Schur-Agler class and Carath\'eodory extremal functions

TL;DR

This paper investigates the role of Carathéodory extremal functions within the Schur-Agler class, revealing domain-specific generation limitations and providing operator-theoretic representations for certain hyperbolic domains.

Contribution

It characterizes when finitely many test functions generate the Schur class and describes Carathéodory extremals in the Drury-Arveson space's multiplier algebra.

Findings

Only the unit disk and bidisk are generated by finitely many test functions under certain conditions.

Provides a description of Carathéodory extremals in the unit ball of the Drury-Arveson space.

Gives operator-theoretic Herglotz representations for Carathéodory hyperbolic domains.

Abstract

We study the role of Carath\'eodory 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 Carath\'eodory extremals in the unit ball of the multiplier algebra of the Drury-Arveson space and give operator-theoretic Herglotz representations for any Carath\'eodory hyperbolic domain.