{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:3FHLK7MDNHZGIDCEW6VJW4NBID","short_pith_number":"pith:3FHLK7MD","schema_version":"1.0","canonical_sha256":"d94eb57d8369f2640c44b7aa9b71a140d96fa2e1a0f181a09a530b57755257e1","source":{"kind":"arxiv","id":"1702.01263","version":2},"attestation_state":"computed","paper":{"title":"Rank of a co-doubly commuting submodule is 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.OA"],"primary_cat":"math.FA","authors_text":"Arup Chattopadhyay, B. Krishna Das, Jaydeb Sarkar","submitted_at":"2017-02-04T10:45:22Z","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1702.01263","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-02-04T10:45:22Z","cross_cats_sorted":["math.CV","math.OA"],"title_canon_sha256":"89aa7b27c9d00021bbbe6bcc96d1f1550bf6b91de97f09f1132de07cacba411a","abstract_canon_sha256":"f511ead108a47a865a364fd7715fe488a4767b7d73836aa094d5cc3715a583da"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:30.071550Z","signature_b64":"uet4ilZYwbl3Rlmn/aeDyqx9ZgqJoeodxr7Z9h4Z0LFooWsqQ3Rmeu9bTkNm27x21NxMZA25Zl4aNDYVeubVDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d94eb57d8369f2640c44b7aa9b71a140d96fa2e1a0f181a09a530b57755257e1","last_reissued_at":"2026-05-18T00:45:30.071165Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:30.071165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rank of a co-doubly commuting submodule is 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.OA"],"primary_cat":"math.FA","authors_text":"Arup Chattopadhyay, B. Krishna Das, Jaydeb Sarkar","submitted_at":"2017-02-04T10:45:22Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.01263","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1702.01263","created_at":"2026-05-18T00:45:30.071222+00:00"},{"alias_kind":"arxiv_version","alias_value":"1702.01263v2","created_at":"2026-05-18T00:45:30.071222+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.01263","created_at":"2026-05-18T00:45:30.071222+00:00"},{"alias_kind":"pith_short_12","alias_value":"3FHLK7MDNHZG","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_16","alias_value":"3FHLK7MDNHZGIDCE","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_8","alias_value":"3FHLK7MD","created_at":"2026-05-18T12:30:58.224056+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3FHLK7MDNHZGIDCEW6VJW4NBID","json":"https://pith.science/pith/3FHLK7MDNHZGIDCEW6VJW4NBID.json","graph_json":"https://pith.science/api/pith-number/3FHLK7MDNHZGIDCEW6VJW4NBID/graph.json","events_json":"https://pith.science/api/pith-number/3FHLK7MDNHZGIDCEW6VJW4NBID/events.json","paper":"https://pith.science/paper/3FHLK7MD"},"agent_actions":{"view_html":"https://pith.science/pith/3FHLK7MDNHZGIDCEW6VJW4NBID","download_json":"https://pith.science/pith/3FHLK7MDNHZGIDCEW6VJW4NBID.json","view_paper":"https://pith.science/paper/3FHLK7MD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1702.01263&json=true","fetch_graph":"https://pith.science/api/pith-number/3FHLK7MDNHZGIDCEW6VJW4NBID/graph.json","fetch_events":"https://pith.science/api/pith-number/3FHLK7MDNHZGIDCEW6VJW4NBID/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3FHLK7MDNHZGIDCEW6VJW4NBID/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3FHLK7MDNHZGIDCEW6VJW4NBID/action/storage_attestation","attest_author":"https://pith.science/pith/3FHLK7MDNHZGIDCEW6VJW4NBID/action/author_attestation","sign_citation":"https://pith.science/pith/3FHLK7MDNHZGIDCEW6VJW4NBID/action/citation_signature","submit_replication":"https://pith.science/pith/3FHLK7MDNHZGIDCEW6VJW4NBID/action/replication_record"}},"created_at":"2026-05-18T00:45:30.071222+00:00","updated_at":"2026-05-18T00:45:30.071222+00:00"}