Functional Models for Commuting Hilbert-space Contractions

TL;DR

This paper develops a functional model for commuting Hilbert-space contractions with a focus on the product operator, introducing new invariants and a pseudo-commutative lift, advancing the understanding of multivariable operator theory.

Contribution

It introduces a comprehensive functional model for commuting contractions with product constraints, including new invariants and a pseudo-commutative lift, extending existing model theories.

Findings

Identifies additional invariants ${ m f G}_lat, { m f W}_lat$ for the product operator.

Establishes a complete unitary invariant for the operator tuple.

Constructs a pseudo-commutative contractive lift for the tuple.

Abstract

We develop a Sz.-Nagy--Foias-type functional model for a commutative contractive operator tuple $\underline{T} = (T_1, \dots, T_d)$ having $T = T_1 \cdots T_d$ equal to a completely nonunitary contraction. We identify additional invariants ${\mathbb G}_\sharp, {\mathbb W}_\sharp$ in addition to the Sz.-Nagy--Foias characteristic function $\Theta_T$ for the product operator $T$ so that the combined triple $({\mathbb G}_\sharp, {\mathbb W}_\sharp, \Theta_T)$ becomes a complete unitary invariant for the original operator tuple $\underline{T}$. For the case $d \ge 3$ in general there is no commutative isometric lift of $\underline{T}$; however there is a (not necessarily commutative) isometric lift having some additional structure so that, when compressed to the minimal isometric-lift space for the product operator $T$, generates a special kind of lift of $\underline{T}$, herein called a {\em pseudo-commutative contractive lift} of $\underline{T}$, which in turn leads to the functional model for $\underline{T}$. This work has many parallels with recently developed model theories for symmetrized-bidisk contractions (commutative operator pairs $(S,P)$ having the symmetrized bidisk $\Gamma$ as a spectral set) and for tetrablock contractions (commutative operator triples $(A, B, P)$ having the tetrablock domain ${\mathbb E}$ as a spectral set).