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Isometric dilations and von Neumann inequality for a class of tuples in the polydisc (1710.07624v2)
Published 20 Oct 2017 in math.FA, math.CV, and math.OA
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.