Schur functions and inner functions on the bidisc

TL;DR

This paper investigates the structure of inner functions on the bidisc using fractional linear transformations, providing conditions for their existence, classifying associated kernels, and analyzing factorization properties.

Contribution

It offers new sufficient conditions for two-variable inner functions, a classification of de Branges-Rovnyak kernels, and insights into one-variable factorization of Schur functions.

Findings

Sufficient conditions for two-variable inner functions via colligation matrices.

Complete classification of de Branges-Rovnyak kernels on the bidisc.

Characterization of Schur functions with one variable factorization.

Abstract

We study representations of inner functions on the bidisc from a fractional linear transformation point of view, and provide sufficient conditions, in terms of colligation matrices, for the existence of two-variable inner functions. Here the sufficient conditions are not necessary in general, and we prove a weak converse for rational inner functions that admit one variable factorization. We present a complete classification of de Branges-Rovnyak kernels on the bidisc (which equally works in the setting of polydisc and the open unit ball of $\mathbb{C}^n$, $n \geq 1$). We also classify, in terms of Agler kernels, two-variable Schur functions that admit one variable factor.