Inner functions on the bidisk and associated Hilbert spaces

TL;DR

This paper explores the structure and properties of matrix-valued inner functions on the bidisk, analyzing associated Hilbert spaces, boundary regularity, and applications to three-variable rational inner functions, with explicit descriptions and examples.

Contribution

It provides a complete description of key spaces linked to matrix rational inner functions and introduces methods to compute their dimensions, advancing understanding of multivariable inner functions.

Findings

Explicit descriptions of spaces associated with matrix rational inner functions

Methods for computing dimensions of these spaces

Illustrative examples highlighting differences between scalar and matrix cases

Abstract

Matrix valued inner functions on the bidisk have a number of natural subspaces of the Hardy space on the torus associated to them. We study their relationship to Agler decompositions, regularity up to the boundary, and restriction maps into one variable spaces. We give a complete description of the important spaces associated to matrix rational inner functions. The dimension of some of these spaces can be computed in a straightforward way, and this ends up having an application to the study of three variable rational inner functions. Examples are included to highlight the differences between the scalar and matrix cases.