Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality
Von Neumann's inequality has fascinated several generations of mathematicians for more than half century. As simple as it is, this inequality turned out to be at the center of dilation theory for contractive linear operators [\textit{B.\,Sz.-Nagy} and \textit{C.\,Foiaş}, ``Analyse harmonique des opérateurs de l'espace de Hilbert'' (Budapest:\ Akadémiaí Kiadó; Paris:\ Masson et Cie) (1967; Zbl 0157.43201)], and, moreover, it had spectacular applications to control theory. However, when trying to prove von Neumann's inequality for systems of operators, one encounters unexpected challenges, which were discovered early [\textit{N.\,Th.\,Varopoulos}, J.~Funct.\ Anal.\ 16, 83--100 (1974; Zbl 0288.47006)]. A good account of the state of the art of multivariate von Neumann type inequalities is the monograph by \textit{J.\,Agler} and \textit{J.\,E.\thinspace McCarthy} [``Pick interpolation and Hilbert function spaces'' (Graduate Studies in Mathematics 44; Providence: AMS) (2002; Zbl 1010.47001)]. The present article proposes a natural approach to enforcing von~Neumann's inequality for all operator valued functions to a class of commuting strict contractions which satisfy a minimal system of operator inequalities. Connections with systems theory, scattering theory and function theory in the polydisk are outlined. The techniques introduced by the authors are natural yet ingenious, and, to the full credit of the authors, they could have been discovered much earlier.
scientific article; zbMATH DE number 5561663
scientific article