Symmetric Schur-class functions on the bidisk and Schur-class functions on the symmetrized bidisk

TL;DR

This paper explores the relationship between symmetric Schur-class functions on the bidisk and Schur-class functions on the symmetrized bidisk, leading to new realization and determinantal representation results.

Contribution

It establishes a connection between these classes of functions, resulting in finite dimensional realizations and determinantal representations for functions on the symmetrized bidisk.

Findings

Finite dimensional realization for rational Schur functions on the symmetrized bidisk

Determinantal representation for zero-free polynomials on the symmetrized bidisk

Enhanced understanding of function relations on complex domains

Abstract

We present some thoughts on the relation between symmetric Schur-class functions on the bidisk and Schur-class functions on the symmetrized bidisk. Among other things, use of this relation leads to a finite dimensional realization result for rational matrix functions in the Schur-class on the symmetrized bidisk and also to a determinantal representation result for polynomials without zeros on the symmetrized bidisk.