Functional Models for Commuting Hilbert-Space Contractions

Abstract

We develop a Sz.-Nagy-Foias-type functional model for a commutative contractive operator tuple T = (T1, . . . , T d ) having T = T1 • • • T d equal to a completely nonunitary contraction. We identify additional invariants G ♯ , W ♯ in addition to the Sz.-Nagy-Foias characteristic function ΘT for the product operator T so that the combined triple (G ♯ , W ♯ , ΘT ) becomes a complete unitary invariant for the original operator tuple T . For the case d ≥ 3 in general there is no commutative isometric lift of 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 T , herein called a pseudo-commutative contractive lift of T , which in turn leads to the functional model for T . This work has many parallels with recently developed model theories for symmetrized-bidisk contractions (commutative operator pairs (S, P ) having the symmetrized bidisk Γ as a spectral set) and for tetrablock contractions (commutative operator triples (A, B, P ) having the tetrablock domain E as a spectral set).