Hadamard multipliers of the Agler class

TL;DR

This paper investigates functions that preserve the Schur-Agler class under Hadamard products, establishing conditions and characterizations for such preservers, and linking them to moments of operator tuples and multivariable inequalities.

Contribution

It proves that Schur class preservers also preserve the Schur-Agler class, characterizes these preservers via moments of operator tuples, and connects counterexamples to multivariable inequalities with Agler class preservers.

Findings

Functions preserving the Schur class also preserve the Schur-Agler class.

Preservers of the Schur-Agler class are characterized by moments of commuting operator tuples.

Counterexamples to the multivariable von Neumann inequality relate to non-trivial Agler class preservers.

Abstract

We prove several results about functions which preserve the Schur-Agler class under Hadamard or coefficient-wise product. First, functions which preserve the Schur class necessarily preserve the Schur-Agler class. Second, ``moments'' of certain commuting operator tuples form coefficients of Schur-Agler class preservers. Finally, any preserver of the full matrix Schur-Agler class must have coefficients given by moments of commuting operator tuples. We also point out that any counterexample to the multivariable von Neumann inequality can be used to derive a non-trivial Agler class preserver.