On the Schur-Agler Norm

TL;DR

This paper provides a new convex optimization-based description of the Schur-Agler norm for holomorphic functions on the polydisc, with theoretical insights and practical methods for approximation and counterexample construction.

Contribution

It introduces a novel convex optimization framework for the Schur-Agler norm, unifies existing proofs, and offers practical tools for approximation and counterexample generation.

Findings

Schur-Agler norm can be tested with diagonalizable or nilpotent matrix tuples.

The predual space of the Schur-Agler space is characterized as a space of analytic functions.

Numerical approximation of the norm via semidefinite programming is feasible.

Abstract

We establish a new description of the Schur-Agler norm of a holomorphic function on the polydisc as the solution of a convex optimization problem. Consequences of this description are explored both from a theoretical and from a practical point of view. Firstly, we give unified proofs of the known facts that the Schur-Agler norm can be tested with diagonalizable or nilpotent matrix tuples, as well as a new proof of the existence of Agler decompositions. Secondly, we describe the predual of the Schur-Agler space as a space of analytic functions on the polydisc. Thirdly, we give a unified treatment of existing counterexamples of von Neumann's inequality in our framework, and exhibit several methods for constructing counterexamples. On the practical side, we explain how the Schur-Agler norm of a homogeneous polynomial can be numerically approximated using semidefinite programming.