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For $n\\ge 3$, we show that if $\\theta \\in H^\\infty(\\mathbb{D}^n)$ is an inner function, then the $n$-tuple of commuting operators $(C_{z_1}, \\ldots, C_{z_n})$ on the Beurling type quotient module $\\mathcal{Q}_{\\theta}$ is not essentially normal, where \\[\\mathcal{Q}_{\\theta} = H^2(\\mathbb{D}^n)/ \\theta H^2(\\mathbb{D}^n) \\quad \\mbox{and} \\quad C_{z_j} = P_{\\mathcal{Q}_{\\theta}} M_{z_j}|_{\\mathcal{Q}_{\\theta}}\\quad (j = 1, \\ldots, n).\\] Rudin's quotient m"},"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":"1410.5633","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-10-21T12:23:10Z","cross_cats_sorted":["math.CV","math.OA"],"title_canon_sha256":"97b316b84a7a605b6c485a718505f1c2a13f9b81aa5232686ab6d062c12858ea","abstract_canon_sha256":"af9db71d49bdb4ca937eaffaa735a3f2f2bfe33440d8837b5ae49e3ba7e8d8ab"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:46.263714Z","signature_b64":"yAq2AO9DyVuDneE9oHCVbae6tckbHOTfyiBXN5dqSyLPdBQVyOgCK9sYnNRCmnCJhQixPTtb3SX9y8x+9QG6DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7f9efaa4ff77e3c3c6fec99cae20198ef3a41df17a740fec4b15e92d544cc0ff","last_reissued_at":"2026-05-18T00:16:46.263121Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:46.263121Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Quotient modules of $H^2(\\mathbb{D}^n)$: Essential Normality and Boundary Representations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV","math.OA"],"primary_cat":"math.FA","authors_text":"B. Krishna Das, Jaydeb Sarkar, Sushil Gorai","submitted_at":"2014-10-21T12:23:10Z","abstract_excerpt":"Let $\\mathbb{D}^n$ be the open unit polydisc in $\\mathbb{C}^n$, $n \\geq 1$, and let $H^2(\\mathbb{D}^n)$ be the Hardy space over $\\mathbb{D}^n$. For $n\\ge 3$, we show that if $\\theta \\in H^\\infty(\\mathbb{D}^n)$ is an inner function, then the $n$-tuple of commuting operators $(C_{z_1}, \\ldots, C_{z_n})$ on the Beurling type quotient module $\\mathcal{Q}_{\\theta}$ is not essentially normal, where \\[\\mathcal{Q}_{\\theta} = H^2(\\mathbb{D}^n)/ \\theta H^2(\\mathbb{D}^n) \\quad \\mbox{and} \\quad C_{z_j} = P_{\\mathcal{Q}_{\\theta}} M_{z_j}|_{\\mathcal{Q}_{\\theta}}\\quad (j = 1, \\ldots, n).\\] Rudin's quotient m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.5633","kind":"arxiv","version":4},"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":"1410.5633","created_at":"2026-05-18T00:16:46.263205+00:00"},{"alias_kind":"arxiv_version","alias_value":"1410.5633v4","created_at":"2026-05-18T00:16:46.263205+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1410.5633","created_at":"2026-05-18T00:16:46.263205+00:00"},{"alias_kind":"pith_short_12","alias_value":"P6PPVJH7O7R4","created_at":"2026-05-18T12:28:43.426989+00:00"},{"alias_kind":"pith_short_16","alias_value":"P6PPVJH7O7R4HRX6","created_at":"2026-05-18T12:28:43.426989+00:00"},{"alias_kind":"pith_short_8","alias_value":"P6PPVJH7","created_at":"2026-05-18T12:28:43.426989+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/P6PPVJH7O7R4HRX6ZGOK4IAZR3","json":"https://pith.science/pith/P6PPVJH7O7R4HRX6ZGOK4IAZR3.json","graph_json":"https://pith.science/api/pith-number/P6PPVJH7O7R4HRX6ZGOK4IAZR3/graph.json","events_json":"https://pith.science/api/pith-number/P6PPVJH7O7R4HRX6ZGOK4IAZR3/events.json","paper":"https://pith.science/paper/P6PPVJH7"},"agent_actions":{"view_html":"https://pith.science/pith/P6PPVJH7O7R4HRX6ZGOK4IAZR3","download_json":"https://pith.science/pith/P6PPVJH7O7R4HRX6ZGOK4IAZR3.json","view_paper":"https://pith.science/paper/P6PPVJH7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1410.5633&json=true","fetch_graph":"https://pith.science/api/pith-number/P6PPVJH7O7R4HRX6ZGOK4IAZR3/graph.json","fetch_events":"https://pith.science/api/pith-number/P6PPVJH7O7R4HRX6ZGOK4IAZR3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/P6PPVJH7O7R4HRX6ZGOK4IAZR3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/P6PPVJH7O7R4HRX6ZGOK4IAZR3/action/storage_attestation","attest_author":"https://pith.science/pith/P6PPVJH7O7R4HRX6ZGOK4IAZR3/action/author_attestation","sign_citation":"https://pith.science/pith/P6PPVJH7O7R4HRX6ZGOK4IAZR3/action/citation_signature","submit_replication":"https://pith.science/pith/P6PPVJH7O7R4HRX6ZGOK4IAZR3/action/replication_record"}},"created_at":"2026-05-18T00:16:46.263205+00:00","updated_at":"2026-05-18T00:16:46.263205+00:00"}