{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:Q3JH3F3Z6BOEKZ2ZPLLCCFJ4A7","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":"78358f70e81f68db75cbc8695ee16548b14fc5b58c500db9e03d602159941219","cross_cats_sorted":["math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-09-04T19:29:18Z","title_canon_sha256":"3f14d043c7bc04aa1faa8dfeee3da4b51499dc4e1180d4130ae37c2d9573950d"},"schema_version":"1.0","source":{"id":"1509.01571","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.01571","created_at":"2026-05-18T01:33:26Z"},{"alias_kind":"arxiv_version","alias_value":"1509.01571v2","created_at":"2026-05-18T01:33:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.01571","created_at":"2026-05-18T01:33:26Z"},{"alias_kind":"pith_short_12","alias_value":"Q3JH3F3Z6BOE","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"Q3JH3F3Z6BOEKZ2Z","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"Q3JH3F3Z","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:94d35e27993a870a4a8825dd9de6f7869099565003302dbeb0c733b8ea0cffc0","target":"graph","created_at":"2026-05-18T01:33:26Z","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":"Let $B$ be a bounded self-adjoint operator and let $A$ be a nonnegative self-adjoint unbounded operator. It is shown that if $BA$ is normal, it must be self-adjoint and so must be $AB$. Commutativity is necessary and sufficient for this result. If $AB$ is normal, it must be self-adjoint and $BA$ is essentially self-adjoint. Although the two problems seem to be alike, two different and quite interesting approaches are used to tackle each one of them.","authors_text":"K. Gustafson, M. H. Mortad","cross_cats":["math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-09-04T19:29:18Z","title":"Conditions Implying Commutativity of Unbounded Self-adjoint Operators and Related Topics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.01571","kind":"arxiv","version":2},"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":"sha256:21499884c2e51bccc9df2e799a4f409b33ac7334a62815520122a319c0286511","target":"record","created_at":"2026-05-18T01:33:26Z","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":"78358f70e81f68db75cbc8695ee16548b14fc5b58c500db9e03d602159941219","cross_cats_sorted":["math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-09-04T19:29:18Z","title_canon_sha256":"3f14d043c7bc04aa1faa8dfeee3da4b51499dc4e1180d4130ae37c2d9573950d"},"schema_version":"1.0","source":{"id":"1509.01571","kind":"arxiv","version":2}},"canonical_sha256":"86d27d9779f05c4567597ad621153c07fa8bc7dd5f2656bc0544929a436a02e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"86d27d9779f05c4567597ad621153c07fa8bc7dd5f2656bc0544929a436a02e5","first_computed_at":"2026-05-18T01:33:26.130902Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:33:26.130902Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DtcERRyr7wEU9+GM9l/xWbPbUwJ74SfZWbkIF9DzPAEAuOjPYYrpLbHE8Lpk42egnUB3VyaB6wkVs576ayc3CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:33:26.131558Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.01571","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:21499884c2e51bccc9df2e799a4f409b33ac7334a62815520122a319c0286511","sha256:94d35e27993a870a4a8825dd9de6f7869099565003302dbeb0c733b8ea0cffc0"],"state_sha256":"43a42361356ef4ae3853ea163ff87afe3b51c1e81be4c4e03a101fa9dfed4428"}