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}$} - T_j T_j^*) < \infty$,$j = 1, 2$,andlet$T_1$ be a pure contraction. Then there exists a variety $V\subseteq {\bar{\mathbb{D}}}^2$ such that for any polynomial $p\in \mathbb{C}[z_1,z_2]$, the inequality $$\mathrm{\parallel }p(T_1,T_2){\mathrm{\parallel }}_{\mathcal{B}(\mathcal{H})}\le \mathrm{\parallel }p{\mathrm{\parallel }}_V$$ holds. If, in addition, $T_2$ is pure, then $$V=\{(z_1,z_2)\in {\mathbb{D}}^2:\det (\mathrm{\Psi }(z_1)-z_2{I}_{{\mathbb{C}}^n})=0\}$$is a distinguished variety, where $\mathrm{\Psi }$ is a matrix-valued analytic function on $\mathbb{D}$ that is unitary on $\mathrm{\partial }\mathbb{D}$. 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.