Isometric dilations and von Neumann inequality for a class of tuples in the polydisc

TL;DR

This paper explores conditions under which certain classes of n-tuples of commuting contractions in the polydisc admit isometric dilations and satisfy the von Neumann inequality, extending classical results to higher dimensions.

Contribution

It provides new results on isometric dilations and von Neumann inequalities for a broad class of n-tuples of commuting contractions, beyond the classical cases of one or two contractions.

Findings

Established isometric dilation results for specific n-tuples

Derived a sharper von Neumann inequality for these classes

Extended classical dilation theory to higher-dimensional cases

Abstract

The celebrated Sz.-Nagy and Foias and Ando theorems state that a single contraction, or a pair of commuting contractions, acting on a Hilbert space always possesses isometric dilation and subsequently satisfies the von Neumann inequality for polynomials in $\mathbb{C}[z]$ or $\mathbb{C}[z_1, z_2]$, respectively. However, in general, neither the existence of isometric dilation nor the von Neumann inequality holds for $n$-tuples, $n \geq 3$, of commuting contractions. The goal of this paper is to provide a taste of the isometric dilations, the von Neumann inequality and a sharper version of von Neumann inequality for a large class of $n$-tuples, $n \geq 3$, of commuting contractions.