The operator-valued Schur-Agler class in infinitely many variables

Abstract

Motivated by a recent work of G. Knese [Canad. Math. Bull., 2026], we introduce the operator-valued Schur-Agler class in infinitely many variables, consisting of functions whose restrictions to every finite-dimensional polydisk lie in the operator-valued Schur-Agler class. We first establish an operator-valued analogue of Montel's theorem and, using it as a key ingredient, establish an Agler decomposition and a realization formula for functions in this class. The series in the Agler decomposition converges in the strong operator topology, and we show that this convergence cannot, in general, be strengthened to norm convergence. The realization formula yields a holomorphic extension of every function in the operator-valued Schur-Agler class to $\mathrm{Ball}(\ell^\infty)$, the open unit ball of $\ell^\infty$. Finally, as an application, we prove an Agler-Pick interpolation theorem in infinitely many variables.