Test functions, Schur-Agler classes and transfer-function realizations: the matrix-valued setting

TL;DR

This paper extends the theory of Schur-Agler classes and transfer-function realizations to matrix- and operator-valued test functions, kernels, and functions, with applications to matrix-valued Hardy spaces and constrained analytic functions.

Contribution

It introduces a generalized framework for matrix- and operator-valued Schur-Agler classes, including new examples where the matrix case is not just a tensor product extension.

Findings

Extended Schur-Agler class framework to matrix-valued functions.

Provided examples involving finitely-connected domains and constrained Hardy algebras.

Highlighted cases where matrix-valued classes are not tensor products of scalar classes.

Abstract

Given a collection of test functions, one defines the associated Schur-Agler class as the intersection of the contractive multipliers over the collection of all positive kernels for which each test function is a contractive multiplier. We indicate extensions of this framework to the case where the test functions, kernel functions, and Schur-Agler-class functions are allowed to be matrix- or operator-valued. We illustrate the general theory with two examples: (1) the matrix-valued Schur class over a finitely-connected planar domain and (2) the matrix-valued version of the constrained Hardy algebra (bounded analytic functions on the unit disk with derivative at the origin constrained to have zero value). Emphasis is on examples where the matrix-valued version is not obtained as a simple higher-multiplicity tensoring of the scalar-valued version.