Ando dilations, von Neumann inequality, and distinguished varieties

Let ID denote the unit disc in the complex plane C and let D-2 = D x D be the unit bidisc in C-2. Let (T-1, T-2) be a pair of commuting contractions on a Hilbert space H. Let dim ran(I-H - TjTj*) (2) such that for any polynomial p is an element of C[z(1) , z(2)], the inequality vertical bar vertical bar p(T-1, T-2)vertical bar vertical bar B(H) <= vertical bar vertical bar p vertical bar vertical bar V holds. If, in addition, T-2 is pure, then V = {(z(1), z(2)) is an element of D-2 : det(Psi(z(1)) - z(2)I(C)(n)) = 0} is a distinguished variety, where Psi is a matrix-valued analytic function on D that is unitary-valued on partial derivative 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. (C) 2016 Elsevier Inc. All rights reserved