Dilation and Model Theory for Pairs of Commuting Contractions

TL;DR

This paper extends the Sz.-Nagy--Foias dilation and model theory from single contractions to pairs of commuting contractions, introducing new functional models, classifications, and invariants for such pairs.

Contribution

It provides new proofs of Ando's Dilation Theorem, classifies minimal isometric lifts, and introduces the characteristic triple as a complete unitary invariant for pairs of commuting contractions.

Findings

Explicit example of non-unitarily equivalent minimal isometric lifts

New functional-model representations for lifts

Introduction of the characteristic triple as a complete invariant

Abstract

This manuscript is an effort to extend the Sz.-Nagy--Foias dilation and model theory for a single contraction to the case of commuting pair of contractions. Fundamental to the Sz.-Nagy--Foias model theory is the functional model for the minimal isometric dilation. The first step in our approach for the pair case is to obtain further information, beyond that in the original paper of Ando, concerning the structure of the plethora of minimal commuting isometric lifts. We exhibit an explicit simple example of two minimal isometric lifts of a commuting contractive pair that are not unitarily equivalent -- see Chapter 5. We provide two constructive new proofs of Ando's Dilation Theorem, each of which leads to a new functional-model representation for such a lift -- see Theorem 4.3.8 and Remark 4.5.7. The construction leads to the identification of a set of additional free parameters which serves to classify the distinct unitary-equivalence classes of minimal Ando lifts. However this lack of uniqueness limits the utility of such minimal Ando lifts for the construction of a functional model for a commuting contractive pair. We identify an intermediate type of lift, called pseudo-commuting contractive lift, which paves the way for a functional model. In the model form, the Sz.-Nagy--Foias characteristic function is augmented by what is called the fundamental operator pair, together with a canonical pair of commuting unitary operators, so that the augmented collection, called the characteristic triple, is a complete unitary invariant for a commuting contractive pair. There is also a notion of admissible triple as the substitute for a purely contractive analytic function in the Sz.-Nagy--Foias theory, from which one can construct a functional model commuting contractive pair having its characteristic triple coinciding with the original admissible triple in an appropriate sense.