{"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"}