Schur norms and the multivariate von Neumann inequality
John Holbrook · Birkhäuser Basel eBooks · 2001
Starting from some classical counterexamples to the von Neumann inequality for several variables, we are led to especially simple examples of this phenomenon. We display three commuting 4-dimensional contractions Ck and a polynomial p(zi, z2, z3) such that 1 $$\parallel p({C_{1}},{C_{2}},{C_{3}})\parallel = \tfrac{6}{5}\max \{ |p({z_{1}},{z_{2}},{z_{3}})| : |{z_{k}}| \leqslant 1\} .$$ We find that this phenomenon depends on the norm of Schur multiplication by certain matrices and on the related Haagerup factorizations. It is easy to perturb the example to a triple of generic commuting contractions and so provide an answer to a question of Lewis and Wermer.