Functional Models for Commuting Hilbert-space Contractions

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 ${\mathbb{G}}_{♯},{\mathbb{W}}_{♯}$ in addition to the Sz.-Nagy--Foias characteristic function ${\Theta }_T$ for the product operator $T$ so that the combined triple $({\mathbb{G}}_{♯},{\mathbb{W}}_{♯},{\Theta }_T)$ 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).

Dedicated to the memory of Ron Douglas. It has appeared in Operator Theory: Advances and Applications (Ronald G. Douglas Memorial Volume)