Hadamard Multipliers of the Agler Class

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.