Ando dilations, von Neumann inequality, and distinguished varieties

Let D denote the unit disc in the complex plane C and let D2=D×D be the unit bidisc in C2. Let (T1,T2) be a pair of commuting contractions on a Hilbert space H. Let dim⁡ran(IH−TjTj⁎)<∞, j=1,2, and let T1 be a pure contraction. Then there exists a variety V⊆D‾2 such that for any polynomial p∈C[z1,z2], the inequality‖p(T1,T2)‖B(H)≤‖p‖V holds. If, in addition, T2 is pure, thenV={(z1,z2)∈D2:det⁡(Ψ(z1)−z2ICn)=0} is a distinguished variety, where Ψ is a matrix-valued analytic function on D that is unitary-valued on ∂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.