# Kummert's approach to realization on the bidisk

## TL;DR

This paper simplifies Kummert's approach to realizing matrix-valued rational inner functions on the bidisk, extends it to isometric functions on the two-torus, and proves the existence of finite-dimensional realizations and special nilpotent structures.

## Contribution

It provides a simplified proof of minimal unitary realizations, extends results to isometric functions, and establishes finite-dimensional realizations for matrix-valued Schur functions.

## Key findings

- Every matrix-valued rational inner function in two variables has a minimal unitary transfer function realization.
- Two-variable matrix-valued rational Schur functions have finite-dimensional contractive realizations.
- Polynomial inner functions in two variables have transfer function realizations with nilpotent linear combinations.

## Abstract

We give a simplified exposition of Kummert's approach to proving that every matrix-valued rational inner function in two variables has a minimal unitary transfer function realization. A slight modification of the approach extends to rational functions which are isometric on the two-torus and we use this to give a largely elementary new proof of the existence of Agler decompositions for every matrix-valued Schur function in two variables. We use a recent result of Dritschel to prove two variable matrix-valued rational Schur functions always have finite-dimensional contractive transfer function realizations. Finally, we prove that two variable matrix-valued polynomial inner functions have transfer function realizations built out of special nilpotent linear combinations.