Schur-Agler class and Carathéodory extremal functions
Anindya Biswas
TL;DR
The paper develops a $\ ext{Ψ}$-Schur-Agler framework to understand when the Schur class on a domain $\\Omega$ can be generated by a finite collection of test functions, tying extremal problems to the Carathéodory distance $c^*_{\\Omega}$. It proves a strong finiteness result: under $\\\ extgamma$-hyperbolicity and $c^*$-finite compactness, having $\ ext{SA}_\\Ψ=\\S(\\Omega)$ with finitely many tests forces $\\Omega$ to be biholomorphic to $\\D$ or $\\D^2$, and it constructs a canonical universal test family that ensures $\\S(\\Omega)\subset\\SAS_\\Ψ(\\Omega)$ for Carathéodory hyperbolic domains. The paper also provides concrete applications: (i) a description of Carathéodory extremals in the unit ball of the Drury–Arveson multiplier algebra, including an explicit relation that determines intermediate values; and (ii) an operator-theoretic Herglotz representation for domains with Carathéodory hyperbolicity. Together, these results illuminate how extremal problems and Agler decompositions interact with domain geometry and operator models in several complex variables and functional analysis.
Abstract
We study the role of Carathéodory extremal functions in the Schur-Agler class generated by a collection of test functions. We show that under certain conditions, $\mathbb{D}$ and $\mathbb{D}^2$ are the only domains where finitely many test functions can generate the Schur class. As applications, we give a description of the Carathéodory extremals in the unit ball of the multiplier algebra of the Drury-Arveson space and give operator-theoretic Herglotz representations for any Carathéodory hyperbolic domain.