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Let $\\mbox{dim } \\mbox{ran}(I_{\\mathcal{H}} - T_j T_j^*) < \\infty$, $j = 1, 2$, and let $T_1$ be a pure contraction. Then there exists a variety $V \\subseteq \\overline{\\mathbb{D}}^2$ such that for any polynomial $p \\in \\mathbb{C}[z_1, z_2]$, the inequality \\[ \\|p(T_1,T_2)\\|_{\\mathcal{B}(\\mathcal{H})} \\leq \\|p\\|_V \\] holds. 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Krishna Das, Jaydeb Sarkar","submitted_at":"2015-10-15T18:23:13Z","abstract_excerpt":"Let $\\mathbb{D}$ denote the unit disc in the complex plane $\\mathbb{C}$ and let $\\mathbb{D}^2 = \\mathbb{D} \\times \\mathbb{D}$ be the unit bidisc in $\\mathbb{C}^2$. Let $(T_1, T_2)$ be a pair of commuting contractions on a Hilbert space $\\mathcal{H}$. Let $\\mbox{dim } \\mbox{ran}(I_{\\mathcal{H}} - T_j T_j^*) < \\infty$, $j = 1, 2$, and let $T_1$ be a pure contraction. Then there exists a variety $V \\subseteq \\overline{\\mathbb{D}}^2$ such that for any polynomial $p \\in \\mathbb{C}[z_1, z_2]$, the inequality \\[ \\|p(T_1,T_2)\\|_{\\mathcal{B}(\\mathcal{H})} \\leq \\|p\\|_V \\] holds. 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