{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:WSOATGYF3FANHYKSVTHTOLI4HZ","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":"f4fa0747836d41e2fd8b8bc6d8f33f9cf3c93a827d1ea12f74d43f82f50043e6","cross_cats_sorted":["math.CV","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-07-20T04:13:57Z","title_canon_sha256":"aec6dda02fdbf1f1dd281dbe7c8fb7e0ff0fb7e4f6bf12512baa0a5d7280ece7"},"schema_version":"1.0","source":{"id":"1607.05815","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.05815","created_at":"2026-05-18T00:32:51Z"},{"alias_kind":"arxiv_version","alias_value":"1607.05815v3","created_at":"2026-05-18T00:32:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.05815","created_at":"2026-05-18T00:32:51Z"},{"alias_kind":"pith_short_12","alias_value":"WSOATGYF3FAN","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"WSOATGYF3FANHYKS","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"WSOATGYF","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:b59e6327d11e3845308f945ecb00ad74f5848b8e25200e77a7e952d0e56e424f","target":"graph","created_at":"2026-05-18T00:32:51Z","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":"The celebrated theorem of Berger, Coburn and Lebow on pairs of commuting isometries can be formulated as follows: a pure isometry $V$ on a Hilbert space $\\mathcal{H}$ is a product of two commuting isometries $V_1$ and $V_2$ in $\\mathcal{B}(\\mathcal{H})$ if and only if there exists a Hilbert space $\\mathcal{E}$, a unitary $U$ in $\\mathcal{B}(\\mathcal{E})$ and an orthogonal projection $P$ in $\\mathcal{B}(\\mathcal{E})$ such that $(V, V_1, V_2)$ and $(M_z, M_{\\Phi}, M_{\\Psi})$ on $H^2_{\\mathcal{E}}(\\mathbb{D})$ are unitarily equivalent, where \\[ \\Phi(z)=(P+zP^{\\perp})U^*\\;\\text{and}\\; \\Psi(z)=U(P^","authors_text":"B. Krishna Das, Jaydeb Sarkar, Srijan Sarkar","cross_cats":["math.CV","math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-07-20T04:13:57Z","title":"Factorizations of Contractions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.05815","kind":"arxiv","version":3},"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:e778054dc676412b1a1086728e8b81e3733d4a5c19929333c396207755e89da2","target":"record","created_at":"2026-05-18T00:32:51Z","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":"f4fa0747836d41e2fd8b8bc6d8f33f9cf3c93a827d1ea12f74d43f82f50043e6","cross_cats_sorted":["math.CV","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-07-20T04:13:57Z","title_canon_sha256":"aec6dda02fdbf1f1dd281dbe7c8fb7e0ff0fb7e4f6bf12512baa0a5d7280ece7"},"schema_version":"1.0","source":{"id":"1607.05815","kind":"arxiv","version":3}},"canonical_sha256":"b49c099b05d940d3e152accf372d1c3e75f8ce2fc022886695cdaa85d759cd78","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b49c099b05d940d3e152accf372d1c3e75f8ce2fc022886695cdaa85d759cd78","first_computed_at":"2026-05-18T00:32:51.220256Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:51.220256Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uYC9LMx2Fhpszz+6+XHSzAgOgw9GpdgZXPCOmqYDD3T44vYVAJUUrOmgMrt+6SHg+NXrjRTo6VeNhy5uxVc0CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:51.220760Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.05815","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e778054dc676412b1a1086728e8b81e3733d4a5c19929333c396207755e89da2","sha256:b59e6327d11e3845308f945ecb00ad74f5848b8e25200e77a7e952d0e56e424f"],"state_sha256":"f115e0ad82ef4ded86b52036fed92089513470418982e37e83ddd49a3a8a02ca"}