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Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality
Journal article   Peer reviewed

Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality

Anatolii Grinshpan, Dmitry S. Kaliuzhnyi-Verbovetskyi, Victor Vinnikov and Hugo J. Woerdeman
Journal of functional analysis, v 256(9), pp 3035-3054
2009

Abstract

Commuting contractions Multivariable Schur class Multivariable von Neumann inequality Nevanlinna–Pick interpolation problem Scattering system Schur–Agler class Unitary dilation
We obtain a decomposition for multivariable Schur-class functions on the unit polydisk which, to a certain extent, is analogous to Agler's decomposition for functions from the Schur–Agler class. As a consequence, we show that d-tuples of commuting strict contractions obeying an additional positivity constraint satisfy the d-variable von Neumann inequality for an arbitrary operator-valued bounded analytic function on the polydisk. Also, this decomposition yields a necessary condition for solvability of the finite data Nevanlinna–Pick interpolation problem in the Schur class on the unit polydisk.

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Mathematics
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