Functional models for $\Gamma_n$-contractions

TL;DR

This paper develops new functional models for $\Gamma_n$-contractions, extending classical dilation and model theories to this class of operator tuples, providing tools for their analysis.

Contribution

It introduces several functional models for $\Gamma_n$-contractions, including Sz.-Nagy-Foias and Sch"affer type models, based on dilation and factorization results.

Findings

Established a Sz.-Nagy-Foias type functional model for non-unitary $\Gamma_n$-contractions.

Developed factorization results linking minimal and arbitrary isometric dilations.

Created Sch"affer type models for $\Gamma_n$-contractions.

Abstract

This article develops several functional models for a given $\Gamma_n$-contraction. The first model is motivated by the Douglas functional model for a contraction. We then establish factorization results that clarify the relationship between a minimal isometric dilation and an arbitrary isometric dilation of a contraction. Using these factorization results, we obtain a Sz.-Nagy-Foias type functional model for a completely non-unitary $\Gamma_n$-contraction, as well as Sch\"affer type functional model for $\Gamma_n$-contraction.