Under certain conditions the unit disk and bidisk are the only domains where finitely many test functions generate the Schur class, yielding descriptions of Carathéodory extremals in the Drury-Arveson multiplier algebra and operator-theoretic Herglotz representations for Carathéodory hyperbolic-dom.
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Schur-Agler class and Carath\'eodory extremal functions
Under certain conditions the unit disk and bidisk are the only domains where finitely many test functions generate the Schur class, yielding descriptions of Carathéodory extremals in the Drury-Arveson multiplier algebra and operator-theoretic Herglotz representations for Carathéodory hyperbolic-dom.