This chapter introduces Ando tuples, a key ingredient for the development of the main results. It is shown that not only do Ando tuples describe every minimal Ando isometric lift of a commuting contractive pair via a Douglas-type model, but also they govern the uniqueness of minimal isometric lifts. Ando tuples come equipped with their own notion of unitary equivalence. Equivalence of Ando tuples corresponds to unitary equivalence of the associated minimal Ando lifts of the given commuting contractive pair. This gives a way of classifying the previously observed lack of uniqueness (up to unitary equivalence). This lack of uniqueness gives part of the explanation as to why the bivariate problem is harder than the univariate classical version. For the development of this chapter, we need what we refer to as Type I Ando tuples. ​An in-depth analysis reveals that Type I Ando tuples exist, which leads to a new proof of Ando’s dilation theorem.

In this chapter, we review the Berger–Coburn–Lebow (BCL) and the Bercovici–Douglas–Foias (BDF) approaches and obtain new results on how one can map a BCL-model commuting isometric pair to a unitarily equivalent BDF-model commuting isometric pair and vice versa. We also present a more complete systematic account of the BDF-model, which fleshes out various scattered results due to the trio. This in-depth study leads to the introduction of what we call a refined BDF-model for a commuting isometric pair, the analysis of which gives a new proof of a Wold-type decomposition result of Gaşper and Gaşper. Here we also make contact with some ideas from discrete-time input/state/output linear systems (e.g., realization of a contractive analytic function as the transfer function of a unitary system node), which are important for understanding how to go from a contractive analytic function (a key invariant for the BDF theory) to a 2 x 2-block unitary operator matrix (a key invariant for the BCL theory). However, it is the BCL-model theory that will be our main tool for obtaining functional models for Ando isometric lifts of a commuting contractive pair.

This chapter introduces a weaker type of lift, called a pseudo-commuting contractive lift, which paves the way for an appropriate parallel of the Sz.-Nagy–Foias model theory for the bivariate setting. Embedded inside a pseudo-commuting contractive lift of a commuting contractive pair is a minimal Sz.-Nagy–Foias isometric lift of the product of components of the commuting contractive pair. The uniqueness of the Sz.-Nagy–Foias isometric lift is then exploited to argue that any two pseudo-commuting contractive lifts are unitarily equivalent. Arguably, the pseudo-commuting contractive lifts for a commuting contractive pair provide a better parallel to the Sz.-Nagy–Foias isometric lifts for a single contraction operator.

The concluding Appendix seeks to understand how the detailed results of the previous chapters fit in with the broader current mathematical landscape. We explain how the Ando lifting problem fits in as a particular instance of the Arveson C-star algebra dilation framework. Within the Arveson framework, we identify an additional distinctive feature, namely the identification of a companion contraction, as an additional piece of structure. We identify some remaining open problems concerning commuting contractive pairs, which should be a fertile area for future research.

Here we give a short Introduction to Sz.-Nagy–Foias classical theory and then briefly describe the main contribution chapterwise.

This chapter extends all the features of the Sz.-Nagy–Foias model theory to the bivariate setting. We introduce the notion of a characteristic triple. The characteristic triple for a commuting contractive pair consists of the Sz.-Nagy–Foias characteristic function for the product of the components of the given commuting contractive pair, the fundamental-operator pair introduced in Chapter 7, and the canonical commuting unitary pair introduced in Chapter 4. The characteristic triple for a commuting contractive pair is shown to be a complete unitary invariant for the commuting contractive pair. Thus the characteristic triple serves as the analogue of the Sz.-Nagy–Foias characteristic function for the univariate case. This chapter also introduces the notion of admissible triples, which is equipped with a natural equivalence relation called "coincidence" inspired by the Sz.-Nagy–Foias notion of coincidence between two contractive analytic functions. It is shown that an admissible triple gives rise to a commuting contractive pair on a functional model space, whose characteristic triple coincides with the prespecified admissible triple. Joint invariant subspaces are characterized.

In this chapter we present a streamlined version of the original proof of Ando. We also discuss in full detail the well-known fact that the commutant lifting theorem and Ando’s dilation theorem are equivalent to each other. Interestingly, while a minimal isomeric lift of a given commuting contractive pair is not unique (up to a natural notion of unitary equivalence of lifts), we show that the unitary part in the Berger–Coburn–Lebow decomposition of every minimal isometric lift of a commuting contractive pair is unique up to unitary equivalence. We also present a class of commuting contractive pairs for which any two minimal isometric lifts are unique. This chapter also constructs a canonical commuting unitary pair corresponding to a commuting contractive pair. The canonical unitary pair plays a key role in later chapters.

This chapter reviews the dilation/model theory for a single contraction operator from four points of view: (i) the coordinate-free geometric picture, (ii) the Douglas model theory, (iii) the Schäffer model theory, and (iv) the Sz.-Nagy–Foias model theory. This chapter also presents certain canonical decomposition theorems for an isometry, a general contraction operator, and the isometric embedding involved in an isometric lifting of a general contraction operator.

This chapter studies a certain strong notion of minimality of an Ando isometric lift. This was inspired by certain concrete examples where such a strong notion of minimality occurs. Various tractable characterizations are provided for a commuting contractive pair to have a strongly minimal Ando isometric lift. One of the characterizations is in terms of what we refer to as the fundamental-operator pair for a commuting contractive pair. Three distinct proofs are given for the existence of the fundamental-operator pair. An intrinsic relation is provided between the fundamental-operator pair and Ando tuples of a commuting contractive pair. The fundamental-operator pair plays a fundamental role in the development of the model theory for commuting contractive pairs in Chapter 9.

This chapter can be considered to be parallel to Chapter 5, but with the objective of constructing a joint Schäffer-type model for Ando isometric lifts of a commuting contractive pair (rather than a Douglas-type model as in Chapter 5). A point of distinction with the model in Chapter 5 is that the isometric embedding involved in the lifting is just the inclusion map. The analysis leads to the notion of a Type II Ando tuple. Again, it turns out that special Ando tuples are automatically also of Type II, so we arrive at an independent second proof of Ando’s dilation theorem. We also establish an intriguing connection between the Schäffer model for a pair of coisometries and the BCL-model for the isometric pair.