La desigualdad de Von Neumann y la teoría de dilatación

Abstract

A famous inequality by von Neumann states that if T is a contraction on a Hilbert space and p is a polynomial, then kp(T)k ≤ sup{|p(z)| : z ∈ C, |z| ≤ 1}. As time went on, this inequality has given rise to a large variety of results estimulating this question. The natural way to generalize this inequality concerns contractions T1, . . . , Tn that commute on a common Hilbert space. Is it true that, for any polynomial p(z1, . . . , zn) in n variables, kp(T1, . . . , Tn)k ≤ sup{|p(z1, . . . , zn)| : zi ∈ C, |zi | ≤ 1, i = 1, . . . , n}? The answer is partial. The major steps in answering this question are due to T. Ando and N. Varopoulos, as much in positive and negative cases, respectively. The aim of this work is to present an elegant proof using dilation theory, whose main forefather is Sz. Nagy, of the original von Neumann’s inequality, as well as describing its generalization for two commuting contractions, and some counterexamples on a finite-dimensional Hilbert space, emphasizing the M. J. Crabb, A. M. Davie and J. A. Holbrook counterexamples.