On von Neumann's inequality for complex triangular Toeplitz contractions

Abstract

We prove that von Neumann’s inequality holds for circulant contractions. We show that every complex polynomial f(z1,…,zn) over 𝔻n is associated to a constant d(f) such that von Neumann’s inequality can hold up to d(f), for n-tuples of commuting contractions on a Hilbert space. We characterise complex polynomials over 𝔻n in which d(f)=2. We introduce the properties of upper (or lower) complex triangular Toeplitz matrices. We show that von Neumann’s inequality holds for n-tuples of upper (or lower) complex triangular Toeplitz contractions. We construct contractive homomorphisms.