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

Barik, Sibaprasad; Das, B.; Haria, Kalpesh; Sarkar, Jaydeb

doi:10.1090/tran/7676

Isometric dilations and von Neumann inequality for a class of tuples in the polydisc
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by Sibaprasad Barik, B. Krishna Das, Kalpesh J. Haria and Jaydeb Sarkar PDF

Trans. Amer. Math. Soc. 372 (2019), 1429-1450 Request permission

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 isometric dilations, von Neumann inequality, and a refined version of von Neumann inequality for a large class of $n$-tuples, $n \geq 3$, of commuting contractions.

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