Testing von Neumann inequalities with nilpotent matrices

Abstract

We give an elementary proof of the folklore result that the Agler norm of a function is determined by its norm on commuting tuples of nilpotent matrices. The proof is variation on a standard cone separation argument. The topic is closely related to the Eschmeier-Patton-Putinar formulation of Carathéodory-Fejér interpolation.

Added some context to the main theorem, included an example and additional reference. Incorporated referee suggestions