Ando dilations, von Neumann inequality, and distinguished varieties

摘要: Let $\mathbb{D}$ denote the unit disc in complex plane $\mathbb{C}$ and let $\mathbb{D}^2 = \mathbb{D} \times \mathbb{D}$ be bidisc $\mathbb{C}^2$. $(T_1, T_2)$ a pair of commuting contractions on Hilbert space $\mathcal{H}$. $\mbox{dim } \mbox{ran}(I_{\mathcal{H}} - T_j T_j^*) < \infty$, $j 1, 2$, $T_1$ pure contraction. Then there exists variety $V \subseteq \overline{\mathbb{D}}^2$ such that for any polynomial $p \in \mathbb{C}[z_1, z_2]$, inequality \[ \|p(T_1,T_2)\|_{\mathcal{B}(\mathcal{H})} \leq \|p\|_V \] holds. If, addition, $T_2$ is pure, then \[V \{(z_1, z_2) \mathbb{D}^2: \det (\Psi(z_1) z_2 I_{\mathbb{C}^n}) 0\}\]is distinguished variety, where $\Psi$ matrix-valued analytic function unitary $\partial \mathbb{D}$. Our results comprise new proof, as well generalization, Agler McCarthy's sharper von Neumann pairs strictly contractive matrices.