{"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"}