Ando dilations, von Neumann inequality, and distinguished varieties

Abstract

Let D denote the unit disc in the complex plane C and let D² = D x D be the unit bidisc in C². Let (T₁, T₂) be a pair of commuting contractions on a Hilbert space H. Let dim ran (I_H - T_j T_j^⁎) < ∞, 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, then V={(z₁, z₂) ∊ D²: det (Ψ (z₁) - z₂ I_Cⁿ) = 0} is a distinguished variety, where Ψ is a matrix-valued analytic function on D that is unitary-valued on δD. 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.