{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:3FHLK7MDNHZGIDCEW6VJW4NBID","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":"f511ead108a47a865a364fd7715fe488a4767b7d73836aa094d5cc3715a583da","cross_cats_sorted":["math.CV","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-02-04T10:45:22Z","title_canon_sha256":"89aa7b27c9d00021bbbe6bcc96d1f1550bf6b91de97f09f1132de07cacba411a"},"schema_version":"1.0","source":{"id":"1702.01263","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.01263","created_at":"2026-05-18T00:45:30Z"},{"alias_kind":"arxiv_version","alias_value":"1702.01263v2","created_at":"2026-05-18T00:45:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.01263","created_at":"2026-05-18T00:45:30Z"},{"alias_kind":"pith_short_12","alias_value":"3FHLK7MDNHZG","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3FHLK7MDNHZGIDCE","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3FHLK7MD","created_at":"2026-05-18T12:30:58Z"}],"graph_snapshots":[{"event_id":"sha256:6afed71cd94d5a15ef1f9c1b9e2af455fa2caa0639a3091326cd847fd483bd4c","target":"graph","created_at":"2026-05-18T00:45:30Z","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":"We prove that the rank of a non-trivial co-doubly commuting submodule is $2$. More precisely, let $\\varphi, \\psi \\in H^\\infty(\\mathbb{D})$ be two inner functions. If $\\mathcal{Q}_{\\varphi} = H^2(\\mathbb{D})/ \\varphi H^2(\\mathbb{D})$ and $\\mathcal{Q}_{\\psi} = H^2(\\mathbb{D})/ \\psi H^2(\\mathbb{D})$, then \\[ \\mbox{rank~}(\\mathcal{Q}_{\\varphi} \\otimes \\mathcal{Q}_{\\psi})^\\perp = 2. \\] An immediate consequence is the following: Let $\\mathcal{S}$ be a co-doubly commuting submodule of $H^2(\\mathbb{D}^2)$. Then $\\mbox{rank~} \\mathcal{S} = 1$ if and only if $\\mathcal{S} = \\Phi H^2(\\mathbb{D}^2)$ for so","authors_text":"Arup Chattopadhyay, B. Krishna Das, Jaydeb Sarkar","cross_cats":["math.CV","math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-02-04T10:45:22Z","title":"Rank of a co-doubly commuting submodule is 2"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.01263","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":"sha256:6767e208919de3ab4f900dfbac15d3da7fbecd615f30de40c21089d73d991922","target":"record","created_at":"2026-05-18T00:45:30Z","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":"f511ead108a47a865a364fd7715fe488a4767b7d73836aa094d5cc3715a583da","cross_cats_sorted":["math.CV","math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-02-04T10:45:22Z","title_canon_sha256":"89aa7b27c9d00021bbbe6bcc96d1f1550bf6b91de97f09f1132de07cacba411a"},"schema_version":"1.0","source":{"id":"1702.01263","kind":"arxiv","version":2}},"canonical_sha256":"d94eb57d8369f2640c44b7aa9b71a140d96fa2e1a0f181a09a530b57755257e1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d94eb57d8369f2640c44b7aa9b71a140d96fa2e1a0f181a09a530b57755257e1","first_computed_at":"2026-05-18T00:45:30.071165Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:30.071165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uet4ilZYwbl3Rlmn/aeDyqx9ZgqJoeodxr7Z9h4Z0LFooWsqQ3Rmeu9bTkNm27x21NxMZA25Zl4aNDYVeubVDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:30.071550Z","signed_message":"canonical_sha256_bytes"},"source_id":"1702.01263","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6767e208919de3ab4f900dfbac15d3da7fbecd615f30de40c21089d73d991922","sha256:6afed71cd94d5a15ef1f9c1b9e2af455fa2caa0639a3091326cd847fd483bd4c"],"state_sha256":"ba166078722f84ba0f682909d1dd753efb338259299216605a372b1f459b2880"}