Schur-Agler class and Carathéodory extremal functions

2025/10/20 by Anindya Biswas, Biswas, Anindya
Mathematics · #32F45 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Holomorphic and Operator Theory #Primary 47A48 #Secondary 32E30

paper · pdf · doi:10.48550/arxiv.2510.17658

openalex publication_date 2025/10/20 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28

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, \mathbbD and \mathbbD² 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.

Citations

• Extension Property and Universal Sets
• On the Nevanlinna problem -- Characterization of all Schur-Agler class solutions affiliated with a given kernel
• Characterizations of some domains via Carathéodory extremals
• Test functions, Schur-Agler classes and transfer-function realizations: the matrix-valued setting
• Multivariable generalizations of the Schur class: positive kernel characterization and transfer function realization
• Test Functions, Kernels, Realizations and Interpolation

Related

• Schur parameters and Carathéodory class
• Characterizations of some domains via Carathéodory extremals
• An improved Julia-Caratheodory theorem for Schur-Agler mappings of the unit ball
• Test functions, Schur-Agler classes and transfer-function realizations: the matrix-valued setting
• The Schur-Agler class in infinitely many variables