{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:O5IXXJBNVLCBVITMTCWXNEBMQ5","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":"6e848d1930ed7e28b77751639cd90be23166b40d07286a7242337f6dc4579b1b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-01-04T18:36:50Z","title_canon_sha256":"177e0c2547305887c3ea01d90681f702827e4694b1435dc607dcc04c1ad699e3"},"schema_version":"1.0","source":{"id":"1301.0797","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.0797","created_at":"2026-05-18T03:37:17Z"},{"alias_kind":"arxiv_version","alias_value":"1301.0797v1","created_at":"2026-05-18T03:37:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.0797","created_at":"2026-05-18T03:37:17Z"},{"alias_kind":"pith_short_12","alias_value":"O5IXXJBNVLCB","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_16","alias_value":"O5IXXJBNVLCBVITM","created_at":"2026-05-18T12:27:54Z"},{"alias_kind":"pith_short_8","alias_value":"O5IXXJBN","created_at":"2026-05-18T12:27:54Z"}],"graph_snapshots":[{"event_id":"sha256:1a221461b3fb1fceeff9c43f0ab79a1e5d5926ec55b4aebe4f40b27d19972c15","target":"graph","created_at":"2026-05-18T03:37:17Z","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 $X,Y$ be normal bounded operators on a Hilbert space such that $e^X=e^Y$. If the spectra of $X$ and $Y$ are contained in the strip $\\s$ of the complex plane defined by $|\\Im(z)|\\leq \\pi$, we show that $|X|=|Y|$. If $Y$ is only assumed to be bounded, then $|X|Y=Y|X|$. We give a formula for $X-Y$ in terms of spectral projections of $X$ and $Y$ provided that $X,Y$ are normal and $e^X=e^Y$. If $X$ is an unbounded self-adjoint operator, which does not have $(2k+1) \\pi$, $k \\in \\ZZ$, as eigenvalues, and $Y$ is normal with spectrum in $\\s$ satisfying $e^{iX}=e^Y$, then $Y \\in \\{\\, e^{iX} \\, \\}\"$.","authors_text":"Eduardo Chiumiento","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-01-04T18:36:50Z","title":"On normal operator logarithms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.0797","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:c374f474ac45d34c8c630f1b82b3b9ae6d80bb720ee6767a407a7aca9b868c82","target":"record","created_at":"2026-05-18T03:37:17Z","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":"6e848d1930ed7e28b77751639cd90be23166b40d07286a7242337f6dc4579b1b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-01-04T18:36:50Z","title_canon_sha256":"177e0c2547305887c3ea01d90681f702827e4694b1435dc607dcc04c1ad699e3"},"schema_version":"1.0","source":{"id":"1301.0797","kind":"arxiv","version":1}},"canonical_sha256":"77517ba42daac41aa26c98ad76902c876c76660e59661bb62f50a7ad680ce6e8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"77517ba42daac41aa26c98ad76902c876c76660e59661bb62f50a7ad680ce6e8","first_computed_at":"2026-05-18T03:37:17.537981Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:37:17.537981Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nIebOrwRSfr0zHsEgSEcYKCu+kgzPCF/ujJEaTVAQKYWjWB9Po/+91jz94+Y4YahDFxs9P9Q5iNPadkMUNH9BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:37:17.538644Z","signed_message":"canonical_sha256_bytes"},"source_id":"1301.0797","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c374f474ac45d34c8c630f1b82b3b9ae6d80bb720ee6767a407a7aca9b868c82","sha256:1a221461b3fb1fceeff9c43f0ab79a1e5d5926ec55b4aebe4f40b27d19972c15"],"state_sha256":"37c3950f5a934757526a67ccc1e5636ea6d80fd32a23c6954bc2070cfeca49ab"}