Fundamental Agler Decompositions

TL;DR

This paper offers an elementary proof of the Agler Decomposition Theorem using shift-invariant subspaces of the Hardy space on the bidisk, and explores their properties, especially for rational inner functions and stable polynomials.

Contribution

It introduces a new elementary proof of the Agler Decomposition Theorem and analyzes shift-invariant subspaces related to rational inner functions and stable polynomials.

Findings

Elementary proof of Agler Decomposition Theorem

Simplified proofs for rational inner functions

Results on stable polynomials on the polydisk

Abstract

We use shift-invariant subspaces of the Hardy space on the bidisk to provide an elementary proof of the Agler Decomposition Theorem. We observe that these shift-invariant subspaces are specific cases of Hilbert spaces that can be defined from Agler decompositions and analyze the properties of such Hilbert spaces. We then restrict attention to rational inner functions and show that the shift-invariant subspaces provide easy proofs of several known results about decompositions of rational inner functions. We use our analysis to obtain a result about stable polynomials on the polydisk.