Interpolation in the noncommutative Schur-Agler class

TL;DR

This paper extends the interpolation theory of the Schur-Agler class from commutative to noncommutative settings, involving operator-valued functions, noncommuting indeterminates, and transfer functions of structured multidimensional systems.

Contribution

It develops a noncommutative interpolation theory for the Schur-Agler class, generalizing previous commutative results to noncommuting variables and operator tuples.

Findings

Established a noncommutative von Neumann inequality

Characterized noncommutative Schur-Agler functions via linear-fractional presentations

Extended interpolation results to noncommutative operator settings

Abstract

The class of Schur-Agler functions over a domain ${\mathcal D} \subset {\mathbb C}^{d}$ is defined as the class of holomorphic operator-valued functions on ${\mathcal D}$ for which a certain von Neumann inequality is satisfied when a commuting tuple of operators satisfying a certain polynomial norm inequality is plugged in for the variables. Such functions are alternatively characterized as those having a linear-fractional presentation which identifies them as transfer functions of a certain type of conservative structured multidimensional linear system. There now has been introduced a noncommutative version of the Schur-Agler class which consists of formal power series in noncommuting indeterminants satisfying a noncommutative version of the von Neumann inequality when a tuple of operators (not necessarily commuting) coming from a noncommutative operator ball are plugged in for the formal indeterminants. Formal power series in this noncommutative Schur-Agler class in turn are characterized as those having a certain linear-fractional presentation in noncommuting variables identifying them as transfer functions of a recently introduced class of conservative structure multidimensional linear systems having evolution along a free semigroup rather than along an integer lattice. The purpose of this paper is to extend the previously developed interpolation theory for the commutative Schur-Agler class to this noncommutative setting.