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 $f\left(z_1,\dots ,z_n\right)$ over ${\mathbb{𝔻}}^n$ is associated to a constant $d\left(f\right)$ such that von Neumann’s inequality can hold up to $d\left(f\right)$, for $n$-tuples of commuting contractions on a Hilbert space. We characterise complex polynomials over ${\mathbb{𝔻}}^n$ in which $d\left(f\right)=2$. 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.