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Varopoulos and the constant $C_2(n)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Rajeev Gupta, Samya Kumar Ray","submitted_at":"2016-11-21T11:17:28Z","abstract_excerpt":"Let $\\mathbb C_k[Z_1,\\ldots, Z_n]$ denote the set of all polynomials of degree at most $k$ in $n$ complex variables and $\\mathscr{C}_n$ denote the set of all $n$ - tuple $\\boldsymbol T=(T_1,\\ldots,T_n)$ of commuting contractions on some Hilbert space $\\mathbb{H}.$ The interesting inequality $$K_{G}^{\\mathbb C}\\leq \\lim_{n\\to \\infty}C_2(n)\\leq 2 K^\\mathbb C_G,$$ where \\[C_k(n)=\\sup\\big\\{\\|p(\\boldsymbol T)\\|:\\|p\\|_{\\mathbb D^n,\\infty}\\leq 1, p\\in \\mathbb C_k[Z_1,\\ldots,Z_n],\\boldsymbol T\\in\\mathscr{C}_n \\big\\}\\] and $K_{G}^{\\mathbb C}$ is the complex Grothendieck constant, is due to Varopoulos. 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