On the Schur-Agler Norm

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.