46:32.822081Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.2036","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6e7c333bff350e9689575eb5a1cb49b31ed88bf1cb72496a6e67d8124881f841","sha256:522e1f2743cd780b897b492246c5901fd90c75feab50b2deb53c90733c1c50f3"],"state_sha256":"88c1da3841c7f6ce20de69148ce66780d8a767dc1ebb0704fcab0dfe73ea561d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Alj9nX9LaPl3154r09Ck3Cd/FS99L4odOnFJKIhNSuC5KznK4BbySDCGMDUPsDQdTSTBwSzpaEeI4JCfUD1TCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-19T23:41:52.291960Z","bundle_sha256":"bf4bfbed3c0e6d9c819b00907d95b440d25d5110721c351408359c9ba811bd1b"}}