Testing von Neumann inequalities with nilpotent matrices

TL;DR

This paper provides an elementary proof that the Agler norm of a function can be characterized by its behavior on commuting nilpotent matrices, connecting to interpolation theory.

Contribution

It offers a new, simplified proof of a known result relating the Agler norm to nilpotent matrices, enhancing understanding of operator norms in function theory.

Findings

Agler norm determined by nilpotent matrices

Elementary proof via cone separation

Connection to Carathéodory-Fejér interpolation

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 Carath\'eodory-Fej\'er interpolation.