{"paper":{"title":"A functional model for pure $\\Gamma$-contractions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Sourav Pal, Tirthankar Bhattacharyya","submitted_at":"2012-02-17T05:12:04Z","abstract_excerpt":"A pair of commuting operators $(S,P)$ defined on a Hilbert space $\\mathcal H$ for which the closed symmetrized bidisc $$ \\Gamma= \\{(z_1+z_2,z_1z_2):: |z_1|\\leq 1,\\, |z_2|\\leq 1 \\}\\subseteq \\mathbb C^2, $$ is a spectral set is called a $\\Gamma$-contraction in the literature. A $\\Gamma$-contraction $(S,P)$ is said to be pure if $P$ is a pure contraction, i.e, ${P^*}^n \\rightarrow 0$ strongly as $n \\rightarrow \\infty $. Here we construct a functional model and produce a complete unitary invariant for a pure $\\Gamma$-contraction. The key ingredient in these constructions is an operator, which is t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.3841","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"}