June 30, 2008
Victor Vinnikov
Title: A multivariable von Neumann inequality
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. Our methods are based on decomposing the scattering subspace of the corresponding $d$-evolution scattering system and analyzing the associated reproducing kernels. This is a joint work with Anatolii Grinshpan, Dmitry Kaliuzhnyi-Verbovetskyi, and Hugo Woerdeman.