Rational inner functions on a square-matrix polyball

TL;DR

This paper proves that all matrix-valued rational inner functions on a square-matrix polyball have finite-dimensional unitary realizations, characterizes their denominators in the scalar case, and extends classical results using a generalized Korányi–Vági theorem.

Contribution

It establishes finite-dimensional unitary realizations for matrix-valued rational inner functions and characterizes denominators in the scalar case, extending classical theory to matrix polyballs.

Findings

Finite-dimensional unitary realization exists for all such functions.

Denominators are characterized in the scalar case.

Polynomials with no zeros in the domain serve as denominators.

Abstract

We establish the existence of a finite-dimensional unitary realization for every matrix-valued rational inner function from the Schur--Agler class on a unit square-matrix polyball. In the scalar-valued case, we characterize the denominators of these functions. We also show that every polynomial with no zeros in the closed domain is such a denominator. One of our tools is the Kor\'{a}nyi--Vagi theorem generalizing Rudin's description of rational inner functions to the case of bounded symmetric domains; we provide a short elementary proof of this theorem suitable in our setting.