Ando dilations, von Neumann inequality, and distinguished varieties

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. If, in addition, $T_2$ is pure, then $$V = \{(z_1, z_2) \in \mathbb{D}^2: \det (\Psi(z_1) - z_2 I_{\mathbb{C}^n}) = 0\}$$is a distinguished variety, where $\Psi$ is a matrix-valued analytic function on $\mathbb{D}$ that is unitary on $\partial \mathbb{D}$. Our results comprise a new proof, as well as a generalization, of Agler and McCarthy's sharper von Neumann inequality for pairs of commuting and strictly contractive matrices.