Functional Models and Invariant Subspaces for Pairs of Commuting Contractions

Abstract

The goal of the present paper is to push Sz.-Nagy--Foias model theory for a completely nonunitary Hilbert-space contraction operator $T$, to the case of a commuting pair of contraction operators $(T_1,T_2)$ having product $T=T_1{T}_2$ which is completely nonunitary. The idea is to use the Sz.-Nagy-Foias functional model for $T$ as the model space also for the commutative tuple ($T_1,T_2)$ with $T=T_1{T}_2$ equal to the usual Sz.-Nagy--Foias model operator, and identify what added structure is required to classify such commutative contractive factorizations $T=T_1{T}_2$ up to unitary equivalence. In addition to the characteristic function ${\Theta }_T$, we identify additional invariants $(\mathbb{G},\mathbb{W})$ which can be used to construct a functional model for the commuting pair $(T_1,T_2)$ and which have good uniqueness properties: if two commutative contractive pairs $(T_1,T_2)$ and $(T_1^{\prime },T_2^{\prime })$ are unitarily equivalent, then their characteristic triples $(\Theta ,\mathbb{G},\mathbb{W}{)}_T$ and $(\Theta ,\mathbb{G},\mathbb{W}{)}_{T^{\prime }}$ coincide in a natural sense. We illustrate the theory with several simple cases where the characteristic triples can be explicitly computed. This work extends earlier results of Berger-Coburn-Lebow \cite{B-C-L} for the case where $(T_1,T_2)$ is a pair of commuting isometries, and of Das-Sarkar \cite{D-S}, Das-Sarkar-Sarkar \cite{D-S-S} and the second author \cite{sauAndo} for the case where $T=T_1{T}_2$ is pure (the operator sequence $T^{\ast n}$ tends strongly to $0$). Finally we use the model to study the structure of joint invariant subspaces for a commutative, contractive operator pair, extending results of Sz.-Nagy--Foias for the single-operator case.

45 pages