# Dilation theory and functional models for tetrablock contractions

Joseph A. Ball  Department of Mathematics, Virginia Tech, Blacksburg, VA
24061-0123, USA<EMAIL_ADDRESS>and  Haripada Sau  Department of
Mathematics, Indian Institute of Science Education and Research Pune,
Maharashtra 411008<EMAIL_ADDRESS><EMAIL_ADDRESS>Dedicated
to the memory of Jörg Eschmeier: a fine mentor, researcher, and colleague

###### Abstract.

A classical result of Sz.-Nagy asserts that a Hilbert space contraction
operator $T$ can be dilated to a unitary ${\mathcal{U}}$, i.e.,
$T^{n}=P_{\mathcal{H}}{\mathcal{U}}^{n}|{\mathcal{H}}$ for all $n=0,1,2,\dots$
. A more general multivariable setting for these ideas is the setup where (i)
the unit disk is replaced by a domain $\Omega$ contained in
${\mathbb{C}}^{d}$, (ii) the contraction operator $T$ is replaced by an
$\Omega$-contraction, i.e., a commutative operator $d$-tuple
${\mathbf{T}}=(T_{1},\dots,T_{d})$ on a Hilbert space ${\mathcal{H}}$ such
that
$\|r(T_{1},\dots,T_{d})\|_{{\mathcal{L}}({\mathcal{H}})}\leq\sup_{\lambda\in\Omega}|r(\lambda)|$
for all rational functions with no singularities in $\overline{\Omega}$ and
the unitary operator ${\mathcal{U}}$ is replaced by an $\Omega$-unitary
operator tuple, i.e., a commutative operator $d$-tuple
${\mathbf{U}}=(U_{1},\dots,U_{d})$ of commuting normal operators with joint
spectrum contained in the distinguished boundary $b\Omega$ of $\Omega$. For a
given domain $\Omega\subset{\mathbb{C}}^{d}$, the rational dilation question
asks: given an $\Omega$-contraction ${\mathbf{T}}$ on ${\mathcal{H}}$, is it
always possible to find an $\Omega$-unitary ${\mathbf{U}}$ on a larger Hilbert
space ${\mathcal{K}}\supset{\mathcal{H}}$ so that, for any $d$-variable
rational function without singularities in $\overline{\Omega}$, one can
recover $r(T)$ as $r(T)=P_{\mathcal{H}}r({\mathbf{U}})|_{\mathcal{H}}$. We
focus here on the case where $\Omega={\mathbb{E}}$, a domain in
${\mathbb{C}}^{3}$ called the tetrablock. (i) We identify a complete set of
unitary invariants for a ${\mathbb{E}}$-contraction $(A,B,T)$ which can then
be used to write down a functional model for $(A,B,T)$, thereby extending
earlier results only done for a special case, (ii) we identify the class of
pseudo-commutative ${\mathbb{E}}$-isometries (a priori slightly larger than
the class of ${\mathbb{E}}$-isometries) to which any
${\mathbb{E}}$-contraction can be lifted, and (iii) we use our functional
model to recover an earlier result on the existence and uniqueness of a
${\mathbb{E}}$-isometric lift $(V_{1},V_{2},V_{3})$ of a special type for a
${\mathbb{E}}$-contraction $(A,B,T)$.

###### Key words and phrases:

commutative contractive operator-tuples; functional model; unitary dilation;
isometric lift; spectral set; pseudo-commutative contractive lift

###### 1991 Mathematics Subject Classification:

Primary: 47A13. Secondary: 47A20, 47A25, 47A56, 47A68, 30H10

The research of the second named author was supported by DST-INSPIRE Faculty
Fellowship DST/INSPIRE/04/2018/002458.

##  1\. Introduction

Suppose that we are given a commutative tuple
${\mathbf{T}}=(T_{1},\dots,T_{d})$ of operators on a Hilbert space
${\mathcal{H}}$ together with a bounded domain $\Omega$ contained in
$d$-dimensional Euclidean space ${\mathbb{C}}^{d}$. We now recall the notion
of $\Omega$ being a spectral set and $\Omega$ being a complete spectral set
for the commutative $d$-tuple ${\mathbf{T}}$ and refer to the original paper
of Arveson [7] for additional details on this and the related matters which
follow. We say that $\Omega$ is a spectral set for ${\mathbf{T}}$ if it is the
case that

$\|r({\mathbf{T}})\|_{{}_{{\mathcal{B}}({\mathcal{H}})}}\leq\sup_{\lambda\in\Omega}|r(\lambda)|\text{
for }r\in\operatorname{Rat}(\Omega)$ (1.1)

where we set $R(\Omega)$ equal to the space of all $d$-variable scalar-valued
rational functions $r$ having no singularities in $\overline{\Omega}$ and
${\mathcal{B}}({\mathcal{H}})$ equal to the Banach algebra of all bounded
linear operators on ${\mathcal{H}}$. Here $r({\mathbf{T}})$ can be defined via
the functional calculus given by

$r({\mathbf{T}})=p(T_{1},\dots,T_{d})q(T_{1},\dots,T_{d})^{-1}$

where $(p,q)$ is a coprime pair of $d$-variable polynomials such that $r=p/q$.
In analogy with what happens for the case $\Omega$ equal to the unit disk
${\mathbb{D}}$ (see the discussion below), we say simply that ${\mathbf{T}}$
is an $\Omega$-contraction if it is the case that $\Omega$ is a spectral set
for ${\mathbf{T}}$.

We say that $\Omega$ is a complete spectral set for ${\mathbf{T}}$ if (1.1)
continues to hold when one substitutes matrix rational functions
$R(\lambda)=[r_{ij}(\lambda)]_{i,j=1,\dots,n}=[p_{ij}(\lambda)q_{ij}(\lambda)^{-1}]$
having no singularities in $\overline{\Omega}$:

$\|R({\mathbf{T}})\|_{{}_{{\mathcal{B}}({\mathcal{H}}^{n})}}\leq\sup_{\lambda\in\Omega}\|R(\lambda)\|_{{}_{{\mathbb{C}}^{n\times
n}}}.$

The seminal result of Arveson (see [7]) is that $\Omega$ is a complete
spectral set for ${\mathbf{T}}$ if and only if there is a commutative
$d$-tuple of normal operators ${\mathbf{N}}=(N_{1},\dots,N_{d})$ on a larger
Hilbert space $\widetilde{\mathcal{K}}\supset{\mathcal{H}}$ with joint
spectrum contained in the distinguished boundary $b\Omega$ of $\Omega$ (in
which case we say that ${\mathbf{N}}$ is a $\Omega$-unitary for short) so
that, for any rational function $r$ with no singularities in
$\overline{\Omega}$ as above, it is the case that $r({\mathbf{T}})$ on
${\mathcal{H}}$ can be represented as the compression of $r({\mathbf{N}})$ to
${\mathcal{H}}$, i.e.,

$r({\mathbf{T}})=P_{\mathcal{H}}r({\mathbf{N}})|_{\mathcal{H}}$

where $P_{\mathcal{H}}$ is the orthogonal projection of ${\mathcal{K}}$ onto
${\mathcal{H}}$. It is easy to see that a necessary condition for $\Omega$ to
be a complete spectral set for a given operator $d$-tuple ${\mathbf{T}}$ is
that $\Omega$ be a spectral set for ${\mathbf{T}}$. The rational dilation
problem for a given domain $\Omega$ is to determine if the converse holds:
given $\Omega$, is it always the case that an operator tuple ${\mathbf{T}}$
having $\Omega$ as a spectral set in fact has $\Omega$ as a complete spectral
set (and hence then any ${\mathbf{T}}$ having $\Omega$ as a spectral set has a
$d\Omega$-normal dilation ${\mathbf{N}}$)?

Let us mention that it is often convenient to reformulate the problem of
existence of an $\Omega$-unitary dilation instead as the problem of existence
of a $\Omega$-isometric lift (see e.g. the introduction of [9]). Here we say
that the operator tuple ${\mathbf{V}}=(V_{1},\dots,V_{d})$ on a Hilbert space
${\mathcal{K}}$ is a $\Omega$-isometry if ${\mathbf{V}}$ extends to a
$\Omega$-unitary operator tuple ${\mathbf{U}}=(U_{1},\dots,U_{d})$ on a
Hilbert spaces $\widetilde{\mathcal{K}}\supset{\mathcal{K}}$. We say that
${\mathbf{V}}=(V_{1},\dots,V_{d})$ on ${\mathcal{K}}$ is a lift of
${\mathbf{T}}=(T_{1},\dots,T_{d})$ on ${\mathcal{H}}$ if
${\mathcal{H}}\subset{\mathcal{K}}$ and
$r({\mathbf{V}})^{*}|_{\mathcal{H}}=r({\mathbf{T}})^{*}$ for
$r\in\operatorname{Rat}(\Omega)$, or equivalently, ${\mathbf{V}}$ is a
coextension of ${\mathbf{T}}$ in the sense that

$P_{\mathcal{H}}r({\mathbf{V}})|_{\mathcal{H}}=r({\mathbf{T}})\text{ and
}r({\mathbf{V}}){\mathcal{H}}^{\perp}\subset{\mathcal{H}}^{\perp}\text{ for
}r\in\operatorname{Rat}(\Omega).$

It suffices to consider only minimal $\Omega$-unitary dilations and minimal
$\Omega$-isometric lifts. It is always the case that the restriction of a
$\Omega$-unitary dilation to the subspace
$\bigvee_{r\in\operatorname{Rat}(\Omega)}r({\mathbf{U}}){\mathcal{H}}$ gives
rise to a minimal $\Omega$-isometric lift, and conversely, the minimal
$\Omega$-unitary extension of a minimal $\Omega$-isometric lift gives rise to
a minimal $\Omega$-unitary dilation for ${\mathbf{T}}$. Finally we point out
that it is often convenient to be more flexible in the definition of an
$\Omega$-isometric lift and of an $\Omega$-unitary dilation by not insisting
that ${\mathcal{H}}$ is a subspace of ${\mathcal{K}}$ or
$\widetilde{\mathcal{K}}$ but rather allow an isometric identification map
$\Pi\colon{\mathcal{H}}\to{\mathcal{K}}$ and
$\widetilde{\Pi}\colon{\mathcal{H}}\to\widetilde{\mathcal{K}}$. Thus we say
that the pair $(\Pi,{\mathbf{V}})$ is an an $\Omega$-isometric lift for
${\mathbf{T}}$ on ${\mathcal{H}}$ if $\Pi\colon{\mathcal{H}}\to{\mathcal{K}}$
is an isometric embedding, ${\mathbf{V}}$ is $\Omega$-isometric on
${\mathcal{K}}$ and $r({\mathbf{V}})^{*}\Pi=\Pi r({\mathbf{T}})^{*}$ for
$r\in\operatorname{Rat}(\Omega)$, while $(\widetilde{\Pi},{\mathbf{U}})$ is a
$\Omega$-unitary dilation of ${\mathbf{T}}$ if
$\widetilde{\Pi}\colon{\mathcal{H}}\to\widetilde{\mathcal{K}}$ is an isometric
embedding, ${\mathbf{U}}$ is $\Omega$-unitary on $\widetilde{\mathcal{K}}$,
and $\Pi^{*}r({\mathbf{U}})\Pi=r({\mathbf{T}})$ for
$r\in\operatorname{Rat}(\Omega)$.

The motivating classical example for this setup is the case where $\Omega$ is
the unit disk ${\mathbb{D}}\subset{\mathbb{C}}$. In this case, the
distinguished boundary $b{\mathbb{D}}$ of ${\mathbb{D}}$ is the same as the
boundary $\partial{\mathbb{D}}$ which is the unit circle ${\mathbb{T}}$ and a
$b{\mathbb{D}}$-normal operator is just a unitary operator. Since
${\mathbb{D}}$ is polynomially convex, it suffices to work with polynomials
rather than rational functions with no poles in $\overline{\mathbb{D}}$. By
choosing the polynomial $p$ to be $p=\chi$ and $\chi(\lambda)=\lambda$, we see
that $\|T\|\leq 1$ (i.e., that $T$ be a contraction) is necessary for
${\mathbb{D}}$ to be a spectral set for $T$. The fact that this condition is
also sufficient, i.e., that the inequality

$\|p(T)\|\leq\sup_{\lambda\in{\mathbb{D}}}|p(\lambda)|$

holds for any contraction operator $T$ and polynomial $p$, is a classical
inequality known as von Neumann’s inequality going back to [36]. to show that
${\mathbb{D}}$ is a complete spectral set for any contraction operator $T$, we
may use the easier side of Arveson’s theorem and show instead that any
contraction operator $T$ has a ${\mathbb{D}}$-unitary dilation. But for the
case $\Omega={\mathbb{D}}$, according to our conventions, a
${\mathbb{D}}$-unitary operator is just a unitary operator $U$ (i.e.,
$U^{*}U=UU^{*}=I_{\widetilde{\mathcal{K}}}$). But any contraction operator $T$
on ${\mathcal{H}}$ dilating to a unitary operator $U$ on
$\widetilde{\mathcal{K}}\supset{\mathcal{H}}$ is exactly the content of the
Sz.-Nagy dilation theorem (see [34, Chapter II]).

Over the ensuing decades there have been sporadic attempts to find other
domains (both contained in ${\mathbb{C}}$ or more generally contained in
${\mathbb{C}}^{d}$) for which one can settle the rational dilation question
one way or the other (i.e., positively or negatively). Among single-variable
domains (as observed in the introduction of [9] where precise references are
given), it is known that rational dilation holds if
$\Omega\subset{\mathbb{C}}$ is a simply connected domain (simply use a
conformal map to reduce to the disk case) or is doubly-connected, but fails if
$\Omega$ has two or more holes (see [2, 23]). As for multivariable domains,
perhaps the first class to be understood are the polydisks ${\mathbb{D}}^{d}$
with $d\geq 2$: for $d=2$ rational dilation holds due to the Andô dilation
theorem [5] while for $d\geq 3$ rational dilation fails (see [31, 37]).

More recently the rational dilation problem has been investigated for other
concrete multivariable domains originally discovered due to connections with
the $\mu$-synthesis problem in Robust Control Theory (see the original Doyle-
Packard paper [22] as well as the book [26] for a more expository treatment).
We mention in particular the symmetrized bidisk

$\Gamma=\\{(s,p)\in{\mathbb{C}}^{2}\colon
s=(\lambda_{1}+\lambda_{2}),\,p=\lambda_{1}\lambda_{2}\text{ for some
}(\lambda_{1},\lambda_{2})\in{\mathbb{D}}^{2}\\}$ (1.2)

and a domain in ${\mathbb{C}}^{3}$ called the tetrablock and denoted by
${\mathbb{E}}$:

$\mathbb{E}:=\left\\{(a,b,\text{det}X):X=\begin{bmatrix}a&a^{\prime}\\\
b^{\prime}&b\end{bmatrix}\text{ with }\lVert X\rVert<1\right\\}.$ (1.3)

As might be expected, the domain $\Gamma$ behaves like ${\mathbb{D}}^{2}$ with
respect to the rational dilation problem as both domains are contained in
${\mathbb{C}}^{2}$: specifically, rational dilation holds for the domain
$\Gamma$ (see [3, 4, 15]) and there is a functional model analogous to the
Sz.-Nagy-Foias model for the disk case (see [4, 16]), at least for the pure
case. The situation of the rational dilation problem for the tetrablock
${\mathbb{E}}$ is less clear: there is a sufficient and a necessary condition
for the existence of a ${\mathbb{E}}$-isometric lift of a certain form [13, 9]
but a definitive resolution of the problem in full generality remains elusive
(see [9, 28]). However it is shown in [32] that, at least in the pure case, it
is still possible to construct a functional representation of a pure
$\Gamma$-contraction as the compression to ${\mathcal{H}}$ of a certain lift
triple $(A_{\ell},B_{\ell},T_{\ell})$ which formally looks like an tetrablock
isometry but is not guaranteed to satisfy all of the required commutativity
conditions. A similar phenomenon holds for the case where
$\Omega={\mathbb{D}}^{d}$ with $d\geq 2$ (see [11]): for this case, as pointed
out above, there are indeed counterexamples to show that rational dilation
fails, but there is nevertheless a weaker type of lift (called pseudo-
commutative ${\mathbb{D}}^{d}$-isometric lift)) which generates a functional
model for the given ${\mathbb{D}}^{d}$-contractive $d$-tuple
${\mathbf{T}}=(T_{1},\dots,T_{d})$ even when rational dilation fails.

In this paper we focus on the case $\Omega={\mathbb{E}}$. As was the case in
[13], the most definitive results are for the case of what we shall call a
special tetrablock contraction, i.e., a tetrablock contraction $(A,B,T)$ which
has a tetrablock isometric lift $(V_{1},V_{2},V_{3})$ such that $V_{3}=V$ is a
Sz.-Nagy-Foias minimal isometric lift for the single contraction operator $T$.
As in [13], we identify the additional commutativity conditions (2.8) which
must be imposed on the Fundamental Operator pair $(G_{1},G_{2})$ of
$(A^{*},B^{*},T^{*})$ which characterizes when $(A,B,T)$ is special. There
results a Douglas-type functional model (as in [21] for the single contraction
operator setting) for the tetrablock contraction which also exhibits the
tetrablock isometric lift $(V_{1},V_{2},V_{3})$, all in a functional-model
form rather than via block-matrix constructions as in [13]. This Douglas-type
model can in turn be converted to a Sz.-Nagy-Foias-type model; the Sz.-Nagy-
Foias characteristic function $\Theta_{T}$ for the contraction operator $T$,
together with the the fundamental operators $(G_{1},G_{2})$ for the adjoint
tetrablock contraction $(A^{*},B^{*},T^{*})$, along with some additional
information needed to handle the case where $T$ is not a pure contraction,
form what we call a characteristic tetrablock data set for $(A,B,T)$ in terms
of which one can write down the functional model. Conversely, we identify a
collection of objects which we call a special tetrablock data set:
specifically, (i) a pure contractive operator function
$({\mathcal{D}},{\mathcal{D}}_{*},\Theta)$, (ii) a pair of operators
$(G_{1},G_{2})$ on the coefficient space ${\mathcal{D}}_{*}$, (iii) a
tetrablock unitary $(R,S,W)$ acting on $\overline{D_{\Theta}\cdot
L^{2}({\mathcal{D}})}$, such that (iv) all these together satisfy a natural
invariant-subspace compatibility condition. From such a characteristic
tetrablock data set we construct a functional model such that the embedded
functional-model operator triple is the most general special tetrablock
contraction up to unitary equivalence, with its special tetrablock isometric
lift also embedded in the functional model. We also are careful to push the
theory as far as we can without the assumption that the original tetrablock
contraction is special. In this case we identify a class of operator triples
$(V_{1},V_{2},V_{3})$ with $V_{3}$ equal to a minimal isometric lift for $T$
to which $(A,B,T)$ can be lifted: here $V_{!}$ and $V_{2}$ commute with
$V_{3}$ but not necessarily with each other and it appears that $V_{1}$,
$V_{2}$ need not be contractions. In this case there is no converse direction:
there is no guarantee that the compression of a general pseudo-commutative
tetrablock isometry $(V_{1},V_{2},V_{3})$ on ${\mathcal{K}}$ back to
${\mathcal{H}}$ will yield a tetrablock contraction.

Let us note that the recent paper of Bisai and Pal [18] contains closely
related results. These authors basically compute the $Z$-transform of the
Schäffer-type construction of the unique special tetrablock isometric lift
$(V_{1},V_{2},V_{3})$ (where $V_{3}$ is equal to the minimal Sz.-Nagy
isometric lift of $T$) to arrive at a functional model for this lift. Our
approach on the other hand uses the Douglas lifting approach to construct the
functional model directly with the existence and uniqueness of the special
tetrablock isometric lift falling out as part of the construction. When the
tetrablock contraction is not special and no such lift is possible, the same
construction still leads to a functional model but $(V_{1},V_{2},V_{3})$ is
only a pseudo-commutative tetrablock isometry and there is no tetrablock
isometric lift constructed in this way. The results for the special case arise
as a special case (the case where the Fundamental Operator pair
$(G_{1},G_{2})$ for the tetrablock contraction $(A^{*},B^{*},T^{*})$ satisfy
the additional commutativity conditions (2.8)) of the general functional-model
construction. The paper [18] also obtains a noncommtative functional model for
a non-special case, based on the work of Durszt [27] (a variation of the
approach of Douglas for the construction of the minimal isometric lift for the
case of a single contraction operator $T$), but with the additional hypothesis
that $A$ and $B$ commute not only with $T$ but also with $T^{*}$. It is clear
that the complete unitary invariant for a pure tetrablock contraction
$(A,B,T)$ consists of the characteristic function $\Theta_{T}$ of $T$ together
with the Fundamental Operator pair $(G_{1},G_{2})$ of $(A^{*},B^{*},T^{*})$;
for the non-pure case (where $\Theta_{T}$ is no longer inner) we add a certain
tetrablock unitary $(R,S,W)$ acting on
$\overline{\Delta_{T}H^{2}({\mathcal{D}}_{T})}$ which is part of our model
(see Theorem 4.5 below), while Bisai-Pal add the Fundamental Operator pair
$(F_{1},F_{2})$ for $(A,B,T)$ and argue that
$(\Theta_{T},(F_{1},F_{2}),G_{1},G_{2}))$ is a complete unitary invariant. It
remains to be seen which is the more relevant and useful in the future.

It is now becoming clear that the domains ${\mathbb{D}}^{d}$ (polydisk),
$\Gamma$ (symmetrized bidisk), ${\mathbb{E}}$ (tetrablock) as well as
${\mathbb{D}}^{d}_{s}$ (symmetrized polydisk) all have common features with
respect to the associated operator theory and applications to the rational
dilation problem for each of these domains. The paper [11] shows how a program
completely parallel to that done here for the tetrablock case can be worked
out equally well for the polydisk case $\Omega={\mathbb{D}}^{d}$ (where
rational dilation is known to fail when $d\geq 3$). In all these settings,
there appear a pair of unitary invariants called Fundamental Operators which
play a key role as part of a set (including the Sz.-Nagy–Foias characteristic
function of an appropriate contraction operator determined by the operator
tuple) of unitary invariants for the operator tuple of whatever class. The
notion of Fundamental Operators as a fundamental object of interest seems to
have appeared first in connection with the symmetrized bidisk $\Gamma$ [16],
then in connection with the un-symmetrized polydisk [33, 11], and now also in
connection with the symmetrized polydisk (see [30]). Often the proper notion
of Fundamental Operators for one setting is found by making a correspondence
of the less understood setting with some other better understood setting, and
then adapting definitions for the first to become definitions for the second.
In particular, many of the results for the tetrablock case were originally
found by adapting from results for the symmetrized bidisk case (see e.g.
[13]), and it has been shown how one can deduce the bidisk functional model
from the tetrablock functional model (see [33]). In this spirit in a future
publication we plan to show how the results from [11] for the polydisk case
(most of which are just statements parallel to what is done here for the
tetrablock case) can alternatively be derived as a corollary of the
corresponding results for the tetrablock case via the simple observation: if
${\mathbf{T}}=(T_{1},\dots,T_{d})$ is a commutative, contractive operator
$d$-tuple, then for $1\leq i\leq d$, if we set $T_{(i)}=\Pi_{1\leq j\leq
d\colon j\neq i}T_{j}$, then for each $i=1,\dots,d$ the $d$-tuple
$(A_{i},B_{i},P)=(T_{i},T_{(i)},\Pi_{1\leq j\leq d}T_{j})$ is a tetrablock
contraction; the $d=2$ case can be found in [33, Section 3, Version 3].

Finally, let us point out that it is possible to reformulate the rational
dilation problem for a given domain $\Omega$ as a problem about unital
representations of a unital function algebra: given a contractive
representation $\pi\colon
f\in{\mathcal{A}}\mapsto\pi(f)\in{\mathcal{B}}({\mathcal{H}})$ which is
contractive
($\|\pi(f)\|_{{\mathcal{B}}({\mathcal{H}})}\leq\|f\|_{{\mathcal{B}}({\mathcal{H}})}$
where the unital representation property is that
$\pi(1_{\mathcal{A}})=I_{\mathcal{H}}$ and $\pi(f_{1}\cdot
f_{2})=\pi(f_{1})\pi(f_{2})$, is it automatically the case that the
representation is completely contractive, i.e., still contractive after
tensoring with ${\mathbb{C}}^{n\times n}$ for any $n\in{\mathbb{N}}$? To
recover the original formulation as a special case, one can take
${\mathcal{A}}=\overline{\operatorname{Rat}(\Omega)}$ where the closure is in
the $C^{*}$-algebra $C(b\Omega)$ (continuous functions on the distinguished
boundary $b\Omega$). However with this more general formulation one can
consider function algebras which go beyond
$\overline{\operatorname{Rat}({\Omega})}$, e.g., the constrained subalgebra
${\mathbb{C}}\cdot 1+z^{2}{\mathbb{A}}({\mathbb{D}})$ of the disk algebra
${\mathbb{A}}({\mathbb{D}})=\overline{\operatorname{Rat}({\mathbb{D}})}$.
Alternatively, it is often possible to represent the algebra ${\mathcal{A}}$
as conformally equivalent to the algebra of all functions analytic on some
algebraic curve $\Omega={\mathbf{C}}$ embedded in some higher-dimensional
closed complex manifold (the Neil parabola intersected with the bidisk for the
case of ${\mathbb{C}}\cdot 1+z^{2}{\mathbb{A}}({\mathbb{D}})$). For the state
of knowledge (up to 2018) on this direction of dilation theory including much
discussion and references on earlier work, we refer to the paper of Dritschel
and Undrakh [25]. We shall not pursue this direction here.

The paper is organized as follows. After the present Introduction, in Section
2 we collect assorted definitions and illustrative results concerning
tetrablock contractions, tetrablock isometries, and tetrablock unitaries,
including a direct proof of the existence of the Fundamental Operator pair for
a given tetrablock contraction, which will be needed in the sequel. Here we
also show how to associate a tetrablock unitary $(R,S,W)$ with a tetrablock
contraction $(A,B,T)$ in a canonical way; this is the key ingredient needed to
eliminate the purity assumption on the contraction operator $T$ required in
earlier work on this problem (see [32]). Section 3 shows how a lifting
framework for the tetrablock-contraction setting can be constructed as an
embellishment of the Douglas-model lifting framework [21] originally
formulated as an approach to the Sz.-Nagy dilation theorem for a single
contraction operator $T$, with the pseudo-commutative tetrablock-isometric
lift $(V_{1},V_{2},V_{3})$ having $V_{3}=T$ and $V_{1}$ and $V_{2}$
constructed by making use of the Fundamental Operator pair for the tetrablock
contraction $(A^{*},B^{*},T^{*})$. The final Section 4 identifies the
invariants required to write down a functional model equipped with a model
operator triple $(A,B,T)$ which is concrete functional-model version of a
general tetrablock contraction.

##  2\. The fundamentals of tetrablock contractions

This section gives a brief introduction to the operator theory associated with
the tetrablock.

###  2.1. Tetrablock contractions

The tetrablock, denoted by ${\mathbb{E}}$, is the non-convex but polynomially
convex domain in $\mathbb{C}^{3}$ given by (1.3). From this formula for
${\mathbb{E}}$ is is easy to read off the following symmetry properties.

######  Proposition 2.1.

The tetrablock ${\mathbb{E}}$ has the following symmetry properties:

  1. (1)

${\mathbb{E}}$ is invariant under complex conjugation:

$(a,b,t)\in{\mathbb{E}}\Leftrightarrow(\overline{a},\overline{b},\overline{t})\in{\mathbb{E}}.$

  2. (2)

${\mathbb{E}}$ is invariant under interchange of the first two coordinates:

$(a,b,t)\in{\mathbb{E}}\Leftrightarrow(b,a,t)\in{\mathbb{E}}.$

The distinguished boundary of ${\mathbb{E}}$, i.e., the Šilov boundary with
respect to the algebra of functions that are analytic in ${\mathbb{E}}$ and
continuous on $\overline{{\mathbb{E}}}$, is given by

$b{\mathbb{E}}:=\left\\{(a,b,\text{det}X):X=\begin{bmatrix}a&a^{\prime}\\\
b^{\prime}&b\end{bmatrix}\text{ is a unitary }\right\\}$

(see [1, Theorem 7.1]). From this characterization it is easy to see that
$b{\mathbb{E}}$ is also invariant under the two involutions
$(a,b,t)\mapsto(\overline{a},\overline{b},\overline{t})$ and
$(a,b,t)\mapsto(b,a,t)$.

Several tractable characterizations of the tetrablock can be found in [1,
Theorem 2.2]; we pick two of these that will be used in what follows.

######  Theorem 2.2.

For a point $(a,b,t)\in{\mathbb{C}}^{3}$, the following are equivalent:

  * (i)

$(a,b,t)\in{\mathbb{E}};$

  * (ii)

with the rational function
$\Psi:\overline{{\mathbb{D}}}\times{\mathbb{C}}^{3}\to{\mathbb{C}}$ defined as

$\displaystyle\Psi(z,(a,b,t))=\frac{a-zt}{1-zb},$ (2.1)

$\sup_{z\in\overline{{\mathbb{D}}}}|\Psi(z,(a,b,t))|<1$; and if $ab=t$ then,
in addition, $|b|<1$;

  * (iii)

with $\Psi$ as in (2.1),
$\sup_{z\in\overline{{\mathbb{D}}}}|\Psi(z,(b,a,t))|<1$; and if $ab=t$ then,
in addition, $|a|<1$.

Moreover, when item (i) is replaced by $(a,b,t)\in\overline{{\mathbb{E}}}$,
then all the strict inequalities in items (ii) and (iii) are replaced by non-
strict inequalities.

######  Remark 2.3.

Note that in Theorem 2.2, the equivalence of (i) $\Leftrightarrow$ (iii) is an
immediate consequence of the equivalence of (i) $\Leftrightarrow$ (ii) in view
of the invariance of ${\mathbb{E}}$ under the involution
$(a,b,t)\mapsto(b,a,t)$.

Recall the notions of tetrablock unitary, tetrablock isometry and tetrablock
contraction given in the Introduction. Several algebraic characterizations of
tetrablock isometries and tetrablock unitaries are known; see Theorems 5.4 and
5.7 in [13]. We recall the ones that are useful for our purposes here. Here we
use the notation $r(X)$ for the spectral radius of a Hilbert-space operator
$X$. It is then not difficult to see that the ${\mathbb{E}}$-symmetries noted
in Proposition 2.1 imply the same symmetries on the respective operator
classes (with respect to the class of ${\mathbb{E}}$-isometries which requires
a little extra care), as noted in the next result. We leave the easy
verification as an exercise for the reader.

######  Proposition 2.4.

Suppose that $(A,B,T)$ is a triple of bounded operators on a Hilbert space
${\mathcal{H}}$. Then:

  1. (1)

$(A,B,T)$ is a ${\mathbb{E}}$-contraction $\Leftrightarrow$
$(A^{*},B^{*},T^{*})$ is a ${\mathbb{E}}$-contraction $\Leftrightarrow$
$(B,A,T)$ is a ${\mathbb{E}}$-contraction.

  2. (2)

$(A,B,T)$ is a ${\mathbb{E}}$-isometry $\Leftrightarrow$ $(B,A,T)$ is a
${\mathbb{E}}$-isometry.

  3. (3)

$(A,B,T)$ is a ${\mathbb{E}}$-unitary $\Leftrightarrow$ $(A^{*},B^{*},T^{*})$
is a ${\mathbb{E}}$-unitary $\Leftrightarrow$ $(B,A,T)$ is a
${\mathbb{E}}$-unitary.

######  Theorem 2.5.

Let $(A,B,T)$ be a commutative triple of bounded Hilbert space operators. Then
the following are equivalent:

  * (i)

$(A,B,T)$ is a tetrablock isometry (respectively unitary);

  * (ii)

$(A,B,T)$ is a tetrablock contraction and $T$ is an isometry (respectively
unitary);

  * (iii)

$A=B^{*}T$, $B$ is a contraction and $T$ is an isometry (respectively
unitary); and

  * (iv)

$B=A^{*}T$, $A$ is a contraction and $T$ is an isometry (respectively
unitary).

  * (v)

$B=A^{*}T$, $r(A)\leq 1$ and $r(B)\leq 1$, and $T$ is an isometry
(respectively unitary).

###  2.2. Pseudo-commutative tetrablock isometries and unitaries

We propose to introduce the notions of pseudo-commutative tetrablock unitary
and pseudo-commutative tetrablock isometry for an operator triple $(A,B,T)$ by
using criterion (iii) or equivalently (iv) in Theorem 2.5 but with the
weakening the commutativity hypothesis imposed on the whole triple $(A,B,T)$
to just the condition that $A$ and $B$ commute with $T$ (but not necessarily
with each other). As we are also dropping the condition that $A$ or $B$ be a
contraction, a more proper term would be noncontractive pseudo-commutative
tetrablock isometry, but, as this term will be consistent throughout, we
settle on the shorter term for brevity. The resulting definition is as
follows. We leave it to the reader to verify that the two formulations are
equivalent.

######  Definition 2.6.

Let $(A,B,T)$ be a triple of bounded Hilbert-space operators. We say that the
triple $(A,B,T)$ is a pseudo-commutative tetrablock isometry (respectively,
unitary) if any of the following equivalent conditions holds:

  1. (1)

$T$ is an isometry (respectively, unitary) and

$AT=TA,\quad BT=TB,\quad A=B^{*}T.$ (2.2)

  2. (2)

$T$ is an isometry (respectively, unitary), and

$AT=TA,\quad BT=TB,\quad B=A^{*}T.$ (2.3)

######  Remark 2.7.

From Definition 2.6 and Theorem 2.5, we see that any tetrablock
isometry/unitary is also a pseudo-commutative tetrablock isometry/unitary but
not conversely. If we wish to emphasize that we are referring to the logically
more special tetrablock isometry/ unitary rather than the more general pseudo-
commutative tetrablock isometry/unitary, we often will say strict tetrablock
isometry/unitary for emphasis.

######  Remark 2.8.

We now present a couple of elementary observations on pseudo-commutative
versus strict tetrablock unitaries unitaries which we hope give the reader
some additional insight.

(1) We remark that if $(A,B,T)$ is a pseudo-commutative tetrablock unitary,
then $A$ and $B$ are not necessarily normal operators as would happen in the
strict case. For example, pick a non-normal contraction $G_{1}$ acting on a
Hilbert space ${\mathcal{E}}$ and consider the triple $(M_{G_{1}^{*}},M_{\zeta
G_{1}},M_{\zeta})$ on $L^{2}({\mathcal{E}})$. It is easy to see that this
triple is a pseudo-commutative tetrablock unitary. However, neither $A$ nor
$B$ is normal unless $G_{1}$ is so. Also note that if $(A,B,T)$ is a pseudo-
commutative tetrablock unitary, then so is the adjoint triple
$(A^{*},B^{*},T^{*})$. This can be seen by observing that the adjoint of the
identities in (2.2) with $T$ unitary can be converted to the identities (2.3)
for $(A^{*},B^{*},T^{*})$ with $T^{*}$ still unitary. Note next that if
$(A,B,T)$ is a pseudo-commutative tetrablock unitary, then

$\displaystyle A^{*}A=T^{*}BB^{*}T=BT^{*}TB^{*}=BB^{*},$ $\displaystyle
B^{*}B=T^{*}AA^{*}T=AT^{*}TA^{*}=AA^{*}.$

Thus we always have

$A^{*}A=BB^{*},\quad AA^{*}=B^{*}B$ (2.4)

for a pseudo-commutative tetrablock unitary $(A,B,T)$. As a first consequence
of (2.4), we see that if $A$ is normal, then

$B^{*}B=AA^{*}=A^{*}A=BB^{*}$

and $B$ is also normal. Similarly if $B$ is normal, then $A$ is also normal.
In conclusion, if $(A,B,T)$ is a pseudo-commutative unitary such that one of
$A$ or $B$ is normal, then so is the other.

(2) We note as a consequence of (iii) $\Rightarrow$ (i) in Theorem 2.5 that in
particular if $(A,B,T)$ is a strict tetrablock unitary (so we also have
$AB=BA$), then the operators $A$ and $B$ are normal.  One can see this
directly from the considerations here as follows. As a strict tetrablock
unitary in particular meets all the requirements for membership in the pseudo-
commutative tetrablock unitary class, we know that (2.4) holds. Combining this
with the commutativity relation $AB=BA$ then gives us

$A^{*}A=A^{*}B^{*}T=B^{*}A^{*}T=B^{*}B=AA^{*}$

showing that $A$ is normal. The same computation with the roles of $A$ and $B$
interchanged then shows that $B$ is also normal. The full strength of (iii)
$\Rightarrow$ (i) in Theorem 2.5 is that in addition the commutative normal
triple $(A,B,T)$ has joint spectrum in the boundary of the tetrablock
${\mathbb{E}}$; for this somewhat deeper fact we refer to [13].

The next result gives a feel for how close pseudo-commutative tetrablock
isometries come to being strict tetrablock isometries.

######  Theorem 2.9.

Let $(A,B,T)$ be a pseudo-commutative tetrablock isometry on a Hilbert space
${\mathcal{H}}$.

  1. (1)

Then the spectral radius $r(AB)$ of the product operator $AB$ is given by

$r(AB)=\operatorname{max}\\{\|A\|^{2},\,\|B\|^{2}\|.$ (2.5)

  2. (2)

Suppose in addition that $AB=BA$ and $r(A)\leq 1$, $r(B)\leq 1$. Then both $A$
and $B$ are contraction operators ($\max\\{\|A\|,\|B\|\\}\leq 1$) and
$(A,B,T)$ is a strict tetrablock isometry.

###### Proof.

The proof follows the ideas of Bhattacharyya [13, pp.1619-1620]. We first
consider statement (1). Form two operators
$X_{1}=\left[\begin{smallmatrix}0&A\\\ B&0\end{smallmatrix}\right]$ and
$X_{2}=\left[\begin{smallmatrix}T&0\\\ 0&T\end{smallmatrix}\right]$ on
$\left[\begin{smallmatrix}{\mathcal{H}}\\\
{\mathcal{H}}\end{smallmatrix}\right]$. From the two relations $B^{*}T=A$ and
$A^{*}T=B$ we deduce that $X_{1}=X_{1}^{*}X_{2}$ where
$X_{2}X_{2}^{*}=\left[\begin{smallmatrix}TT^{*}&0\\\
0&TT^{*}\end{smallmatrix}\right]\preceq\left[\begin{smallmatrix}I&0\\\
0&I\end{smallmatrix}\right]$ since $T$ is an isometry. Hence

$X_{1}X_{1}^{*}=X_{1}^{*}X_{2}X_{2}^{*}X_{1}\preceq X_{1}^{*}X_{1},$

i.e., $X_{1}$ is a hyponormal operator. By a theorem of Stampfli (see [19,
Proposition 4.6], it follows that $r(X_{1})=\|X_{1}\|$. We compute the
operator norm of $X_{1}$ as follows:

$\|X_{1}\|^{2}=\bigg{\|}\begin{bmatrix}0&A\\\
B&0\end{bmatrix}\begin{bmatrix}0&B^{*}\\\
A^{*}&0\end{bmatrix}\bigg{\|}=\bigg{\|}\begin{bmatrix}AA^{*}&0\\\
0&BB^{*}\end{bmatrix}\bigg{\|}=\operatorname{max}\\{\|A\|^{2},\|B\\\ ^{2}\\}.$

and hence $\|X_{1}\|=\max\\{\|A\|,\|B\|\\}$. To compute $r(X_{1})$, note first
that $X_{1}^{*}X_{1}=\left[\begin{smallmatrix}AB&0\\\
0&BA\end{smallmatrix}\right]$ and hence

$X_{1}^{2n}=\begin{bmatrix}(AB)^{n}&0\\\ 0&(BA)^{n}\end{bmatrix}.$

Consequently,

$r(X_{1})=\lim_{n\to\infty}\max\|\|(AB)^{n}\|^{\frac{1}{2n}},\|(BA)^{n}\|^{\frac{1}{2n}}\\}=\max\\{r(AB)^{\frac{1}{2}},r(BA)^{\frac{1}{2}}\\}.$

However a general fact is that the nonzero spectrum of $AB$ is the same as the
nonzero spectrum of $BA$, and hence $r(AB)=r(BA)$. Thus $r(X_{1})=\|X_{1}\|$
gives us (2.5) and the proof of statement (1) is complete.

As for statement (2), a known fact is that if $A$ and $B$ commute, then the
spectrum of the product operator $AB$ is given by

$\sigma(AB)=\\{\lambda\cdot\mu\colon\lambda\in\sigma(A),\,\mu\in\sigma(B)\\}.$

Hence the hypothesis that $r(A)\leq 1$ and $r(B)\leq 1$ implies that
$r(AB)\leq 1$ as well. But then from the conclusion of statement (1) already
proved, we conclude that both $A$ and $B$ are contraction operators, and the
proof of statement (2) is now complete. (Note that this also proves (v)
$\Rightarrow$ (iii) or (iv) in Theorem 2.5.) ∎

######  Example 2.10.

(1) A pseudo-commutative/strict tetrablock isometry. Let ${\mathcal{E}}$ be a
coefficient Hilbert space and $H^{2}({\mathcal{E}})=H^{2}\otimes{\mathcal{E}}$
be the associated Hardy space of ${\mathcal{E}}$-valued functions. Let $G_{1}$
and $G_{2}$ be operators on ${\mathcal{E}}$ and set

$A=M_{G_{1}^{*}+zG_{2}},\quad B=M_{G_{2}^{*}+zG_{1}},\quad T=M_{z}\text{ on
}H^{2}({\mathcal{E}}).$ (2.6)

Then it is immediate that $T$ is an isometry and that $A$ and $B$ commute with
$T$. The special coupled form of the pencils defining $A$ and $B$ enables us
to show that $A=B^{*}T$:

$\displaystyle B^{*}T$ $\displaystyle=(I_{H^{2}}\otimes G_{2}^{*}+M_{z}\otimes
G_{1})^{*}\cdot(M_{z}\otimes I_{\mathcal{G}})$
$\displaystyle=(I_{H^{2}}\otimes G_{2}+M_{z}^{*}\otimes
G_{1}^{*})\cdot(M_{z}\otimes I_{\mathcal{G}})$
$\displaystyle=(I_{H^{2}}\otimes G_{1}^{*})+(M_{z}\otimes
G_{2})=M_{G_{1}^{*}+zG_{2}}=A$

and similarly $B=A^{*}T$.

For $(A,B,T)$ to be a strict tetrablock isometry, we need in addition that
$AB=BA$ and that $\|A\|\leq 1$ (in which case also $\|B\|=\|A^{*}T\|\leq 1$ as
well). To ensure that $\|A\|\leq 1$ requires that $G_{1}$ and $G_{2}$ are not
too large in the precise sense that

$\sup_{z\in{\mathbb{T}}}\|G_{1}^{*}+zG_{2}\|\leq 1.$ (2.7)

To check the condition $AB=BA$, we compute

$\displaystyle AB$ $\displaystyle=M_{G_{1}^{*}+zG_{2}}M_{G_{2}^{*}+zG_{1}}$
$\displaystyle=(I_{H^{2}}\otimes G_{1}^{*}+M_{z}\otimes
G_{2})\cdot(I_{H^{2}}\otimes G_{2}^{*}+M_{z}\otimes G_{1})$
$\displaystyle=I_{H^{2}}\otimes
G_{1}^{*}G_{2}^{*}+M_{z}\otimes(G_{1}^{*}G_{1}+G_{2}G_{2}^{*})+M_{z}^{2}\otimes
G_{1}G_{2}$

while a similar computation gives us

$BA=I_{H^{2}}\otimes
G_{2}^{*}G_{1}^{*}+M_{z}\otimes(G_{1}G_{1}^{*}+G_{2}^{*}G_{2})+M_{z}^{2}\otimes
G_{2}G_{1}.$

We conclude that in this example, $(A,B,T)$ is a strict tetrablock isometry
exactly when (2.7) together with the following commutativity conditions hold:

$G_{1}G_{2}=G_{2}G_{1},\quad
G_{1}^{*}G_{1}+G_{2}G_{2}^{*}=G_{1}G_{1}^{*}+G_{2}^{*}G_{2},$ (2.8)

sometimes also written more compactly in terms of commutators as

$[G_{1},G_{2}]=0,\quad[G_{1}^{*},G_{1}]=[G_{2}^{*},G_{2}]$

where in general $[X,Y]$ is the commutator:

$[X,Y]=XY-YX.$

(2) A pseudo-commutative/strict tetrablock unitary. It is easy to use the
spectral theory for unitary operators (a particular case of the spectral
theory of normal operators) to write down a model for the general pseudo-
commutative/strict tetrablock unitary $(R,S,W)$. as follows. By Definition 2.6
we see in particular that $W$ is unitary. By the spectral theory for general
normal operators (see e,g, any of [20, 6] or [8, Chapter 2]), after a unitary
change of coordinates, we can represent $W$ as the operator $M_{\zeta}$ of
multiplication by the coordinate function ($M_{\zeta}\colon
h(\zeta)\mapsto\zeta h(\zeta)$) on a direct-integral space
$\bigoplus\int_{\mathbb{T}}{\mathcal{H}}_{\zeta}\,\nu({\tt d}\zeta)$
determined by a scalar spectral measure $\nu$ supported on ${\mathbb{T}}$ and
a measurable multiplicity function $\zeta\mapsto\dim{\mathcal{H}}_{\zeta}$..
Since the operators $R$ and $S$ commute with $W=M_{\zeta}$, it follows that
$R$ and $S$ are represented as decomposable operators on
$\bigoplus\int_{\mathbb{T}}{\mathcal{H}}_{\zeta}\,\tt{d}\nu(\zeta)$, i.e.,
$R=M_{\phi}\colon h(\zeta)\mapsto\phi(\zeta)h(\zeta)$ and $S=M_{\psi}\colon
h(\zeta)\mapsto\psi(\zeta)h(\zeta)$ for measurable functions such that
$\phi(\zeta)\in{\mathcal{B}}({\mathcal{H}}_{\zeta})$,
$\psi(\zeta)\in{\mathcal{B}}({\mathcal{H}}_{\zeta})$ for a.e. $\zeta$. The
fact that in addition $R=S^{*}W$ then forces
$\phi(\zeta)=\psi(\zeta)^{*}\cdot\zeta$ for a.e. $\zeta$, Thus any pseudo-
commutative tetrablock unitary has the form

$(R,S,W)=(M_{\psi^{*}\cdot\zeta},M_{\psi},M_{\zeta})\text{ acting on
}\bigoplus\int_{\mathbb{T}}{\mathcal{H}}_{\zeta}\,\nu(\tt{d}\zeta)$ (2.9)

If $(R,S,W)$ is a strict tetrablock unitary, then in addition we must have
that $\psi(\zeta)$ is a contractive normal operator on ${\mathcal{H}}_{\zeta}$
for a.e. $\zeta$ in order to guarantee in addition that $RS=SR$ and that
$\|R\|\leq 1$, $\|S\|\leq 1$. By this analysis we conclude that (2.9) (with
$\psi(\zeta)$ constrained to be contractive normal for a.e. $\zeta$ for the
strict case) is the general form for a pseudo-commutative/strict tetrablock
unitary. In a less-functional form, to write a pseudo-contractive tetrablock
triple $(R,S,W)$, the free parameters are: (i) a unitary operator $W$, and
(ii) an operator $S$ commuting with $W$; then the associated pseudo-
commutative tetrablock contraction is $(W^{*}S,S,W)$; for this to be strict,
one must require in addition that the operator $S$ in the commutant of $W$ be
a normal contraction.

To deduce the von Neumann-Wold decomposition for a tetrablock isometry, the
next lemma is useful. Only the special case where the operator $S$ in the
statement is a shift will be needed for our application, in which case the
result is well-known (see e.g. [35, page 22]). For completeness we present
here a proof of the general result.

######  Lemma 2.11.

Let $W$ be a unitary operator on ${\mathcal{H}}_{2}$, $S$ an operator on
${\mathcal{H}}_{1}$ such that $S^{*n}\to 0$ in the strong operator topology as
$n\to\infty$. If $X$ is a bounded operator from ${\mathcal{H}}_{2}$ to
${\mathcal{H}}_{1}$ such that $XW=SX$, then $X=0$.

###### Proof.

From $XW=SX$ we get by iteration that $XW^{n}=S^{n}X$ for $n=1,2,\dots$.
Taking adjoints gives then $W^{*n}X^{*}=X^{*}S^{*n}$. Apply this identity to
an arbitrary fixed vector $x\in{\mathcal{H}}_{2}$ to get
$W^{*n}X^{*}x=X^{*}S^{*n}x$ for all $n\geq 1$. Apply $W^{n}$ to both sides of
this equation to get $W^{n}W^{*n}X^{*}x=W^{n}X^{*}S^{*n}x$ for $n=1,2,\dots$.
As $W$ is unitary, this becomes $X^{*}x=W^{n}X^{*}S^{*n}x$. Taking norms then
gives

$\|X^{*}x\|=\|W^{n}X^{*}S^{*n}x\|=\|X^{*}S^{*n}x\|\leq\|X^{*}\|\|S^{*n}x\|\to
0\text{ as }n\to\infty$

by the assumed strong convergence of powers of $S^{*}$ to zero, forcing
$X^{*}$ (and hence also $X$) to be the zero operator. ∎

The von Neumann-Wold decomposition (see [38, 36, 34]) ensures that if $T$ is
an isometry acting on a Hilbert space ${\mathcal{H}}$, then $T$ can be
represented as an operator as the external direct sum $M_{z}\oplus U$ of a
shift operator $M_{z}$ acting on a Hardy space $H^{2}({\mathcal{E}})$ and a
unitary operator $W$ on ${\mathcal{F}}$ for some coefficient Hilbert spaces
${\mathcal{E}}$ and ${\mathcal{F}}$. The following result not only gives a
model for an arbitrary pseudo-commutative/strict tetrablock isometry
$(A,B,T)$, but also can be seen as a pseudo-commutative/strict tetrablock-
isometry analogue of the classical von Neumann–Wold decomposition for a single
isometric Hilbert space operator $T$.

######  Theorem 2.12.

Let $(A,B,T)$ be an operator-triple on the Hilbert space ${\mathcal{H}}$.

(1) Then $(A,B,T)$ is a pseudo-commutative tetrablock isometry on
${\mathcal{H}}$ if and only if there exist Hilbert spaces ${\mathcal{E}}$,
${\mathcal{F}}$, operators $G_{1},G_{2}$ acting on ${\mathcal{E}}$ subject to
(2.7), along with a pseudo-commutative tetrablock unitary $(R,S,W)$ acting on
${\mathcal{F}}$, such that ${\mathcal{H}}$ is isomorphic to
$\left[\begin{smallmatrix}H^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{smallmatrix}\right]$ and under the same isomorphism
$(A,B,T)$ is unitarily equivalent to

$\displaystyle\left(\begin{bmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\
0&R\end{bmatrix},\begin{bmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&S\end{bmatrix},\begin{bmatrix}M_{z}&0\\\ 0&W\end{bmatrix}\right).$ (2.10)

(2) Then $(A,B,T)$ is a strict tetrablock isometry on ${\mathcal{H}}$ if and
only if $(A,B,T)$ is unitarily equivalent to the operator triple as in (2.10)
(with $G_{1}$, $G_{2}$ subject to (2.7)) acting on a space
$\left[\begin{smallmatrix}H^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{smallmatrix}\right]$, where in addition the operator-pencil
coefficients $(G_{1},G_{2})$ satisfy the system of operator identities (2.8),
and the triple $(R,S,W)$ is a strict tetrablock unitary (i.e., we also have
the relation $RS=SR$ with $R$ and $S$ contraction operators).

######  Remark 2.13.

We shall think of a triple of operators on
$\left[\begin{smallmatrix}H^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{smallmatrix}\right]$ as in (2.10) as a functional model for
a pseudo-commutative/strict tetrablock isometry/unitary. The
$H^{2}({\mathcal{E}})$-component clearly has a functional form while the
second component can be brought to a measure-theoretic functional form as in
item (2) in Example 2.10.

###### Proof.

The sufficiency (for both the pseudo-commutative and the strict case) follows
from Example 2.10.

We now suppose that $(A,B,T)$ is a strict tetrablock isometry. Let us apply
the Wold decomposition to the isometry $T$: there exist Hilbert spaces
${\mathcal{E}}$, ${\mathcal{F}}$, and a unitary
$\tau:{\mathcal{H}}\to\left[\begin{smallmatrix}H^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{smallmatrix}\right]$ such that

$\tau T\tau^{*}=\begin{bmatrix}M_{z}&0\\\
0&W\end{bmatrix}:\begin{bmatrix}H^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{bmatrix}\to\begin{bmatrix}H^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{bmatrix}$

for some unitary $W$ on ${\mathcal{F}}$. Next assume that

$\tau(A,B)\tau^{*}=\left(\begin{bmatrix}A_{11}&A_{12}\\\
A_{21}&R\end{bmatrix},\begin{bmatrix}B_{11}&B_{12}\\\
B_{21}&S\end{bmatrix}\right):\begin{bmatrix}H^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{bmatrix}\to\begin{bmatrix}H^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{bmatrix}.$

Now use these matrix representations and equate the (12)-entries of the
relation $AT=TA$ to get $A_{12}W=M_{z}A_{12}$. Therefore $A_{12}=0$ by Lemma
2.11. Similarly, from the relation $BT=TB$ we have $B_{12}=0$. Compare the
(21)-entries of the relation $A=B^{*}T$ to get $A_{21}=0$. The same treatment
for the relation $B=A^{*}T$ gives $B_{21}=0$. Therefore we are left with the
following relations

$\displaystyle
A_{11}M_{z}=M_{z}A_{11},\,B_{11}M_{z}=M_{z}B_{11},\,A_{11}=B_{11}^{*}M_{z}\text{
(and $B_{11}=A_{11}^{*}M_{z}$);}$ $\displaystyle
RW=WR,\,SW=WS,\,R=S^{*}W\text{ (and $S=WR^{*}$).}$

The second set of the above relations together with the fact that $W$ is a
unitary implies that $(R,S,W)$ is a pseudo-commutative tetrablock unitary. The
first two intertwining relations in the first set implies that there exist
bounded analytic functions
$\Phi,\Psi:\mathbb{D}\to{\mathcal{B}}({\mathcal{E}})$ such that
$A_{11}=M_{\Phi}$, $B_{11}=M_{\Psi}$. The remaining relations in the first set
then give us

$M_{\Phi}=M_{\Psi}^{*}M_{z},\quad M_{\Psi}=M_{\Phi}^{*}M_{z}.$

There now only remains a tedious computation with the power series expansions
of $\Phi$ and $\Psi$ to see that the remaining relations in the second set
forces $\Phi$ and $\Psi$ to have the coupled linear forms

$\Phi(z)=G_{1}^{*}+zG_{2}\quad\mbox{and}\quad\Psi(z)=G_{2}^{*}+zG_{1}$

for some operators $G_{1},\,G_{2}\in{\mathcal{B}}({\mathcal{E}})$. Again the
relation (2.7) is equivalent to $M_{\Phi}$ being a contraction operator. From
Definition 2.6 it follows that $(M_{\Phi},M_{\Psi},M_{z})$ is a pseudo-
commutative tetrablock isometry. The completes the proof for the pseudo-
commutative setting.

Suppose now that $(A,B,T)$ is a tetrablock isometry. Then in particular
$(A,B,T)$ satisfies all the requirements to be a pseudo-commutative tetrablock
isometry, so all the preceding analysis applies. We then see that $(A,B,T)$ is
unitarily equivalent to the triple in (2.10). As $(A,B,T)$ now is actually a
tetrablock isometry, we have that $AB=BA$. The unitary equivalence then forces

$\begin{bmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\
0&R\end{bmatrix}\begin{bmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&S\end{bmatrix}=\begin{bmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&S\end{bmatrix}\begin{bmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\ 0&R\end{bmatrix}$

which can be split up as two commutativity conditions

$\displaystyle
M_{G_{1}^{*}+zG_{2}}M_{G_{2}^{*}+zG_{1}}=M_{G_{2}^{*}+zG_{1}}M_{G_{1}^{*}+zG_{2}}$
(2.11) $\displaystyle RS=SR.$ (2.12)

By reversing the computations done in item (1) of Example 2.10, we see that
the intertwining (2.11) forces the set of conditions (2.8) Moreover, the
condition (2.12) is exactly the missing ingredient needed to promote $(R,S,W)$
from a pseudo-commutative tetrablock unitary to a strict tetrablock unitary.
This completes the proof. ∎

######  Remark 2.14.

It will be useful to have a terminology for an intermediate class of operator
triples $(A,B,T)$ which sits between strict tetrablock isometries and general
pseudo-commutative tetrablock isometries. Let us say that the triple $(A,B,T)$
is a semi-strict tetrablock isometry if $(A,B,T)$ is a pseudo-commutative
isometry with Wold decomposition as in (2.10) is such that the pseudo-
commutative tetrablock unitary component $(R,S,W)$ is actually a strict
tetrablock unitary, i.e., $R$ and $S$ are contractions which commute with each
other as well as with $W$ which is unitary.

We next present an analogue of the single-variable operator theory fact that
any isometry can always be extended to a unitary.

######  Corollary 2.15.

Pseudo-commutative/strict tetrablock isometries can be extended to pseudo-
commutative/strict tetrablock unitaries. More precisely:

(1) A triple $(A,B,T)$ is a pseudo-commutative tetrablock isometry if and only
if it extends to a pseudo-commutative tetrablock unitary. Moreover there
exists an extension that acts on the space of minimal unitary extension of the
isometry $T$.

(2) A triple $(A,B,T)$ is a strict tetrablock isometry if and only if it
extends to a strict tetrablock unitary acting on the space of the minimal
unitary extension of the isometry $T$.

###### Proof.

If a triple extends to a pseudo-commutative tetrablock unitary, then from
Definition 2.6 we can read off that it is also a pseudo-commutative tetrablock
isometry. Similarly, if a triple extends to a strict tetrablock unitary, we
can read off from criterion (iii) or (iv) in Theorem 2.5 that the triple
itself must be a strict tetrablock isometry.

We now address the converse. In view of Theorem 2.12, we can assume without
loss of generality that a pseudo-commutative tetrablock isometry $(A,B,T)$ is
given in the form:

$\left(\begin{bmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\
0&R\end{bmatrix},\begin{bmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&S\end{bmatrix},\begin{bmatrix}M_{z}&0\\\
0&W\end{bmatrix}\right):\begin{bmatrix}H^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{bmatrix}\to\begin{bmatrix}H^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{bmatrix}$

for some operators $G_{1}$, $G_{2}$ on ${\mathcal{E}}$ and for some pseudo-
commutative tetrablock unitary $(R,S,W)$ acting on ${\mathcal{F}}$. Consider
$H^{2}({\mathcal{E}})\oplus{\mathcal{F}}$ as a subspace of
$L^{2}({\mathcal{E}})\oplus{\mathcal{F}}$ in the natural way. Then the triple

$\left(\begin{bmatrix}M_{G_{1}^{*}+\zeta G_{2}}&0\\\
0&R\end{bmatrix},\begin{bmatrix}M_{G_{2}^{*}+\zeta G_{1}}&0\\\
0&S\end{bmatrix},\begin{bmatrix}M_{\zeta}&0\\\
0&W\end{bmatrix}\right)\colon\begin{bmatrix}L^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{bmatrix}\to\begin{bmatrix}L^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{bmatrix}$

is an extension of $(A,B,T)$. The unitary $M_{\zeta}\oplus W$ is clearly a
minimal unitary extension of the isometry $M_{z}\oplus W$. And since
$M_{G_{1}^{*}+\zeta G_{2}}=M_{G_{2}^{*}+\zeta G_{1}}^{*}M_{\zeta}$ and
$(R,S,W)$ is a pseudo-commutative tetrablock unitary, the above triple is a
pseudo-commutative tetrablock unitary by Definition 2.6.

If we start with a strict tetrablock isometry, then we shall also have that
$\left[\begin{smallmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\
0&R\end{smallmatrix}\right]$ commutes with
$\left[\begin{smallmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&S\end{smallmatrix}\right]$ on
$\left[\begin{smallmatrix}H^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{smallmatrix}\right]$, or equivalently, $RS=SR$ and the
Toeplitz operator symbols equal to the pencils $G_{1}^{*}+zG_{2}$ and
$G_{2}^{*}+zG_{1}$ commute:

$(G_{1}^{*}+zG_{2})(G_{2}^{*}+zG_{1})=(G_{2}^{*}+zG_{1})(G_{1}^{*}+zG_{2}).$

But then it is straightforward to see that this implies the commutativity of
the associated Laurent operators acting on $L^{2}({\mathcal{E}})$:

$M_{G_{1}^{*}+\zeta G_{2}}M_{G_{2}^{*}+\zeta G_{1}}=M_{G_{2}^{*}+\zeta
G_{1}}M_{G_{1}^{*}+\zeta G_{2}}.$

and hence also

$\begin{bmatrix}M_{G_{1}^{*}+\zeta G_{2}}&0\\\
0&R\end{bmatrix}\begin{bmatrix}M_{G_{2}^{*}+\zeta G_{1}}&0\\\
0&S\end{bmatrix}=\begin{bmatrix}M_{G_{2}^{*}+\zeta G_{1}}&0\\\
0&S\end{bmatrix}\begin{bmatrix}M_{G_{1}^{*}+\zeta G_{2}}&0\\\
0&R\end{bmatrix}.$

on $\left[\begin{smallmatrix}L^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{smallmatrix}\right]$. Moreover the extension of
$M_{G_{1}^{*}+zG_{2}}$ on $H^{2}({\mathcal{E}})$ to $M_{G_{1}^{*}+\zeta
G_{2}}$ on $L^{2}({\mathcal{E}})$ is norm-preserving, and hence the latter is
contractive whenever the former is contractive, and similarly for
$M_{G_{2}^{*}+zG_{1}}$ and $M_{G_{2}^{*}+\zeta G_{1}}$. We now have enough
observations to conclude by criterion (iii) or (iv) in Theorem 2.5 that
$\big{(}\left[\begin{smallmatrix}M_{G_{1}^{*}+\zeta G_{2}}&0\\\
0&R\end{smallmatrix}\right],\left[\begin{smallmatrix}M_{G_{2}^{*}+\zeta
G_{1}}&0\\\
0&S\end{smallmatrix}\right],\left[\begin{smallmatrix}M_{\zeta}&0\\\
0&W\end{smallmatrix}\right]\big{)}$ on
$\left[\begin{smallmatrix}L^{2}({\mathcal{E}})\\\
{\mathcal{F}}\end{smallmatrix}\right]$ is a tetrablock unitary as required. ∎

Another one-variable fact is the result due to Sz.-Nagy-Foias (see [34,
Theorem I.3.2]): any contraction operator $T$ on a Hilbert space
${\mathcal{H}}$ can be decomposed as $T=\left[\begin{smallmatrix}T_{cnu}&0\\\
0&U\end{smallmatrix}\right]$ where $T_{cnu}$ is a completely nonunitary
(c.n.u.) contraction operator (meaning there is no reducing subspace of
${\mathcal{H}}$ such that $T|_{\mathcal{H}}$ is unitary) and where $U$ is
unitary. There is a analogous result for the setting of tetrablock unitaries
and tetrablock contractions. We say that the tetrablock contraction $(A,B,T)$
on ${\mathcal{H}}$ is a c.n.u. tetrablock contraction if there is no
nontrivial jointly reducing subspace ${\mathcal{H}}_{u}\subset{\mathcal{H}}$
for $(A,B,T)$ such that $(A,B,T)|_{{\mathcal{H}}_{u}}$ is a tetrablock
unitary. The following result appears in [29]

######  Theorem 2.16.

Let $(A,B,T)$ be a tetrablock contraction on ${\mathcal{H}}$. Then
${\mathcal{H}}$ has an internal orthogonal direct-sum decomposition
${\mathcal{H}}={\mathcal{H}}_{c.n.u.}\oplus{\mathcal{H}}_{u}$ with
${\mathcal{H}}_{c.n.u.}$ and ${\mathcal{H}}_{u}$ jointly reducing for
$(A,B,T)|_{{\mathcal{H}}_{c.n.u.}}$ equal to a c.n.u. tetrablock contraction
and $(A,B,T)|_{{\mathcal{H}}_{u}}$ equal to a tetrablock unitary.

The at first surprising fact is that the same decomposition
${\mathcal{H}}={\mathcal{H}}_{c.n.u.}\oplus{\mathcal{H}}_{u}$ inducing the
canonical decomposition of the contraction operator $T$ into is c.n.u. part
$T_{c.n.u.}$ and its unitary part $T_{u}$ turns out to also be jointly
reducing for the whole operator triple $(A,B,T)$ and induces the canonical
tetrablock decomposition for the tetrablock contraction $(A,B,T)$, i.e.,
$(A,B,T)|_{{\mathcal{H}}_{c.n.u.}}$ is a c.n.u. tetrablock contraction, and
$(A,B,T)|_{{\mathcal{H}}_{u}}$ is a tetrablock unitary.

######  Remark 2.17.

The import of Theorem 2.16 for model theory is the same as is the case for the
classical case: since the model theory for a (strict) tetrablock unitary is
already well understood (see item (2) in Remark 2.8, it follows that it is
perfectly satisfactory to focus on the case where $(A,B,T)$ is a c.n.u.
tetrablock contraction for the purposes of model theory.

###  2.3. A canonical construction of a tetrablock unitary from a tetrablock
contraction

In this section we start with a tetrablock contraction $(A,B,T)$ and construct
a tetrablock unitary which is uniquely associated with $(A,B,T)$ in a sense
that will be made precise later in this subsection. This will be used later to
construct a concrete functional model for a pseudo-commutative tetrablock-
isometric lift for $(A,B,T)$ which can be viewed as a functional model for
$(A,B,T)$ itself.

We start by using the fact that the last entry $T$ in our tetrablock
contraction $(A,B,T)$ is a contraction operator. Hence there exist a positive
semidefinite operator $Q_{T^{*}}$ such that

$\displaystyle Q_{T^{*}}^{2}:=\operatorname{SOT-}\lim T^{n}T^{*n}.$ (2.13)

Define the operator
$X_{T^{*}}^{*}:\overline{\operatorname{Ran}}\;Q_{T^{*}}\to\overline{\operatorname{Ran}}\;Q_{T^{*}}$
densely by

$\displaystyle X_{T^{*}}^{*}Q_{T^{*}}h=Q_{T^{*}}T^{*}h.$ (2.14)

This is an isometry because for all $h\in{\mathcal{H}}$,

$\displaystyle\|X_{T^{*}}^{*}Q_{T^{*}}h\|^{2}=\langle
Q_{T^{*}}^{2}T^{*}h,T^{*}h\rangle=\lim_{n\to\infty}\langle
T^{n}T^{*n}T^{*}h,T^{*}h\rangle$ $\displaystyle\quad=\lim_{n\to\infty}\langle
T^{n+1}T^{*(n+1)}h,h\rangle=\langle
Q_{T^{*}}^{2}h,h\rangle=\|Q_{T^{*}}h\|^{2}.$ (2.15)

Since $A$ is a contraction, we have for all $h\in{\mathcal{H}}$

$\displaystyle\langle
AQ_{T^{*}}^{2}A^{*}h,h\rangle=\langle\lim_{n}T^{n}AA^{*}T^{*n}h,h\rangle\leq\lim_{n}\langle
T^{n}T^{*n}h,h\rangle=\langle Q_{T^{*}}^{2}h,h\rangle.$

The same computation for the contraction $B$ will yield the same inequality
involving $B$ in place of $A$. Consequently, the operators
$A_{T^{*}}.B_{T^{*}}:\overline{\operatorname{Ran}}\;Q_{T^{*}}\to\overline{\operatorname{Ran}}\;Q_{T^{*}}$
defined densely by

$\displaystyle A_{T^{*}}^{*}Q_{T^{*}}h=Q_{T^{*}}A^{*}h\quad\text{and}\quad
B_{T^{*}}^{*}Q_{T^{*}}h=Q_{T^{*}}B^{*}h$ (2.16)

are contractions, and extend contractively to all of
$\overline{\operatorname{Ran}}\,Q_{T^{*}}$ by a limiting process. Furthermore,
from the definitions it is easy to see that $(A_{T^{*}},B_{T^{*}},X_{T^{*}})$
is a commutative triple since by assumption we know that $(A^{*},B^{*},T^{*})$
is commutative. This and (2.14) imply that if $f$ is a three-variable
polynomial, then for all $h\in{\mathcal{H}}$,

$\displaystyle\|f(A_{T^{*}}^{*},B_{T^{*}}^{*},X_{T^{*}}^{*})Q_{T^{*}}h\|=\|Q_{T^{*}}f(A^{*},B^{*},T^{*})h\|$
$\displaystyle\quad\leq\|f(A^{*},B^{*},T^{*})\|\|h||\leq\|f(A^{*},B^{*},T^{*})\|\|h\|\leq\big{(}\sup_{\mathbb{E}}|f|\big{)}\|h\|$

where in the last inequality we used the fact that $(A,B,T)$ is a tetrablock
contraction. This inequality together with the fact that $X_{T^{*}}^{*}$ is an
isometry implies that $(A_{T^{*}}^{*},B_{T^{*}}^{*},X_{T^{*}}^{*})$ is a
tetrablock isometry. By Corollary 2.15,
$(A_{T^{*}}^{*},B_{T^{*}}^{*},X_{T^{*}}^{*})$ has a tetrablock unitary
extension $(R_{D}^{*},S_{D}^{*},W_{D}^{*})$ acting on a space which we shall
call ${\mathcal{Q}}_{T^{*}}\supseteq\overline{\operatorname{Ran}}\;Q_{T^{*}}$,
where $W_{D}^{*}$ acting on ${\mathcal{Q}}_{T^{*}}$ is the minimal unitary
extension of $X_{T^{*}}^{*}$.

######  Definition 2.18.

Let $(A,B,T)$ be a tetrablock contraction and let $(R_{D},S_{D},W_{D})$ be the
tetrablock unitary constructed from $(A,B,T)$ as above. We say that
$(R_{D},S_{D},W_{D})$ is the canonical tetrablock unitary associated with the
tetrablock contraction $(A,B,T)$.

The next result assures us that canonical tetrablock unitaries associated with
the same tetrablock contraction $(A,B,T)$ are the same up to unitary
equivalence.

######  Theorem 2.19.

Let $(A,B,T)$ on ${\mathcal{H}}$ and $(A^{\prime},B^{\prime},T^{\prime})$ on
${\mathcal{H}}^{\prime}$ be two tetrablock contractions with
$(R_{D},S_{D},W_{D})$ and $(R_{D}^{\prime},S_{D}^{\prime},W_{D}^{\prime})$
equal to the respective canonical tetrablock unitaries. If $(A,B,T)$ and
$(A^{\prime},B^{\prime},T^{\prime})$ are unitarily equivalent via $\tau$, then
$(R_{D},S_{D},W_{D})$ and $(R_{D}^{\prime},S_{D}^{\prime},W_{D})$ are
unitarily equivalent via
$\omega_{\tau}:{\mathcal{Q}}_{T^{*}}\to{\mathcal{Q}}_{T^{{}^{\prime}*}}$

$\displaystyle\omega_{\tau}:W_{D}^{n}Q_{T^{*}}h\mapsto
W_{D}^{{}^{\prime}n}Q_{T^{\prime*}}\tau h$ (2.17)

for all $n\geq 0$ and $h\in{\mathcal{H}}$.

###### Proof.

Let the spaces ${\mathcal{Q}}_{T^{*}},{\mathcal{Q}}_{T^{{}^{\prime}*}}$ and
the operators $\\{A_{T^{*}},B_{T^{*}},Q_{T^{*}}\\}$,
$\\{A_{T^{\prime*}},B_{T^{\prime*}},Q_{T^{\prime*}}\\}$ be obtained as above
from $(A,B,T)$ and $(A^{\prime},B^{\prime},T^{\prime})$, respectively. Since
$\tau$ is a unitary intertwining $T$ and $T^{\prime}$, it intertwines $T^{*}$
and $T^{\prime*}$ and thus $\tau Q_{T^{*}}=Q_{T^{\prime*}}\tau$. Therefore by
definition (2.16) it follows that
$\tau(A_{T^{*}},B_{T^{*}},Q_{T^{*}})=(A_{T^{\prime*}},B_{T^{\prime*}},Q_{T^{\prime*}})\tau$.
By definition of $\omega_{\tau}$ it is clear that
$\omega_{\tau}W_{D}=W_{D}^{\prime}\omega_{\tau}$. Therefore for every
$h\in{\mathcal{H}}$ and $n\geq 0$,

$\displaystyle\omega_{\tau}R_{D}W_{D}^{n}Q_{T^{*}}h$
$\displaystyle=\omega_{\tau}W_{D}^{n+1}W_{D}^{*}R_{D}Q_{T^{*}}h$
$\displaystyle=\omega_{\tau}W_{D}^{n+1}S_{D}^{*}Q_{T^{*}}h\quad\mbox{[using
Theorem \ref{T:IsoChar}, part (iv)]}$ $\displaystyle=W_{D}^{\prime(n+1)}\tau
B_{T^{*}}^{*}Q_{T^{*}}h=W_{D}^{\prime(n+1)}B_{T^{\prime*}}^{*}Q_{T^{\prime*}}\tau
h$ $\displaystyle=W_{D}^{\prime(n+1)}S_{D}^{\prime*}Q_{T^{\prime*}}\tau h$
$\displaystyle=S_{D}^{\prime*}W_{D}^{\prime(n+1)}Q_{T^{*}}\quad\mbox{[since
$S_{D}^{\prime*}W_{D}^{\prime}=W_{D}^{\prime}S_{D}^{\prime*}$]}$
$\displaystyle=R_{D}^{\prime}W_{D}^{\prime n}Q_{T^{\prime*}}\tau
h\quad\mbox{[using Theorem \ref{T:IsoChar}, part (iii)]}$
$\displaystyle=R_{D}^{\prime}\omega_{\tau}W_{D}^{n}Q_{T^{*}}h.$

A similar computation shows that
$\omega_{\tau}S_{D}=S_{D}^{\prime}\omega_{\tau}$. ∎

###  2.4. The fundamental operators

Much of the theory of tetrablock contractions is heavily based on a pair of
operators that is uniquely associated with a tetrablock contraction. These are
called the fundamental operators, the existence of which was proved in [13]
with appeal to connections between tetrablock contractions and symmetrized-
bidisk contractions. We state the result and sketch a more self-contained
proof with the appeal to symmetrized-bidisk theory eliminated. In the sequel
we shall use the notation $\nu(X)$ to denote the numerical radius of the
operator $X$ on the Hilbert space ${\mathcal{H}}$:

$\nu(X):=\sup_{x\in{\mathcal{H}}\colon\|x\|=1}|\langle
Xx,x\rangle_{\mathcal{H}}|.$

######  Theorem 2.20.

Let $(A,B,T)$ be a tetrablock contraction on a Hilbert space $\mathcal{H}$.

  1. (i)

(See [13, Theorem 3.4]) There exist two unique operators $F_{1}$ and $F_{2}$
acting on ${\mathcal{D}}_{T}$ with the numerical radii at most one such that

$\displaystyle A-B^{*}T=D_{T}F_{1}D_{T}\quad\text{and}\quad
B-A^{*}T=D_{T}F_{2}D_{T}.$ (2.18)

Moreover, the operators $F_{1},F_{2}$ are such that $\nu(F_{1}+zF_{2})\leq 1$
for all $z\in\overline{\mathbb{D}}$.

  2. (ii)

(See [13, Corollary 4.2]) The operators $F_{1},F_{2}$ are alternatively
characterized as the unique bounded operators on ${\mathcal{D}}_{T}$ such that
$(X_{1},X_{2})=(F_{1},F_{2})$ satisfies the system of operator equations

$\displaystyle D_{T}A=X_{1}D_{T}+X_{2}^{*}D_{T}T\quad\text{and}\quad
D_{T}B=X_{2}D_{T}+X_{1}^{*}D_{T}T.$ (2.19)

###### Proof.

Let $(A,B,T)$ be a tetrablock contraction on ${\mathcal{H}}$. Since for every
$z\in{\mathbb{D}}$, $\Psi(z,\cdot)$ as in item (ii) of Theorem 2.2 is analytic
in an open set containing $\overline{{\mathbb{E}}}$, and
$\overline{{\mathbb{E}}}$ is polynomially convex, a limiting argument implies
that $\Psi(\zeta,(A,B,T))$ is a contraction for every $\zeta\in{\mathbb{T}}$,
or equivalently, on simplifying
$I-\Psi(\zeta,(A,B,T))^{*}\Psi(\zeta,(A,B,T))\succeq 0$ we get

$\displaystyle(I-T^{*}T)+(B^{*}B-A^{*}A)-\zeta(B-A^{*}T)-\overline{\zeta}(B-A^{*}T)^{*}\succeq
0.$

Similarly, applying item (iii) of Theorem 2.2 we have for every
$\alpha\in{\mathbb{T}}$,

$\displaystyle(I-T^{*}T)+(A^{*}A-B^{*}B)-\alpha(A-B^{*}T)-\overline{\alpha}(A-B^{*}T)^{*}\succeq
0.$

On adding the above two positive operators and then simplifying we get

$\displaystyle
D_{T}^{2}\succeq\operatorname{Re}\alpha\left[(A-B^{*}T)+\beta(B-A^{*}T)\right]$
(2.20)

for every $\alpha,\beta\in{\mathbb{T}}$. We now make use the following lemma
of independent interest:

######  Lemma 2.21.

(See [15, Lemma 4.1]) Let $\Sigma$ and $D$ be two operators such that

$DD^{*}\succeq\operatorname{Re}\alpha\Sigma\quad\mbox{for all
}\alpha\in{\mathbb{T}}.$

Then there exists an operator $F$ acting on
$\overline{\operatorname{Ran}}\,D^{*}$ with numerical radius at most one such
that $\Sigma=DFD^{*}$.

###### Sketch of proof..

Apply the Fejer-Riesz factorization theorem of Dritschel-Rovnyak [24, Theorem
2.1] to the Laurent operator-valued polynomial
$P(e^{i\theta})=2DD^{*}-e^{i\theta}\Sigma-e^{-i\theta}\Sigma^{*}$. Along the
way one makes use of the standard Douglas lemma ($\exists$
$X\in{\mathcal{B}}({\mathcal{H}})$ with $AX=B$ $\Leftrightarrow$
$BB^{*}\preceq AA^{*}$) and a criterion for a Hilbert space operator to have
numerical radius at most $1$: $X\in{\mathcal{B}}({\mathcal{H}})$ has
$\nu(X)\leq 1$ $\Leftrightarrow$ $\operatorname{Re}\,(\beta X)\preceq
I_{\mathcal{H}}$ for all $\beta\in{\mathbb{T}}$. Note that Lemma 2.21 can
itself be viewed as a quadratic, numerical-radius version of the Douglas
lemma. ∎

We apply Lemma 2.21 to the case (2.20) for each $\beta$ to get a numerical
contraction $F(\beta)$ such that

$\displaystyle(A-B^{*}T)+\beta(B-A^{*}T)=D_{T}F(\beta)D_{T}.$ (2.21)

On adding equations (2.21) for the cases $\beta=1$ and $-1$, we get

$\displaystyle A-B^{*}T=D_{T}F_{1}D_{T}\quad\mbox{where}\quad
F_{1}:=\frac{F(1)+F(-1)}{2}.$ (2.22)

Thus putting $\beta=1$ in (2.21) and combining with (2.21) gives us

$B-A^{*}T=D_{T}(F(1)-F_{1})D_{T}=D_{T}F_{2}D_{T}\quad\mbox{where}\quad
F_{2}:=\frac{F(1)-F(-1)}{2}.$ (2.23)

We conclude that $F_{1}$ and $F_{2}$ so constructed satisfy equations (2.18).
It is easy to see that in general

$X\in{\mathcal{B}}({\mathcal{D}}_{T})\text{ with }D_{T}XD_{T}=0\Rightarrow
X=0.$ (2.24)

Applying this observation to the homogeneous version of equations (2.18)
implies that the solutions $(F_{1},F_{2})$ of (2.18) must be unique whenever
they exist.

On the other hand, if we combine (2.22) with (2.23) we see that
$\widetilde{F}(\beta):=F_{1}+\beta F_{2}$ gives us a second solution of
(2.21). Hence

$D_{T}(\widetilde{F}(\beta)-F(\beta))D_{T}=0\text{ where
}\widetilde{F}(\beta)-F(\beta)\in{\mathcal{B}}({\mathcal{D}}_{T})\text{ for
all }\beta\in{\mathbb{T}}.$

Again by (2.24), we see that $\widetilde{F}(\beta)=F(\beta)$ for all
$\beta\in{\mathbb{T}}$. But we saw above (as a consequence of Lemma 2.21) that
$F(\beta)$ is a numerical contraction for all $\beta\in{\mathbb{T}}$. As we
now know that $\widetilde{F}(\beta)=F(\beta)$, we conclude that the pencil
$\widetilde{F}(\beta)=F_{1}+\beta F_{2}$ is a numerical contraction for all
$\beta\in{\mathbb{T}}$. By applying the Maximum Modulus Theorem to the
holomorphic function $\langle(F_{1}+\beta F_{2})h,h\rangle$ for each fixed
$h\in{\mathcal{H}}$, we see that $F_{1}+zF_{2}$ is a numerical contraction for
all $z\in\overline{\mathbb{D}}$. This completes the proof of item (i) in
Theorem 2.20.

To see that $F_{1}$ and $F_{2}$ satisfy equations (2.19), simply multiply
$D_{T}$ on the left of each equation and use the identities (2.18) to
simplify. To show that $F_{1},F_{2}$ are the unique operators on
${\mathcal{D}}_{T}$ satisfying these two equations, it is enough to show that
$X=0$ and $Y=0$ are the only operators in ${\mathcal{B}}({\mathcal{D}}_{T})$
satisfying

$XD_{T}+Y^{*}D_{T}T=0,\quad YD_{T}+X^{*}D_{T}T=0.$

To show that $X=0$, compute

$\displaystyle D_{T}XD_{T}=-D_{T}Y^{*}D_{T}T=T^{*}D_{T}XD_{T}T$
$\displaystyle=-T^{*}D_{T}Y^{*}D_{T}T^{2}$
$\displaystyle=T^{*2}D_{T}XD_{T}T^{2}.$

Thus by iteration of the above process $D_{T}XD_{T}=T^{*n}D_{T}XD_{T}T^{n}$.
This shows that $X=0$ because for every $h\in{\mathcal{H}}$

$\lim_{n}\|D_{T}T^{n}h\|^{2}=\lim_{n}\|T^{n}h\|^{2}-\lim_{n}\|T^{n+1}h\|^{2}=0.$

A similar argument gives $Y=0$. This is the idea of the proof due by
Bhattacharyya [13]. ∎

The unique operators $F_{1},F_{2}$ in item (i) of Theorem 2.20 will be
referred to as the fundamental operators for the tetrablock contraction
$(A,B,T)$, as in [13].

As we have seen in Proposition 2.1, if $(A,B,T)$ is a
${\mathbb{E}}$-contraction, so also is $(A^{*},B^{*},T^{*})$. For the
construction of the functional model for a tetrablock contraction $(A,B,T)$,
it turns out to be more convenient to work with the fundamental operators for
the adjoint tetrablock contraction $(A^{*},B^{*},T^{*})$ which we denote as
$(G_{1},G_{2})$. Thus there exists exactly one solution
$(X_{1},X_{2})=(G_{1},G_{2})$ of the system of equations

$A^{*}-BT^{*}=D_{T^{*}}X_{1}D_{T^{*}},\quad
B^{*}-AT^{*}=D_{T^{*}}X_{2}D_{T^{*}}$ (2.25)

with equivalent characterization as the unique solution
$(X_{1},X_{2})=(G_{1},G_{2})$ of the second system of equations

$D_{T^{*}}A^{*}=X_{1}D_{T^{*}}+X_{2}^{*}D_{T^{*}}T^{*},\quad
D_{T^{*}}B^{*}=X_{2}D_{T^{*}}+X_{1}^{*}D_{T^{*}}T^{*}$ (2.26)

Then from Example 2.10 we see immediately that the operator pair
$(M_{G_{1}^{*}+zG_{2}},M_{G_{2}^{*}+zG_{1}},M_{z})$ acting on the Hardy space
$H^{2}({\mathcal{D}}_{T^{*}})$ is of the correct form to be a pseudo-
commutative tetrablock isometry. We would like to establish conditions under
which this a priori only pseudo-commutative ${\mathbb{E}}$-isometry is
actually a strict ${\mathbb{E}}$-isometry. This follows from the following
result.

######  Theorem 2.22.

(See [13].) Suppose that $(G_{1},G_{2})$ is the Fundamental Operator pair for
the ${\mathbb{E}}$-contraction $(A^{*},B^{*},T^{*})$. Let
$(V_{1},V_{2},V_{3})$ be the operator triple

$(V_{1},V_{2},V_{2})=(M_{G_{1}^{*}+zG_{2}},M_{G_{2}^{*}+zG_{1}},M_{z})\text{
on }H^{2}({\mathcal{D}}_{T^{*}}).$ (2.27)

Then:

  1. (1)

$(V_{1},V_{2},V_{3})$ is a pseudo-commutative tetrablock isometry having the
additional property that

$r(V_{1})\leq 1,\quad r(V_{2})\leq 1.$

  2. (2)

Suppose in addition that the Fundamental Operator pair $(G_{1},G_{2})$ satisfy
the commutativity conditions. Then $(V_{1},V_{2},V_{3})$ is a strict
tetrablock isometry.

######  Corollary 2.23.

Let $(G_{1},G_{2})$ be the Fundamental Operator pair for the tetrablock
contraction $(A^{*},B^{*},T^{*})$ and set $(V_{1},V_{2},V_{3})$ as in (2.27).
Then $(V_{1},V_{2},V_{3})$ is a (strict) tetrablock isometry if and only if
the commutativity conditions (2.8) hold.

###### Proof of Corollary 2.23.

If the commutativity conditions (2.8) are satisfied, then statement (2) of
Theorem 2.22 says that $(V_{1},V_{2},V_{3})$ is a tetrablock isometry.
Conversely, if $(V_{1},V_{2},V_{3})$ is a tetrablock isometry, in particular
$(V_{1},V_{2},V_{3})$ is commutative so the commutativity conditions (2.8) are
satisfied. ∎

###### Proof of Theorem 2.22.

We first consider statement (1). That $(V_{1},V_{2},V_{3})$ is a pseudo-
commutative tetrablock isometry follows from the fact that it has the required
form (2.6) as presented in Example 2.10. It remains to use the fact that
$(G_{1},G_{2})$ is a Fundamental Operator pair for the tetrablock contraction
$(A^{*},B^{*},T^{*})$ (to see that we also have $r(V_{1})\leq 1$ and
$r(V_{2})\leq 1$.

By Theorem 2.20 (applied to $(A^{*},B^{*},T^{*})$ in place of $(A,B,T)$), we
know that $\nu(G_{1}+zG_{2})\leq 1$ for all $z\in\overline{\mathbb{D}}$. By
the Andô criterion for the numerical radius of an operator to be no more than
1, this means that

$\beta(G_{1}+\alpha
G_{2})+\overline{\beta}(G_{1}^{*}+\overline{\alpha}G_{2}^{*})\preceq
2I_{{\mathcal{D}}_{T^{*}}}\forall\alpha,\beta\in\overline{\mathbb{D}}.$

Rearrange this inequality as

$\displaystyle(\overline{\beta}G_{1}^{*}+\beta\alpha G_{2})+(\beta
G_{1}+\overline{\beta}\overline{\alpha}G_{2}^{*})$
$\displaystyle\quad=\overline{\beta}(G_{1}^{*}+\beta^{2}\alpha
G_{2})+\beta(G_{1}+\overline{\beta}^{2}\overline{\alpha}G_{2}^{*})\preceq
2I_{{\mathcal{D}}_{T^{*}}}.$

By the Andô criterion applied in the reverse direction, this tells us that

$\nu(G_{1}^{*}+zG_{2})\leq 1\text{ for all }z\in\overline{\mathbb{D}}.$

But in general the numerical radius dominates the spectral radius; thus

$r(G_{1}^{*}+zG_{2})\leq\nu(G_{1}^{*}+zG_{2})\leq 1\text{ for all
}z\in\overline{\mathbb{D}}.$

If we choose $\lambda\in{\mathbb{C}}$ with $|\lambda|>1$, then $\lambda
I_{{\mathcal{D}}_{T^{*}}}-(G_{2}^{*}+zG_{1})$ is invertible, and in fact the
operator-valued function $z\mapsto(\lambda
I_{{\mathcal{D}}_{T^{*}}}-(G_{2}^{*}+zG_{1}))^{-1}$ is in
$H^{\infty}({\mathcal{B}}({\mathcal{D}}_{T^{*}})$. We thus conclude that in
fact

$r(M_{G_{2}^{*}+zG_{1}})\leq 1.$

All the above analysis applies to the pair $(G_{2},G_{1})$ in place of
$(G_{1},G_{2})$, as $(G_{2},G_{1})$ is the Fundamental Operator pair for the
${\mathbb{E}}$-contraction $(B^{*},A^{*},T^{*})$; hence we also have also

$r(M_{G_{2}^{*}+zG_{1}})\leq 1.$

This completes the proof of statement (1).

We now consider statement (2). As we are now assuming that $(G_{1},G_{2})$
satisfy the commutativity conditions (2.8), it follows that
$M_{G_{1}^{*}+zG_{2}}$ commutes with $M_{G_{2}^{*}+zG_{1}}$. We are now in a
position to apply statement (3) in Remark 2.8 (or, what is the same, statement
(2) in Theorem 2.9 to conclude that the a priori only pseudo-commutative
${\mathbb{E}}$-contraction $(M_{G_{1}^{*}+zG_{2}},M_{G_{2}^{*}+zG_{1}},$
$M_{z})$ is in fact a strict ${\mathbb{E}}$-contraction. ∎

##  3\. Functional models for tetrablock contractions

In this section we produce two functional models for tetrablock contractions,
the first inspired by model theory of Douglas [21], and the second by the
model theory of Sz.-Nagy and Foias [34]. We note that so far only a functional
model is known for the special case when the last entry is a pure contraction;
see [32, Theorem 4.2].

###  3.1. A Douglas-type functional model

Let $T$ be any contraction on ${\mathcal{H}}$. Define the operators
${\mathcal{O}}_{D_{T^{*}},T^{*}}:{\mathcal{H}}\to
H^{2}({\mathcal{D}}_{T^{*}})$ as

$\displaystyle{\mathcal{O}}_{D_{T^{*}},T^{*}}(z)h=\sum_{n=0}^{\infty}z^{n}D_{T^{*}}T^{*n}h,\text{
for every }h\in{\mathcal{H}},$ (3.1)

and
$\Pi_{D}:{\mathcal{H}}\to\left[\begin{smallmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
{\mathcal{Q}}_{T^{*}}\end{smallmatrix}\right]$ by

$\displaystyle\Pi_{D}h=\begin{bmatrix}{\mathcal{O}}_{D_{T^{*}},T^{*}}(z)h\\\
Q_{T^{*}}h\end{bmatrix}\quad\text{for all }h\in{\mathcal{H}},$ (3.2)

where $Q_{T^{*}}$ is as in (2.13). Then the computation

$\displaystyle\|\Pi_{D}h\|^{2}=$
$\displaystyle\|{\mathcal{O}}_{D_{T^{*}},T^{*}}(z)h\|_{H^{2}({\mathcal{D}}_{T^{*}})}^{2}+\|Q_{T^{*}}h\|^{2}$
$\displaystyle=\sum_{n=0}^{\infty}\|D_{T^{*}}T^{*n}h\|^{2}+\lim_{n\to\infty}\|T^{*n}h\|^{2}$
$\displaystyle=(\|h\|^{2}-\|T^{*}h\|^{2})+(\|T^{*}h\|^{2}-\|T^{*2}h\|^{2})+\cdots)+\lim_{n\to\infty}\|T^{*n}h\|^{2}$
$\displaystyle=\|h\|^{2}$

shows that $\Pi_{D}$ is an isometry. Note the following intertwining property
of ${\mathcal{O}}_{D_{T^{*}},T^{*}}$:

$\displaystyle{\mathcal{O}}_{D_{T^{*}},T^{*}}(z)T^{*}h$
$\displaystyle=\sum_{n=0}^{\infty}z^{n}D_{T^{*}}T^{*n+1}h=M_{z}^{*}\sum_{n=0}^{\infty}z^{n}D_{T^{*}}T^{*n}h$
$\displaystyle=M_{z}^{*}{\mathcal{O}}_{D_{T^{*}},T^{*}}(z)h.$ (3.3)

This together with intertwining (2.14) of $Q_{T^{*}}$ implies

$\displaystyle\Pi_{D}T^{*}=\begin{bmatrix}M_{z}&0\\\
0&W_{D}\end{bmatrix}^{*}\Pi_{D}.$ (3.4)

This shows that the the pair

$V_{D}:=\begin{bmatrix}M_{z}&0\\\
0&W_{D}\end{bmatrix}:\begin{bmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
{\mathcal{Q}}_{T^{*}}\end{bmatrix}\to\begin{bmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
{\mathcal{Q}}_{T^{*}}\end{bmatrix}$

is an isometric lift of $T$. This construction is by Douglas, where he also
showed that this lift is minimal (see [21]).

Now let $(A,B,T)$ be a tetrablock contraction acting on ${\mathcal{H}}$ and
$G_{1},G_{2}$ be the fundamental operators of $(A^{*},B^{*},T^{*})$. Let
$(R_{D},S_{D},W_{D})$ acting on ${\mathcal{Q}}_{T^{*}}$ be the canonical
tetrablock unitary associated with $(A,B,T)$. Consider the operators

$\displaystyle\left(\begin{bmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\
0&R_{D}\end{bmatrix},\begin{bmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&S_{D}\end{bmatrix},\begin{bmatrix}M_{z}&0\\\
0&W_{D}\end{bmatrix}\right)\mbox{ on
}\begin{bmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
{\mathcal{Q}}_{T^{*}}\end{bmatrix}.$ (3.5)

We claim that

$\displaystyle\Pi_{D}(A^{*},B^{*},T^{*})$
$\displaystyle=\left(\begin{bmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\
0&R_{D}\end{bmatrix}^{*},\begin{bmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&S_{D}\end{bmatrix}^{*},\begin{bmatrix}M_{z}&0\\\
0&W_{D}\end{bmatrix}^{*}\right)\Pi_{D},$ (3.6)

where
$\Pi_{D}:{\mathcal{H}}\to\left[\begin{smallmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
{\mathcal{Q}}_{T^{*}}\end{smallmatrix}\right]$ is the isometry as in (3.2).

Recalling the definition
$\Pi_{D}=\left[\begin{smallmatrix}{\mathcal{O}}_{D_{T^{*}},T^{*}}\\\
{\mathcal{Q}}_{T^{*}}\end{smallmatrix}\right]\colon{\mathcal{H}}\to\left[\begin{smallmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
{\mathcal{Q}}_{T^{*}}\end{smallmatrix}\right]$, we see that the three-fold
intertwining condition (3.1) splits into two three-fold intertwining
conditions

$\displaystyle{\mathcal{O}}_{D_{T^{*}},T^{*}}(A^{*},B^{*},T^{*})=(M^{*}_{G_{1}^{*}+zG_{2}},M^{*}_{G_{2}^{*}+zG_{1}},M^{*}_{z}){\mathcal{O}}_{D_{T^{*}},T^{*}},$
(3.7) $\displaystyle
Q_{T^{*}}(A^{*},B^{*},T^{*})=(R^{*}_{D},S^{*}_{D},W^{*}_{D})Q_{T^{*}}.$ (3.8)

The last equation in (3.7) combined with the last equation in (3.8) we have
already seen as the condition that $\Pi_{D}$ is the isometric identification
map implementing $\left[\begin{smallmatrix}M_{z}&0\\\
0&W_{D}\end{smallmatrix}\right]$ as the Douglas minimal isometric lift of $T$
(see (3.4)). We shall next check only the first equation in (3.7) and the
first equation in (3.8) as the verification of the respective second equations
is completely analogous. Thus it remains to check

$\displaystyle{\mathcal{O}}_{D_{T^{*}},T^{*}}A^{*}=M^{*}_{G_{1}^{*}+zG_{2}}{\mathcal{O}}_{D_{T^{*}},T^{*}},$
(3.9) $\displaystyle Q_{T^{*}}A^{*}=R^{*}_{D}Q_{T^{*}}.$ (3.10)

Note that (3.10) is part of the construction of the canonical tetrablock
unitary associated with the original tetrablock contraction $(A,B,T)$ (see
(2.16)). To check (3.9), let us rewrite the condition in function form:

$D_{T^{*}}(I-zT^{*})^{-1}A^{*}=G_{1}D_{T^{*}}(I-zT^{*})^{-1}+G_{2}^{*}D_{T^{*}}(I-zT^{*})^{-1}T^{*}$

As $A$ commutes with $T$, we can rewrite this as

$D_{T^{*}}A^{*}(I-zT^{*})^{-1}=(G_{1}D_{T^{*}}+G_{2}^{*}D_{T^{*}}T^{*})(I-zT^{*})^{-1}.$

We may now cancel off the resolvent term $(I-zT^{*})^{-1}$ to get a pure
operator equation

$D_{T^{*}}A^{*}=G_{1}D_{T^{*}}+G_{2}^{*}D_{T^{*}}T^{*}.$ (3.11)

Let us now recall that the operators $(G_{1},G_{2})$ on
${\mathcal{D}}_{T^{*}}$ were chosen to be the Fundamental Operators for the
tetrablock contraction $(A^{*},B^{*},T^{*})$. Thus by our earlier discussion
of Fundamental Operators for tetrablock contractions (see Theorem 2.20), we
know that $(X_{1},X_{2})=(G_{1},G_{2})$ satisfies the identities (2.26), the
first of which gives the same condition on $(G_{1},G_{2})$ as (3.11). Thus
this choice of $(G_{1},G_{2})$ indeed leads to a solution of (3.9), and the
proof of (3.1) is complete. (Of course the second equation in (2.26) amounts
to the verification of the second equation in (3.7).)

This then means that the operator triple with isometric embedding operator
$\Pi_{D}$

$\displaystyle(V_{1},V_{2},V_{3}):=$
$\displaystyle\quad\bigg{(}\Pi_{D},\bigg{(}\begin{bmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\
0&R_{D}\end{bmatrix},\begin{bmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&S_{D}\end{bmatrix},\begin{bmatrix}M_{z}&0\\\
0&W_{D}\end{bmatrix}\bigg{)}\bigg{)}$ (3.12)

is a lift of the tetrablock contraction $(A,B,T)$. As $(R_{D},S_{D},W_{D})$ is
${\mathbb{E}}$-unitary as part of the canonical construction in Section 2.3,
we see from the form of the top components of $(V_{1},V_{2},V_{3})$ in (3.12)
that $(V_{1},V_{2},V_{3})$ is a semi-strict ${\mathbb{E}}$-isometry and is a
strict ${\mathbb{E}}$-isometry exactly when the top-component triple
$(M_{G_{1}^{*}+zG_{2}},M_{G_{2}^{*}+zG_{1}},M_{z})$ is a strict
${\mathbb{E}}$-isometry. By Theorem 2.22, this in turn happens exactly when
the Fundamental Operator pair $(G_{1},G_{2})$ satisfies the commutativity
conditions (2.8). In summary we have verified most of the following result. We
note that item (2) recovers a result of Bhattacharyya-Sau [17] via functional-
model methods rather than by block-matrix-construction methods.

######  Theorem 3.1.

(1) Let $(A,B,T)$ be a tetrablock contraction on ${\mathcal{H}}$ and let
$V_{3}$ on ${\mathcal{K}}\supset{\mathcal{H}}$ be the (essentially unique)
minimal isometric lift for the contraction operator $T$. Then there is a
unique choice of operators $(V_{1},V_{2})$ on ${\mathcal{K}}$ so that the
triple ${\mathbf{V}}=(V_{1},V_{2},V_{3})$ is a semi-strict tetrablock
isometric lift for $(A,B,T)$.

(2) A necessary and sufficient condition that there be a strict tetrablock
isometric lift ${\mathbf{V}}=(V_{1},V_{2},V_{3})$ for $(A,B,T)$ with the
isometry $V_{3}$ equal to a minimal isometric lift for $T$ is that the
fundamental operators $(G_{1},G_{2})$ for the adjoint tetrablock contraction
$(A^{*},B^{*},T^{*})$ satisfy the system of operator equations (2.8). In this
case the operator pair $(V_{1},V_{2})$ on ${\mathcal{K}}$ is uniquely
determined once one fixes a choice (essentially unique) for a minimal
isometric lift $V_{3}$ for $T$.

(3) The lift $(V_{1},V_{2},V_{3})$ can be given in functional form in the
coordinates of the Douglas model as follows. For $(A,B,T)$ equal to a
tetrablock contraction on a Hilbert space $\mathcal{H}$ let $(G_{1},G_{2})$ be
the fundamental operators of $(A^{*},B^{*},T^{*})$, let $(R_{D},S_{D},W_{D})$
be the tetrablock unitary canonically associated with $(A,B,T)$ as in
Definition 2.18, and let
$\Pi_{D}=\left[\begin{smallmatrix}{\mathcal{O}}_{D_{T^{*}},T^{*}}\\\
Q_{T^{*}}\end{smallmatrix}\right]$ be the Douglas isometric embedding map from
${\mathcal{H}}$ into $\left[\begin{smallmatrix}H^{2}({\mathcal{D}}_{T*})\\\
{\mathcal{Q}}_{T^{*}}\end{smallmatrix}\right]$. Then

$\bigg{(}\Pi_{D},\bigg{(}\begin{bmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\
0&R_{D}\end{bmatrix},\begin{bmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&S_{D}\end{bmatrix},\begin{bmatrix}M_{z}&0\\\
0&W_{D}\end{bmatrix}\bigg{)}\bigg{)}$ (3.13)

is a semi-strict (strict exactly when $(G_{1},G_{2})$ satisfies (2.8))
tetrablock isometric lift for $(A,B,T)$. In particular $(A,B,T)$ is jointly
unitarily equivalent to

$\displaystyle
P_{{\mathcal{H}}_{D}}\left(\begin{bmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\
0&R_{D}\end{bmatrix},\begin{bmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&S_{D}\end{bmatrix},\begin{bmatrix}M_{z}&0\\\
0&W_{D}\end{bmatrix}\right)\bigg{|}_{{\mathcal{H}}_{D}},$ (3.14)

where ${\mathcal{H}}_{D}$ is the functional model space given by

$\displaystyle{\mathcal{H}}_{D}:=\operatorname{Ran}\Pi_{D}\subset\begin{bmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
{\mathcal{Q}}_{T^{*}}\end{bmatrix}.$ (3.15)

and any other semi-strict ${\mathbb{E}}$-isometric lift
$(V^{\prime}_{1},V^{\prime}_{2},V^{\prime}_{3})$ with
$V^{\prime}_{3}=\left[\begin{smallmatrix}M_{z}&0\\\
0&W_{D}\end{smallmatrix}\right]$ on
$\left[\begin{smallmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
{\mathcal{Q}}_{T^{*}}\end{smallmatrix}\right]$ is equal to (3.13).

###### Proof.

The discussion preceding the statement of the theorem amounts to a proof of
statement (3) in Theorem 3.1. Statements (1) and (2) apart from the uniqueness
assertion amounts to a coordinate-free (abstract, model-free) interpretation
of the results of statement (3). It remains only to discuss the uniqueness
assertion in statements (1) and (2). This can also be formulated in terms of
the model as follows: Given a tetrablock contraction $(A,B,T)$, let
$\Pi_{D}=\left[\begin{smallmatrix}{\mathcal{O}}_{D_{T^{*}},T^{*}}\\\
Q_{T^{*}}\end{smallmatrix}\right]$ be the Douglas embedding map and let
${\mathcal{K}}_{D}=\left[\begin{smallmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
{\mathcal{Q}}_{T^{*}}\end{smallmatrix}\right]$ be the Douglas minimal
isometric lift space for $T$ with $V_{D}=\left[\begin{smallmatrix}M_{z}&0\\\
0&W_{D}\end{smallmatrix}\right]$ on ${\mathcal{K}}_{D}$ equal to the Douglas
minimal isometric lift for $T$. Suppose that

$\widetilde{A}=\begin{bmatrix}\widetilde{A}_{11}&\widetilde{A}_{12}\\\
\widetilde{A}_{21}&\widetilde{A}_{22}\end{bmatrix},\quad\widetilde{B}=\begin{bmatrix}\widetilde{B}_{11}&\widetilde{B}_{12}\\\
\widetilde{B}_{21}&\widetilde{B}_{22}\end{bmatrix}$

are two operators on
${\mathcal{K}}_{D}=\left[\begin{smallmatrix}H^{2}({\mathcal{D}}_{T^{*}}\\\
{\mathcal{Q}}_{T^{*}}\end{smallmatrix}\right]$ such that
$\big{(}\widetilde{A},\widetilde{B},\left[\begin{smallmatrix}M_{z}&0\\\
0&W_{D}\end{smallmatrix}\right]\big{)}$ is a pseudo-commutative tetrablock
isometric lift for $T$. Then necessarily

$\widetilde{A}=\begin{bmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\
0&R_{D}\end{bmatrix},\,\widetilde{B}=\begin{bmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&S_{D}\end{bmatrix}.$

where the pair $(G_{1},G_{2})$ is equal to the pair of Fundamental Operators
for the tetrablock contraction $(A^{*},B^{*},T^{*})$, and where
$(R_{D},S_{D},W_{D})$ is the tetrablock unitary canonically associated with
the tetrablock contraction $(A,B,T)$ as in Definition 2.18.

To prove this model-theoretic reformulation of the uniqueness problem, we
proceed as follows. We are given first of all that the triple

$\displaystyle\bigg{(}\widetilde{A}=\begin{bmatrix}\widetilde{A}_{11}&\widetilde{A}_{12}\\\
\widetilde{A}_{21}&\widetilde{A}_{22}\end{bmatrix},\quad\widetilde{B}=\begin{bmatrix}\widetilde{B}_{11}&\widetilde{B}_{12}\\\
\widetilde{B}_{21}&\widetilde{B}_{22}\end{bmatrix},\quad\begin{bmatrix}M_{z}&0\\\
0&W_{D}\end{bmatrix}\bigg{)}$ (3.16)

is a pseudo-commutative tetrablock isometry. According to Definition 2.6 we
have the following:

  * •

(i) $\widetilde{A}\left[\begin{smallmatrix}M_{z}&0\\\
0&W_{D}\end{smallmatrix}\right]=\left[\begin{smallmatrix}M_{z}&0\\\
0&W_{D}\end{smallmatrix}\right]\widetilde{A}$ and
$\widetilde{B}\left[\begin{smallmatrix}M_{z}&0\\\
0&W_{D}\end{smallmatrix}\right]=\left[\begin{smallmatrix}M_{z}&0\\\
0&W_{D}\end{smallmatrix}\right]\widetilde{B}$;

  * •

(ii) $\widetilde{B}=\widetilde{A}^{*}\left[\begin{smallmatrix}M_{z}&0\\\
0&W_{D}\end{smallmatrix}\right],$ and
$\widetilde{B}=\widetilde{A}^{*}\left[\begin{smallmatrix}M_{z}&0\\\
0&W_{D}\end{smallmatrix}\right]$; and

  * •

(iii) $\|\widetilde{A}\|\leq 1$.

As in the proof of Theorem 2.12, conditions (i) ad (ii) force $\widetilde{A}$
and $\widetilde{B}$ to have the block-diagonal form

$(\widetilde{A},\widetilde{B})=\left(\begin{bmatrix}M_{\widetilde{G}_{1}^{*}+z\widetilde{G}_{2}}&0\\\
0&\widetilde{A}_{22}\end{bmatrix},\begin{bmatrix}M_{\widetilde{G}_{2}^{*}+z\widetilde{G}_{1}}&0\\\
0&\widetilde{B}_{22}\end{bmatrix}\right).$

for some pseudo-commutative tetrablock unitary
$(\widetilde{A}_{22},\widetilde{B}_{22},W_{D})$, and operators
$\widetilde{G}_{1},\widetilde{G}_{2}\in{\mathcal{B}}({\mathcal{D}}_{T^{*}})$
so that the linear pencils $\widetilde{G}_{1}^{*}+z\widetilde{G}_{2}$ and
$\widetilde{G}_{1}^{*}+z\widetilde{G}_{2}$ are contraction-valued for all
$z\in{\mathbb{D}}$. We now use the fact that the triple (3.16) is a pseudo-
commutative lift of $(A,B,T)$, i.e., the operators
$\widetilde{A},\widetilde{B}$ satisfy the conditions

$\begin{bmatrix}M_{\widetilde{G}_{1}^{*}+z\widetilde{G}_{2}}^{*}&0\\\
0&\widetilde{A}_{22}^{*}\end{bmatrix}\begin{bmatrix}{\mathcal{O}}_{D_{T^{*}},T^{*}}\\\
Q_{T^{*}}\end{bmatrix}=\begin{bmatrix}{\mathcal{O}}_{D_{T^{*}},T^{*}}\\\
Q_{T^{*}}\end{bmatrix}A^{*}$

and

$\begin{bmatrix}M_{\widetilde{G}_{2}^{*}+z\widetilde{G}_{1}}^{*}&0\\\
0&\widetilde{B}_{22}^{*}\end{bmatrix}\begin{bmatrix}{\mathcal{O}}_{D_{T^{*}},T^{*}}\\\
Q_{T^{*}}\end{bmatrix}=\begin{bmatrix}{\mathcal{O}}_{D_{T^{*}},T^{*}}\\\
Q_{T^{*}}\end{bmatrix}B^{*}.$

Equivalently,

$\displaystyle
M_{\widetilde{G}_{1}^{*}+z\widetilde{G}_{2}}^{*}{\mathcal{O}}_{D_{T^{*}},T^{*}}={\mathcal{O}}_{D_{T^{*}},T^{*}}A^{*},\;M_{\widetilde{G}_{1}^{*}+z\widetilde{G}_{2}}^{*}{\mathcal{O}}_{D_{T^{*}},T^{*}}={\mathcal{O}}_{D_{T^{*}},T^{*}}A^{*}$
(3.17)
$\displaystyle\mbox{and}\quad(\widetilde{A}_{22}^{*},\widetilde{B}_{22}^{*},W_{D}^{*})Q_{T^{*}}=Q_{T^{*}}(A^{*},B^{*},T^{*}).$
(3.18)

We first show that $(\widetilde{A}_{22},\widetilde{B}_{22})=(R_{D},S_{D})$.
Note that since $\widetilde{A}_{22}$ commutes with $W_{D}$ and $W_{D}$ is a
unitary, $\widetilde{A}_{22}$ commutes with $W_{D}^{*}$ as well, and so we use
(3.18) to compute

$\displaystyle\widetilde{A}_{22}^{*}(W_{D}^{n}Q_{T^{*}}h)=W_{D}^{n}\widetilde{A}_{22}^{*}Q_{T^{*}}h$
$\displaystyle=W_{D}^{n}Q_{T^{*}}A^{*}h$
$\displaystyle=W_{D}^{n}R_{D}^{*}Q_{T^{*}}h=R_{D}^{*}(W_{D}^{n}Q_{T^{*}}h).$

Since $\\{W_{D}^{n}Q_{T^{*}}h:n\geq 0\mbox{ and }h\in{\mathcal{H}}\\}$ is
dense in ${\mathcal{Q}}_{T^{*}}$, we have $\widetilde{A}_{22}=R_{D}$.
Similarly, $\widetilde{B}_{22}=S_{D}$. Next we show that
$(\widetilde{G}_{1},\widetilde{G}_{2})$ are the fundamental operators of
$(A^{*},B^{*},T^{*})$. To this end, we can use (3.17) and the power series
expansion of ${\mathcal{O}}_{D^{*},T^{*}}(z)=\sum_{n\geq 0}D_{T^{*}}T^{*n}h$
to arrive at the equations

$\widetilde{G}_{1}D_{T^{*}}+\widetilde{G}_{2}^{*}D_{T^{*}}T^{*}=D_{T^{*}}A^{*}\quad\text{and}\quad\widetilde{G}_{2}D_{T^{*}}+\widetilde{G}_{1}^{*}D_{T^{*}}T^{*}=D_{T^{*}}B^{*}.$

By part (ii) of Theorem 2.20 applied to the tetrablock contraction
$(A^{*},B^{*},T^{*})$, $(\widetilde{G}_{1},\widetilde{G}_{2})$ must be equal
to the Fundamental Operator pair for $(A^{*},B^{*},T^{*})$. ∎

###  3.2. A Sz.-Nagy–Foias type functional model

Sz.-Nagy and Foias gave a function-space realization of
${\mathcal{Q}}_{T^{*}}$ and thereby produced a concrete functional model for a
contraction $T$. In their analysis a crucial role is played by what they
called the characteristic function associated with $T$:

$\displaystyle\Theta_{T}(z):=-T+z{\mathcal{O}}_{D_{T^{*}},T^{*}}T|_{{\mathcal{D}}_{T}}:{\mathcal{D}}_{T}\mapsto{\mathcal{D}}_{T^{*}}.$
(3.19)

The name suggests the fact the characteristic function $\Theta_{T}$ enables
one to write down an explicit functional model on which there is a compressed
multiplication operator ${\mathbf{T}}$ which recovers the original operator
$T$ up to unitary equivalence in case $T$ is a c.n.u. contraction (see Chapter
VI of [34]). Let $\Theta_{T}(\zeta)$ be the radial limit of the characteristic
function defined almost everywhere on ${\mathbb{T}}$. Consider

$\Delta_{T}(\zeta):=(I-\Theta_{T}(\zeta)^{*}\Theta_{T}(\zeta))^{1/2}.$ (3.20)

Sz.-Nagy and Foias showed in [34] that

$V_{\rm NF}:=\begin{bmatrix}M_{z}&0\\\
0&M_{\zeta}|_{\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}}\end{bmatrix}:\begin{bmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}\end{bmatrix}\to\begin{bmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}\end{bmatrix}$

is a minimal isometric lift of $T$ via some isometric embedding

$\Pi_{\rm NF}:{\mathcal{H}}\to\begin{bmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}\end{bmatrix}=:{\mathcal{K}}_{\rm
NF}$

such that

$\displaystyle{\mathcal{H}}_{\rm NF}:=\operatorname{Ran}\Pi_{\rm
NF}=\begin{bmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}\end{bmatrix}\ominus\begin{bmatrix}\Theta_{T}\\\
\Delta_{T}\end{bmatrix}\cdot H^{2}({\mathcal{D}}_{T}).$ (3.21)

Any two minimal isometric lifts of a given contraction $T$ are unitarily
equivalent; see Chapter I of [34]. In [10] an explicit unitary
$u_{\text{min}}:{\mathcal{Q}}_{T^{*}}\to\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}$
is found that intertwines $W_{D}$ and
$M_{\zeta}|_{\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}}$ and

$\Pi_{\rm NF}=\begin{bmatrix}I_{H^{2}({\mathcal{D}}_{T^{*}})}&0\\\
0&u_{\text{min}}\end{bmatrix}\Pi_{D}.$ (3.22)

It is possible to give a concrete Sz.-Nagy–Foias type functional model using
the transition map
$u_{\rm{min}}:{\mathcal{Q}}_{T^{*}}\to\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}$
as appeared (see (3.22)) in the case of a single contractive operator above.
We must replace the canonical tetrablock unitary $(R_{D},S_{D},W_{D})$ by its
avatar on the function space $\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}$:

$\displaystyle(R_{\rm{NF}},S_{\rm{NF}},W_{\rm{NF}})=u_{\rm{min}}^{*}(R_{D},S_{D},W_{D})u_{\rm{min}}.$
(3.23)

Then the following functional model is a straightforward consequence of
Theorem 3.1 and (3.22).

######  Theorem 3.2.

Let $(A,B,T)$ be a tetrablock contraction on a Hilbert space $\mathcal{H}$
such that $T$ is c.n.u., and $(G_{1},G_{2})$ be the fundamental operators of
$(A^{*},B^{*},T^{*})$. Then $(A,B,T)$ is jointly unitarily equivalent to

$\displaystyle
P_{{\mathcal{H}}_{\rm{NF}}}\left(\begin{bmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\
0&R_{\rm{NF}}\end{bmatrix},\begin{bmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&S_{\rm{NF}}\end{bmatrix},\begin{bmatrix}M_{z}&0\\\
0&W_{\rm{NF}}\end{bmatrix}\right)\bigg{|}_{{\mathcal{H}}_{\rm{NF}}},$ (3.24)

where ${\mathcal{H}}_{\rm{NF}}$ is the functional model space given by (see
(3.21))

${\mathcal{H}}_{\rm{NF}}=\operatorname{Ran}\Pi_{\rm{NF}}=\begin{bmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}\end{bmatrix}\ominus\begin{bmatrix}\Theta_{T}\\\
\Delta_{T}\end{bmatrix}\cdot H^{2}({\mathcal{D}}_{T}).$

Note that in the special case when $T^{*n}\to 0$ strongly as $n\to\infty$, the
space ${\mathcal{Q}}_{T^{*}}=0$ and hence also
$\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}=0$, i.e., $\Theta_{T}$ is
inner. Therefore in this special case the models above simply boil down to the
following which was obtained in [32].

######  Theorem 3.3.

(See [32, Theorem 4.2]) Let $(A,B,T)$ be a pure tetrablock contraction on a
Hilbert space $\mathcal{H}$ and $(G_{1},G_{2})$ be the fundamental operators
of $(A^{*},B^{*},T^{*})$. Then $(A,B,T)$ is jointly unitarily equivalent to

$P_{\operatorname{Ran}{\mathcal{O}}_{D_{T^{*}},T^{*}}}(M_{G_{1}^{*}+zG_{2}},M_{G_{2}^{*}+zG_{1}},M_{z}))|_{\operatorname{Ran}{\mathcal{O}}_{D_{T^{*}},T^{*}}}.$

##  4\. Tetrablock data sets: characteristic and special

In this section we provide some preliminary results towards a Sz.-Nagy-Foias-
type model theory for tetrablock-contraction operator tuples $(A,B,T)$. Note
that if $T$ is unitary, then the characteristic function $\Theta_{T}$, as in
(3.19), is trivial (i.e., equal to the zero operator between the zero spaces).
As the most general contraction operator $T$ is the direct sum of a unitary
$T_{u}$ with a completely nonunitary (c.n.u.) part $T_{\rm cnu}$ and the model
theory for unitary operators is easily handled by spectral theory, it is
natural for model theory purposes to restrict to the case where $T$ is c.n.u.
One then associates a functional model spaces

${\bm{\mathcal{K}}}_{T}=\begin{bmatrix}H^{2}({\mathcal{D}}_{T^{*}}\\\
\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}\end{bmatrix}\quad{\bm{\mathcal{H}}}_{T}={\bm{\mathcal{K}}}_{T}\ominus\begin{bmatrix}\Theta_{T}\\\
\Delta_{T}\end{bmatrix}H^{2}({\mathcal{D}}_{T})\subset{\bm{\mathcal{K}}}_{T}$

together with functional-model operators ${\mathbf{T}}_{T}$ and
${\mathbf{V}}_{T}$ by

${\mathbf{T}}_{T}=P_{{\bm{\mathcal{H}}}_{T}}\begin{bmatrix}M_{z}&0\\\
0&M_{\zeta}\end{bmatrix}\bigg{|}_{{\bm{\mathcal{H}}}_{T}}\text{ on
}{\bm{\mathcal{H}}}_{T},\quad{\mathbf{V}}_{T}=\begin{bmatrix}M_{z}&0\\\
0&M_{\zeta}\end{bmatrix}\text{ on }{\bm{\mathcal{K}}}_{\mathbb{T}}.$

Then it is immediate that:

  1. (NF1)

${\mathbf{V}}_{T}$ is an isometry on ${\bm{\mathcal{K}}}_{T}$.

  2. (NF2)

${\bm{\mathcal{H}}}_{\mathbf{T}}^{\perp}=\left[\begin{smallmatrix}\Theta_{T}\\\
\Delta_{T}\end{smallmatrix}\right]H^{2}({\mathcal{D}}_{T})$ is invariant for
${\mathbf{V}}_{T}$, and hence ${\mathbf{V}}_{T}$ is an isometric lift for
${\mathbf{T}}_{T}$.

Less immediately obvious are other features of the model:

  1. (NF3)

(See [34, Theorem VI.2.3]) If $T$ is c.n.u., then ${\mathbf{T}}_{T}$ is
unitarily equivalent to $T$.

  2. (NF4)

(See [34, Theorem VI.3.4].) If $T$ on ${\mathcal{H}}$ and $T^{\prime}$ on
${\mathcal{H}}^{\prime}$ are two c.n.u. contraction operators, then $T$ is
unitarily equivalent to $T^{\prime}$ if and only if $\Theta_{T}$ coincides
with $\Theta_{T^{\prime}}$ in the following sense: there exist unitary change-
of-coordinate maps $\phi\colon{\mathcal{D}}_{T}\to{\mathcal{D}}_{T^{\prime}}$
and $\phi_{*}\colon{\mathcal{D}}_{T^{*}}\to{\mathcal{D}}_{T^{\prime*}}$ so
that $\phi_{*}\Theta_{T}(\lambda)=\Theta_{T^{\prime}}(\lambda)\phi$ for all
$\lambda\in{\mathbb{D}}$. Often in the literature this property is described
simply as: the characteristic function $\Theta_{T}$ is a complete unitary
invariant for c.n.u. contraction operator $T$.

The Sz.-Nagy-Foias theory goes still further by identifying the coincidence-
envelop of the characteristic functions (LABEL:ThetaT), i.e., the set of all
contractive operator functions $({\mathcal{D}},{\mathcal{D}}_{*},\Theta)$
coinciding with the characteristic function $\Theta_{T}$ for some c.n.u.
contraction operator $T$, as simply any contractive operator function
$({\mathcal{D}},{\mathcal{D}}_{*},\Theta)$ which is pure in the sense that

$\|\Theta(0)u\|<\|u\|\text{ for any }u\in{\mathcal{D}}\text{ such that }u\neq
0.$

Then we can start with any pure COF
$({\mathcal{D}},{\mathcal{D}}_{*},\Theta)$, form ${\bm{\mathcal{K}}}(\Theta)$
and ${\bm{\mathcal{H}}}(\Theta)$ according to

${\bm{\mathcal{K}}}(\Theta)=\begin{bmatrix}H^{2}({\mathcal{D}}_{*})\\\
\overline{D_{\Theta}\cdot
L^{2}({\mathcal{D}})}\end{bmatrix},\quad{\bm{\mathcal{H}}}(\Theta)={\bm{\mathcal{K}}}(\Theta)\ominus\begin{bmatrix}\Theta\\\
D_{\Theta}\end{bmatrix}H^{2}({\mathcal{D}})\subset{\bm{\mathcal{K}}}(\Theta)$

where $D_{\Theta}$ is the $\Theta$-defect operator function
$D_{\Theta}(\zeta)=(I_{\mathcal{D}}-\Theta(\zeta)^{*}\Theta(\zeta))^{\frac{1}{2}}$.
Then we can form the model operators

${\mathbf{V}}(\Theta)=\begin{bmatrix}M_{z}&0\\\ 0&M_{\zeta}\end{bmatrix}\text{
on
}{\bm{\mathcal{K}}}(\Theta),\quad{\mathbf{T}}(\Theta)=P_{{\bm{\mathcal{H}}}(\Theta)}{\mathbf{V}}(\Theta)\bigg{|}_{{\bm{\mathcal{H}}}(\Theta)}.$

Then we have the additional results:

  1. (NF5)

(See [34, Theorem VI.3.1].) Given any pure COF $\Theta$,
${\mathbf{T}}(\Theta)$ is a c.n.u. contraction operator on
${\bm{\mathcal{H}}}(\Theta)$ with characteristic operator function
$\Theta_{{\mathbf{T}}(\Theta)}$ coinciding with $\Theta$.

In this way the loop is closed: the study of c.n.u. contraction operators is
the same as the study of pure COFs.

To explain generalizations to the setting of tetrablock contractions, we first
introduce some useful terminology. For this discussion, just as in the
classical Sz.-Nagy-Foias settings, it makes sense to restrict to c.n.u.
tetrablock contractions (see Theorem 2.16 and Remark 2.17).

######  Definition 4.1.

Let us say that any collection of objects

$\Xi=(\Theta,(G_{1},G_{2}),\psi\\}$

consisting of

  1. (i)

a pure COF function $({\mathcal{D}},{\mathcal{D}}_{*},\Theta)$,

  2. (ii)

a pair of operators $(G_{1},G_{2})$ on the coefficient space
${\mathcal{D}}_{*}$, and

  3. (iii)

a measurable function $\psi$ on ${\mathbb{T}}$ such that, for a.e.
$\zeta\in{\mathbb{T}}$, $\psi(\zeta)$ is a contractive normal operator on
${\mathcal{D}}_{\Theta}(\zeta)=\overline{\operatorname{Ran}\,D_{\Theta}(\zeta)}$.

is a tetrablock data set.

######  Remark 4.2.

Canonically associated with any such $\psi$ as in item (iii) in Definition 4.1
is the tetrablock unitary triple $(R,S,W)$ on the direct integral space
$\oplus\int_{\mathbb{T}}{\mathcal{D}}_{\Theta(\zeta)}\frac{|{\tt
d}\zeta|}{2\pi}$ given by

$R=M_{\psi^{*}\cdot\zeta},\quad S=M_{\psi},\quad W=M_{\zeta}$

and (as one sweeps over all possible such $\psi$), this is the general
tetrablock unitary operator triple $(R,S,W)$ on the space
$\oplus\int_{\mathbb{T}}{\mathcal{D}}_{\Theta(\zeta)}\frac{|{\tt
d}\zeta|}{2\pi}$ with the unitary operator $W$ equal to $W=M_{\zeta}$
(multiplication by the coordinate function) (see Example 2.10 (2)). Thus item
(iii) in the definition of tetrablock data set can be equivalently rephrased
as:

  1. (iii’)

a tetrablock unitary operator-triple $(R,S,W)$ on the direct-integral space
$\int_{\mathbb{T}}{\mathcal{D}}_{\Theta(\zeta)}\frac{|{\tt d}\zeta|}{2\pi}$
such that the last unitary component $W$ is equal to multiplication by the
coordinate function $W=M_{\zeta}$.

However, for convenience of notation, we shall continue to use the notation
$(R,S,W)$ for the third component of a tetrablock data set
$\Xi=(\Theta,(G_{1},G_{2}),(R,S,W))$ with the convention (iii’) also part of
the definition.

######  Definition 4.3.

Given a c.n.u. tetrablock contraction $(A,B,T)$ we say that
$\Xi_{A,B,T}=(\Theta,(G_{1},G_{2}),\psi)$ is the characteristic tetrablock
data set for $(A,B,T)$ if

  1. (i)

$({\mathcal{D}},{\mathcal{D}}_{*},\Theta)$ is equal to the Sz.-Nagy-Foias
characteristic function $({\mathcal{D}}_{T},{\mathcal{D}}_{T^{*}},\Theta_{T})$
for the c.n.u. contraction operator $T$,

  2. (ii)

$(G_{1},G_{2})$ is equal to the Fundamental Operator pair for the adjoint
tetrablock contraction $(A^{*},B^{*},T^{*})$, and

  3. (iii)

$(R,S,D)$ is given by

$((R,S,W))=(R_{\rm NF},S_{\rm NF},W_{\rm NF}):=u_{\rm
min}^{*}(R_{D},S_{D},W_{D})u_{\rm min}$

where $(R_{D},S_{D},W_{D})$ is the tetrablock unitary on
${\mathcal{Q}}_{T^{*}}$ determined by tetrablock contraction $(A,B,T)$
according to Definition 2.18, and where $u_{\rm
min}\colon\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}\to{\mathcal{Q}}_{T^{*}}$
is the unitary identification map identifying the Sz.-Nagy-Foias lifting
residual space $\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}$ with the
Douglas lifting residual space ${\mathcal{Q}}_{T^{*}}$.

Then it is clear that the characteristic tetrablock data set $\Xi_{A,B,T}$ for
a c.n.u. tetrablock contraction $(A,B,T)$ is a tetrablock data set. The
natural notion of equivalence for tetrablock-data sets is the following.

######  Definition 4.4.

Let $(\mathcal{D},\mathcal{D}_{*},\Theta)$,
$(\mathcal{D^{\prime}},\mathcal{D^{\prime}_{*}},\Theta^{\prime})$ be two
purely contractive analytic functions. Let
$G_{1},G_{2}\in{\mathcal{B}}({\mathcal{D}}_{*})$,
$G_{1}^{\prime},G_{2}^{\prime}\in{\mathcal{B}}(\mathcal{D^{\prime}_{*}})$, and
$(R,S,W)$ on $\overline{\Delta_{\Theta}L^{2}(\mathcal{D})}$ and
$(R^{\prime},S^{\prime},W^{\prime})$ on
$\overline{\Delta_{\Theta^{\prime}}L^{2}(\mathcal{D^{\prime}})}$ be two
tetrablock unitaries (with $W$ and $W^{\prime}$ equal to $M_{\zeta}$ on their
respective spaces). We say that the two triples
$(\Theta,(G_{1},G_{2}),(R,S,W))$ and
$(\Theta^{\prime},(G_{1}^{\prime},G_{2}^{\prime}),(R^{\prime},S^{\prime},W^{\prime}))$
coincide if:

  1. (i)

$(\mathcal{D},\mathcal{D}_{*},\Theta)$ and
$(\mathcal{D^{\prime}},\mathcal{D^{\prime}_{*}},\Theta^{\prime})$ coincide,

  2. (ii)

the unitary operators $\phi$, $\phi_{*}$ involved in the coincidence of
$(\mathcal{D},\mathcal{D}_{*},\Theta)$ and
$(\mathcal{D^{\prime}},\mathcal{D^{\prime}_{*}},\Theta^{\prime})$ satisfy the
additional intertwining conditions:

$\displaystyle\phi_{*}(G_{1},G_{2})=(G_{1}^{\prime},G_{2}^{\prime})\phi_{*}\quad\mbox{and}\quad\omega_{\phi}(R,S,W)=(R^{\prime},S^{\prime},W^{\prime})\omega_{\phi},$

where
$\omega_{\phi}:\overline{\Delta_{\Theta}L^{2}(\mathcal{D})}\to\overline{\Delta_{\Theta^{\prime}}L^{2}(\mathcal{D^{\prime}})}$
is the unitary map induced by $\phi$ according to the formula

$\omega_{\phi}:=(I_{L^{2}}\otimes\phi)|_{\overline{\Delta_{\Theta}L^{2}(\mathcal{D})}}.$
(4.1)

Given a characteristic tetrablock data set
$\Xi_{(A,B,T)}=(\Theta,(G_{1},G_{2}),(R,S,W))$ for a tetrablock contraction
$(A,B,T)$ we can write down a functional model:

${\bm{\mathcal{K}}}(\Xi)=\begin{bmatrix}H^{2}({\mathcal{D}}_{*})\\\
\overline{D_{\Theta}L^{2}({\mathcal{D}})}\end{bmatrix},\quad{\bm{\mathcal{H}}}(\Xi)={\bm{\mathcal{K}}}\ominus\begin{bmatrix}\Theta\\\
D_{\Theta}\end{bmatrix}H^{2}({\mathcal{D}})$

with functional-model operators

$\displaystyle{\mathbf{V}}(\Xi)=\bigg{(}\begin{bmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\
0&R\end{bmatrix},\begin{bmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&S\end{bmatrix},\begin{bmatrix}M_{z}&0\\\
0&M_{\zeta}\end{bmatrix}\bigg{)}\text{ on }{\bm{\mathcal{K}}}(\Xi),$
$\displaystyle{\mathbf{T}}(\Xi)=P_{{\bm{\mathcal{H}}}(\Xi)}{\mathbf{V}}(\Xi)|_{{\bm{\mathcal{H}}}(\Xi)}.$

The tetrablock analogue of items (NF1)-(NF2) in our discussion of the
Sz.-Nagy-Foias model above fails without the additional assumptions: namely,
it is not the case that ${\mathbf{V}}$ is a tetrablock isometry as well as
that ${\mathbf{V}}$ is a lift for ${\mathbf{T}}$ unless we also impose the
condition (2.8) on $(G_{1},G_{2})$. Nevertheless, the analogue of (NF3) does
hold: given that $\Xi$ is the characteristic tetrablock triple for $(A,B,T)$,
it is the case that $(A,B,T)$ is unitarily equivalent to ${\mathbf{T}}(\Xi)$:
this is the content of the last part of item (3) in Theorem 3.1 (after a
conversion to Sz.-Nagy-Foias rather than Douglas coordinates): see (3.14) and
(3.15). The next theorem amounts to the analogue of (NF4) in our list of
features for the Sz.-Nagy-Foias model above.

######  Theorem 4.5.

Let $(A,B,T)$ and $(A^{\prime},B^{\prime},T^{\prime})$ be two tetrablock
contractions acting on ${\mathcal{H}}$ and ${\mathcal{H}}^{\prime}$,
respectively. Let

$(\Theta_{T},(G_{1},G_{2}),(R_{\rm NF},S_{\rm NF},W_{\rm
NF}),\quad(\Theta_{T^{\prime}},(G_{1}^{\prime},G_{2}^{\prime}),(R_{\rm
NF}^{\prime},S_{\rm NF}^{\prime},W_{\rm NF}^{\prime}))$

be the characteristic tetrablock data sets for $(A,B,T)$ and
$(A^{\prime},B^{\prime},T^{\prime})$, respectively.

(1) If $(A,B,T)$ and $(A^{\prime},B^{\prime},T^{\prime})$ are unitarily
equivalent, then their characteristic tetrablock data sets coincide.

(2) Conversely, if $T$ and $T^{\prime}$ are c.n.u. contractions and the
characteristic tetrablock data sets of $(A,B,T)$ and
$(A^{\prime},B^{\prime},T^{\prime})$ coincide, then $(A,B,T)$ and
$(A^{\prime},B^{\prime},T^{\prime})$ are unitarily equivalent.

###### Proof.

First suppose that $(A,B,T)$ and $(A^{\prime},B^{\prime},T^{\prime})$ are
unitarily equivalent via a unitary
$\tau:{\mathcal{H}}\to{\mathcal{H}}^{\prime}$. The fact that $\Theta_{T}$ and
$\Theta_{T^{\prime}}$ coincide is a part of the Sz.-Nagy–Foias theory [34].
Indeed, note that

$\displaystyle\tau(I-T^{*}T)=(I-T^{\prime*}T^{\prime})\tau\text{ and
}\tau(I-TT^{*})=(I-T^{\prime}T^{\prime*})\tau$

and therefore by the functional calculus for positive operators,

$\displaystyle\tau D_{T}=D_{T^{\prime}}\tau\quad\mbox{and}\quad\tau
D_{T^{*}}=D_{T^{\prime*}}\tau$ (4.2)

and thereby inducing two unitary operators

$\displaystyle\phi:=\tau|_{{\mathcal{D}}_{T}}:{\mathcal{D}}_{T}\to{\mathcal{D}}_{T^{\prime}}\text{
and
}\phi_{*}:=\tau|_{{\mathcal{D}}_{T^{*}}}:{\mathcal{D}}_{T^{*}}\to{\mathcal{D}}_{T^{\prime*}}.$
(4.3)

Consequently $\phi_{*}\Theta_{T}=\Theta_{T^{\prime}}\phi$. Next, since the
fundamental operators are the unique operators that satisfy the equations for
$(X_{1},X_{2})=(G_{1},G_{2})$:

$\displaystyle A^{*}-BT^{*}=D_{T^{*}}X_{1}D_{T^{*}}\quad\mbox{and}\quad
B^{*}-AT^{*}=D_{T^{*}}X_{2}D_{T^{*}},$

one can easily obtain using (4.2) that

$\displaystyle\phi_{*}(G_{1},G_{2})=(G_{1}^{\prime},G_{2}^{\prime})\phi_{*}.$
(4.4)

Finally the proof of the forward direction will be complete if we establish
that

$\displaystyle(R_{\rm NF},S_{\rm NF},W_{\rm
NF})=\omega_{\phi}^{*}(R^{\prime}_{\rm NF},S^{\prime}_{\rm NF},W^{\prime}_{\rm
NF})\omega_{\phi}$ (4.5)

where
$\omega_{\phi}=(I_{L^{2}}\otimes\phi)|_{\overline{\Delta_{T}L^{2}(\mathcal{D}_{T})}}:\overline{\Delta_{T}L^{2}(\mathcal{D}_{T})}\to\overline{\Delta_{T^{\prime}}L^{2}(\mathcal{D}_{T^{\prime}})}$.
For this we first note that

$\displaystyle\begin{bmatrix}I&0\\\ 0&u_{\rm
min}\end{bmatrix}\begin{bmatrix}{\mathcal{O}}_{D_{T^{*}},T^{*}}\\\
Q_{T^{*}}\end{bmatrix}=\Pi_{\rm NF}$
$\displaystyle=\begin{bmatrix}I_{H^{2}}\otimes\phi_{*}^{*}&0\\\
0&\omega_{\phi}^{*}\end{bmatrix}\Pi_{\rm NF}^{\prime}\tau$
$\displaystyle=\begin{bmatrix}I_{H^{2}}\otimes\phi_{*}^{*}&0\\\
0&\omega_{\phi}^{*}\end{bmatrix}\begin{bmatrix}I&0\\\ 0&u_{\rm
min}^{\prime}\end{bmatrix}\begin{bmatrix}{\mathcal{O}}_{D_{T^{\prime*}},T^{\prime*}}\\\
Q_{T^{\prime*}}\end{bmatrix}\tau,$

from which we read off that

$\displaystyle u_{\rm min}Q_{T^{*}}=\omega_{\phi}^{*}u_{\rm
min}^{\prime}Q_{T^{\prime*}}\tau.$ (4.6)

Now since
${\mathcal{Q}}_{T^{\prime*}}=\overline{\operatorname{span}}\\{W_{D}^{\prime
n}Q_{T^{\prime*}}\tau h:h\in{\mathcal{H}},n\geq 0\\}$ and $u_{\rm min}$ has
the intertwining property $u_{\rm min}W_{D}=M_{\zeta}u_{\rm min}$, we use
(4.6) to compute

$\displaystyle\omega_{\phi}\cdot u_{\rm
min}\cdot\omega_{\tau}^{*}(W_{D}^{\prime n}Q_{T^{\prime*}}\tau h)$
$\displaystyle=\omega_{\phi}\cdot u_{\rm min}(W_{D}^{n}Q_{T^{*}}h)$
$\displaystyle=\omega_{\phi}M_{\zeta}^{n}u_{\rm min}Q_{T^{*}}h$
$\displaystyle=M_{\zeta}^{n}\omega_{\phi}\cdot u_{\rm min}Q_{T^{*}}h$
$\displaystyle=M_{\zeta}^{n}u_{\rm min}^{\prime}Q_{T^{\prime*}}\tau h=u_{\rm
min}^{\prime}(W_{D}^{\prime n}Q_{T^{\prime*}}\tau h).$

Consequently

$\displaystyle\omega_{\phi}\cdot u_{\rm min}\cdot\omega_{\tau}^{*}=u_{\rm
min}^{\prime}.$ (4.7)

Using this identity and the intertwining properties of the unitaries involved,
it is now easy to establish (4.5).

Conversely, suppose $T$ and $T^{\prime}$ are c.n.u. contractions,
$\phi:{\mathcal{D}}_{T}\to{\mathcal{D}}_{T^{\prime}}$ and
$\phi_{*}:{\mathcal{D}}_{T^{*}}\to{\mathcal{D}}_{T^{\prime*}}$ be the unitary
operators involved in the coincidence of the characteristic tetrablock data
sets

$((G_{1},G_{2}),(R_{\rm NF},S_{\rm NF},W_{\rm
NF}),\Theta_{T}),\quad((G_{1}^{\prime},G_{2}^{\prime}),(R_{\rm
NF}^{\prime},S_{\rm NF}^{\prime},W_{\rm NF}^{\prime}),\Theta_{T^{\prime}}).$

By Definition 4.4, it follows that the unitary

$U=\begin{bmatrix}I_{H^{2}}\otimes\phi_{*}&0\\\
0&\omega_{\phi}\end{bmatrix}\colon\begin{bmatrix}H^{2}({\mathcal{D}}_{T^{*}})\\\
\overline{\Delta_{T}L^{2}({\mathcal{D}}_{T})}\end{bmatrix}\to\begin{bmatrix}H^{2}({\mathcal{D}}_{T^{\prime*}})\\\
\overline{\Delta_{T^{\prime}}L^{2}({\mathcal{D}}_{T^{\prime}})}\end{bmatrix}$

identifies the model spaces ${\mathcal{H}}_{\rm NF}$ and ${\mathcal{H}}_{\rm
NF}^{\prime}$ and intertwines the model operators as in (3.24) associated with
$(A,B,T)$ and $(A^{\prime},B^{\prime},T^{\prime})$, respectively. This
completes the proof of Theorem 4.5. ∎

The question remains as to what additional coupling conditions must be imposed
on a tetrablock data set $\Xi$ to assure that $\Xi$ coincides with the
characteristic tetrablock data set for a c.n.u. tetrablock contraction
$(A,B,T)$. In the Sz.-Nagy-Foias theory, the data set (or invariant) consists
of a single COF, and the only additional requirement is that it must be pure.

From the results of Section 2.4 we see that any characteristic tetrablock data

$\Xi_{A,B,T}=(\Theta,(G_{1},G_{2}),(R,S,W))$

for a c.n.u. tetrablock contraction operator-triple $(A,B,T)$ satisfies the
additional conditions (expressed directly in terms of the components of
$\Xi_{A,B,T}$ rather than in terms of $(A,B,T)$):

  1. (i)

$\Theta$ is a pure COF (see [34, Theorem VI.3.1]),

  2. (ii)

the numerical radius conditions

$\nu(G_{1}^{*}+zG_{2})\leq 1,\quad\nu(G_{2}^{*}+zG_{1})\leq 1\text{ for all
}z\in\overline{\mathbb{D}}$

hold, implying that also the spectral radius conditions

$r(M_{G_{1}^{*}+zG_{2}})\leq 1,\quad r(M_{G_{2}^{*}+zG_{1}})\leq 1$

(see Theorems 2.22 (1)).

  3. (iii)

It is almost the case that the spectral radius and norm agree for
$M_{G_{1}^{*}+zG_{2}}$ and $M_{G_{2}^{*}+zG_{1}}$ in the following sense (see
Theorem 2.9 (1)):

$r(M_{G_{1}^{*}+zG_{2}}\cdot
M_{G_{2}^{*}+zG_{1}})=\max\\{\|M_{G_{1}^{*}+zG_{2}}\|^{2},\,\|M_{G_{2}^{*}+zG_{1}}\|^{2}\\}.$

However we do not expect that just imposing these conditions is sufficient to
guarantee that such a tetrablock data set $\Xi$ will coincide with the
characteristic tetrablock data set for some tetrablock contraction, so we do
not expect to have an analogue of (NF5) at this level of generality.

Let us now specialize our class of tetrablock contractions to what we call
special tetrablock contractions, i.e., any tetrablock contraction $(A,B,T)$
with the special property that the Fundamental Operator pair $(G_{1},G_{2})$
for $(A^{*},B^{*},T^{*})$ satisfies the additional pair of operator equations
(2.8). By Theorem 3.1, this is equivalent to $(A,B,T)$ having a minimal
tetrablock isometric lift $(V_{1},V_{2},V_{3})$ acting on a minimal Sz.-Nagy
isometric-lift space for $T$ with $V_{3}$ equal to a minimal isometric lift
for the single contraction operator $T$.

This suggests that we define a special tetrablock data set as follows. For
convenience in later discussion we shall now write the last component
$(R,S,W)$ simply as $\psi$ for a measurable contractive-normal operator-valued
function
$\zeta\mapsto\psi(\zeta)\in{\mathcal{B}}({\mathcal{D}}_{\Theta(\zeta)})$
according to the convention explained in Remark 4.2.

######  Definition 4.6.

We say that the tetrablock data set

$\Xi=(({\mathcal{D}},{\mathcal{D}}_{*},\Theta),(G_{1},G_{2}),\psi)$ (4.8)

is a special tetrablock data set if the following conditions hold:

  1. (i)

The operators $G_{1},G_{2}\in{\mathcal{B}}({\mathcal{D}}_{*})$ satisfy the
commutativity conditions (2.8), i.e.,

$[G_{1},G_{2}]=0,\quad[G_{1}^{*},G_{1}]=[G_{2}^{*},G_{2}]$

as well as the pencil-contractivity condition

$\|G_{1}^{*}+zG_{2}\|\leq 1\text{ for all }z\in\overline{{\mathbb{D}}}$

and then also

$\|G_{2}^{*}+zG_{1}\|\leq 1\text{ for all }z\in\overline{{\mathbb{D}}}.$

  2. (ii)

the space $\left\\{\left[\begin{smallmatrix}\Theta\\\
D_{\Theta}\end{smallmatrix}\right]f:f\in H^{2}({\mathcal{D}})\right\\}$ is
jointly invariant under the operator triple

$\displaystyle\bigg{(}\begin{bmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\
0&M_{\psi(\zeta)^{*}\cdot\zeta}\end{bmatrix},\begin{bmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&M_{\psi(\zeta)}\end{bmatrix},\begin{bmatrix}M_{z}&0\\\
0&M_{\zeta}\end{bmatrix}\bigg{)}.$ (4.9)

Given a special tetrablock data set $(\Theta,(G_{1},G_{2}),\psi)$, we say that
the space

$\displaystyle{\bm{{\mathcal{H}}}}=\begin{bmatrix}H^{2}({\mathcal{D}}_{*})\\\
\overline{D_{\Theta}L^{2}({\mathcal{D}})}\end{bmatrix}\ominus\begin{bmatrix}\Theta\\\
D_{\Theta}\end{bmatrix}H^{2}({\mathcal{D}})$ (4.10)

is the functional model space and the (commutative) operator triple
$({\mathbf{A}},{\mathbf{B}},{\mathbf{T}})$ given by

$\displaystyle
P_{{\bm{\mathcal{H}}}}\bigg{(}\begin{bmatrix}M_{G_{1}^{*}+zG_{2}}&0\\\
0&M_{\psi(\zeta)^{*}\cdot\zeta}\end{bmatrix},\begin{bmatrix}M_{G_{2}^{*}+zG_{1}}&0\\\
0&M_{\psi(\zeta)}\end{bmatrix},\begin{bmatrix}M_{z}&0\\\
0&M_{\zeta}\end{bmatrix}\bigg{)}\bigg{|}_{\bm{\mathcal{H}}}$ (4.11)

the functional-model operator triple associated with the data set. The
following theorem is our one analogue of item (NF5) in our list of features of
the Sz.-Nagy-Foias model.

######  Theorem 4.7.

If $\Xi=(\Theta,(G_{1},G_{2}),\psi)$ is a special tetrablock data set, then
the associated model operator triple $(\bf A,\bf B,\bf T)$ as in (4.11), is a
tetrablock contraction that lifts to the tetrablock isometry as in (4.9).
Moreover, the tetrablock data set $\Xi$ coincides with the characteristic
triple of $(\bf A,\bf B,\bf T)$.

###### Proof.

By part (i) of Definition 4.6, the triple as in (4.9) is a strict tetrablock
isometry, and by part (ii), it lifts the model triple $(\bf A,\bf B,\bf T)$ as
in (4.11). Thus in particular, $(\bf A,\bf B,\bf T)$ is a tetrablock
contraction and the first part of the theorem follows. For the second part, we
use the Sz.-Nagy–Foias model theory for single contractions and Theorem 3.1 as
follows. Apply Theorem VI.3.1 in [34] to the purely contractive analytic
function $\Theta$ to conclude that the characteristic function $\Theta_{\bf
T}$ of $\bf T$ coincides with $\Theta$, i.e., there exists unitary operators
$u:{\mathcal{D}}\to{\mathcal{D}}_{\bf T}$ and
$u_{*}:{\mathcal{D}}_{*}\to{\mathcal{D}}_{\bf T^{*}}$ such that
$u_{*}\cdot\Theta(z)=\Theta_{\bf T}(z)\cdot u$ for all $z\in{\mathbb{D}}$. Let
us set $(G_{1}^{\prime},G_{2}^{\prime}):=u_{*}(G_{1},G_{2})u_{*}^{*}$ and
$(R^{\prime},S^{\prime},W^{\prime}):=\omega_{u}(M_{\psi(\zeta)^{*}\cdot\zeta},M_{\psi(\zeta)},M_{\zeta})\omega_{u}^{*}$.
Since $G_{1},G_{2}$ satisfy the commutativity conditions,
$G_{1}^{\prime},G_{2}^{\prime}$ also satisfy the same conditions, and
consequently the triple

$\displaystyle\bigg{(}\begin{bmatrix}M_{G_{1}^{\prime*}+zG_{2}^{\prime}}&0\\\
0&\omega_{u}M_{\psi(\zeta)^{*}\cdot\zeta}\omega_{u}^{*}\end{bmatrix},\begin{bmatrix}M_{G_{2}^{\prime*}+zG_{1}^{\prime}}&0\\\
0&\omega_{u}M_{\psi}\omega_{u}^{*}\end{bmatrix},\begin{bmatrix}M_{z}&0\\\
0&M_{\zeta}\end{bmatrix}\bigg{)}$ (4.12)

is a strict tetrablock isometry. Note that since

$\displaystyle\begin{bmatrix}u_{*}&0\\\
0&\omega_{u}\end{bmatrix}\begin{bmatrix}\Theta\\\
\Delta_{\Theta}\end{bmatrix}=\begin{bmatrix}\Theta_{\bf T}\\\
\Delta_{\Theta_{\bf T}}\end{bmatrix}\begin{bmatrix}u&0\\\
0&\omega_{u}\end{bmatrix},$

the unitary operator

$\tau:=\begin{bmatrix}u_{*}&0\\\
o&\omega_{u}\end{bmatrix}:\begin{bmatrix}H^{2}({\mathcal{D}}_{*})\\\
\overline{\Delta_{\Theta}L^{2}({\mathcal{D}})}\end{bmatrix}\to\begin{bmatrix}H^{2}({\mathcal{D}}_{\bf
T^{*}})\\\ \overline{\Delta_{\Theta_{\bf T}}L^{2}({\mathcal{D}}_{\bf
T})}\end{bmatrix}$

takes the functional model space ${\bm{\mathcal{H}}}$ as in (4.10) onto

$\displaystyle\begin{bmatrix}H^{2}({\mathcal{D}}_{\bf T^{*}})\\\
\overline{\Delta_{\Theta_{\bf T}}L^{2}({\mathcal{D}}_{\bf
T})}\end{bmatrix}\ominus\begin{bmatrix}\Theta_{\bf T}\\\ \Delta_{\Theta_{\bf
T}}\end{bmatrix}H^{2}({\mathcal{D}}_{\bf T}).$

Therefore by part (ii) of Definition 4.6, the tetrablock isometry as in (4.12)
is a lift of $(\bf A,\bf B,\bf T)$ via the embedding
$\iota\cdot\tau|_{\bm{\mathcal{H}}}$, where

$\iota:\begin{bmatrix}H^{2}({\mathcal{D}}_{\bf T^{*}})\\\
\overline{\Delta_{\Theta_{\bf T}}L^{2}({\mathcal{D}}_{\bf
T})}\end{bmatrix}\ominus\begin{bmatrix}\Theta_{\bf T}\\\ \Delta_{\Theta_{\bf
T}}\end{bmatrix}\to\begin{bmatrix}H^{2}({\mathcal{D}}_{\bf T})\\\
\overline{\Delta_{\Theta_{\bf T}}L^{2}({\mathcal{D}}_{\bf T})}\end{bmatrix}$

is the inclusion map. Since

$\begin{bmatrix}M_{z}&0\\\
0&M_{\zeta}\end{bmatrix}:\begin{bmatrix}H^{2}(D_{\bf T})\\\
\overline{\Delta_{\Theta_{\bf T}}L^{2}({\mathcal{D}}_{\bf
T})}\end{bmatrix}\to\begin{bmatrix}H^{2}(D_{\bf T})\\\
\overline{\Delta_{\Theta_{\bf T}}L^{2}({\mathcal{D}}_{\bf T})}\end{bmatrix}$

is a minimal isometric lift of $\bf T$, by part (2) of Theorem 3.1, there is a
unique such tetrablock isometric lift with the last entry of the lift fixed.
If $\bf G_{1},\bf G_{2}$ are the Fundamental Operators of $(\bf A^{*},\bf
B^{*},\bf T^{*})$, then by Theorem 3.2,

$\displaystyle\left(\begin{bmatrix}M_{\bf G_{1}^{*}+z\bf G_{2}}&0\\\
0&R_{\rm{NF}}\end{bmatrix},\begin{bmatrix}M_{\bf G_{2}^{*}+z\bf G_{1}}&0\\\
0&S_{\rm{NF}}\end{bmatrix},\begin{bmatrix}M_{z}&0\\\
0&W_{\rm{NF}}\end{bmatrix}\right)$

is another tetrablock isometric lift of $(\bf A,\bf B,\bf T)$, where $(R_{\rm
NF},S_{\rm NF},W_{\rm NF})$ is the canonical tetrablock unitary associated
with $(\bf A,\bf B,\bf T)$. Therefore we must have $({\bf G}_{1},{\bf
G}_{2})=(G_{1}^{\prime},G_{2}^{\prime})=u_{*}(G_{1},G_{2})u_{*}^{*}$ and

$(R_{\rm NF},S_{\rm NF},W_{\rm
NF})=\omega_{u}(M_{\psi(\zeta)^{*}\cdot\zeta},M_{\psi(\zeta)},M_{\zeta})\omega_{u}^{*}.$

This is what was needed to be shown. ∎

Epilogue. It is easy to write down tetrablock data sets as in (4.8). Given
such a tetrablock data set, it may not be very tractable to determine if in
addition it satisfies conditions (i) and (ii) in Definition 4.6.

However it is not so difficult to cook up viable examples. For example, we
note that the commutativity conditions in (i) are automatic if we choose
$G_{1}$ and $G_{2}$ to be scalar operators on ${\mathcal{D}}_{*}$. We can
arrange the pencil contractivity condition to hold just by choosing $G_{1}$
and $G_{2}$ to be sufficiently small. If we choose the operator $\psi(\zeta)$
to be a scalar for each $\zeta$, then we are forced to choose
$\psi(\zeta)=G_{2}^{*}\zeta+G_{1}$. Since we have already chosen $G_{2}$ and
$G_{1}$ so that the pencil contractivity condition holds, then $\psi(\zeta)$
is contractive and of course a scalar operator is normal. Then all conditions
are satisfied. In this way we get a whole class of tractable examples of
special tetrablock contractions for any pure COF $\Theta$. If
${\mathcal{D}}_{*}$ and ${\mathcal{D}}$ are both at most one-dimensional, all
the examples are of this form.

Conflict of Interest/Dataset Statement. The authors state that there are no
conflicts of interest. No data sets were generated or analyzed during the
current study.

## References

  * [1] A. A. Abouhajar, M. C. White and N. J. Young, A Schwarz lemma for a domain related to $\mu$-synthesis, J. Geom. Anal. 17 (2007), 717-750.
  * [2] J. Agler, J. Harland and B. Raphael, Classical function theory, operator dilation theory, and machine computation on multiply-connected domains, Mem. Amer. Math. Soc. 191 (2008), no. 892, viii+159 pp.
  * [3] J. Agler and N. J. Young, Operators having the symmetrized bidisc as a spectral set, Proc. Edinburgh Math. Soc. (2) 43 (2000), 195-210.
  * [4] J. Agler and N. J. Young, A model theory for $\Gamma$-contractions, J. Operator Theory 49 (2003), 45-60. (2004), 375-403.
  * [5] T. Andô, On a Pair of Commuting Contractions, Acta Sci. Math. (Szeged) 24 (1963), 88-90.
  * [6] W.B. Arveson, An Invitation to $C^{*}$-algebras, Graduate Texts in Mathematics 39, Springer-Verlag, Heidelberg, New York, 1976.
  * [7] W. B. Arveson, Subalgebras of $C^{*}$-algebra. II., Acta Math. 128 (1972), 271-308.
  * [8] W.B. Arveson, A short course in spectral theory, Graduate Texts in Mathematics 209, Springer-Verlag, New York, 2002.
  * [9] J.A. Ball and H. Sau, Rational dilation of tetrablock contractions revisited, J. Funct. Anal. 278 (2020) 108275.
  * [10] J.A. Ball and H. Sau, Sz.-Nagy–Foias functional models for pairs of commuting Hilbert-space contraction operators, in preparation.
  * [11] J.A. Ball and H. Sau, Functional models for commuting Hilbert-space contractions, Operator theory, operator algebras and their interactions with geometry and topology: Ronald G. Douglas memorial volume, 11-54, Oper. Theory Adv. Appl., 278, Birkhäuser/Springer, Cham, 2020.
  * [12] C. A. Berger, L. A. Coburn and A. Lebow, Representation and index theory for $C^{*}$-algebras generated by commuting isometries, J. Funct. Anal. 27 (1978), no. 1, 51-99.
  * [13] T. Bhattacharyya, The tetrablock as a spectral set, Indiana Univ. Math. J. 63 (2014), 1601-1629.
  * [14] T. Bhattacharyya, S. Lata and H. Sau, Admissible fundamental operators, J. Math. Anal. Appl. 425 (2015), no. 2, 983-1003.
  * [15] T. Bhattacharyya, S. Pal and S. Shyam Roy, Dilations of $\Gamma$-contractions by solving operator equations, Adv. Math. 230 (2012) 577-606.
  * [16] T. Bhattacharyya and S. Pal, A functional model for pure $\Gamma$-contractions, J. Operator Theory 71 (2014), 327-339.
  * [17] T. Bhattacharyya and H. Sau, Explicit and unique construction of tetrablock unitary dilation in a certain case, Complex Anal. Oper. Theory, 10 (2016), 749-768.
  * [18] B. Bisai and S. Pal, A model theory for operators associated with a domain related to $\mu$-synthesis, Collect. Math (2021). https://doi.org/10.1007/s13348-021-00341-6
  * [19] J.B. Conway, The Theory of Subnormal Operators, Mathematical Surveys ad Monographs, vol. 36, American Mathematical Society, Providence, RI, 1991.
  * [20] J. Dixmier, von Neumann Algebras (with a preface by E.C. Lance), translated from the second French edition by F. Jellett, North-Holland Math. Library 27, North Holland Publishing Co., Amsterdam/New York, 1981.
  * [21] R. G. Douglas, Structure theory for operators. I., J. Reine Angew. Math. 232 (1968) 180-193.
  * [22] J.C. Doyle and A. Packard, The complex structured singular value, Automatica 29 no. 1, (1993), 71-109.
  * [23] M. A. Dritschel and S. McCullough, Failure of rational dilation on a triply connected domain, J. Amer. Math. Soc., 18 (2005), 873-918.
  * [24] M. A. Dritschel and J. Rovnyak, The operator Fejér-Riesz theorem, in: A Glimpse at Hilbert Space Operators: Paul R. Halmos in Memoriam, Oper. Theory Adv. Appl., 207, pages 223 - 254, Birkhäuser Verlag, Basel, 2010.
  * [25] M.A. Dritschel and B. Undrakh, Rational dilation problems associated with constrained algebras, J. Math. Anal. Appl. 467 (2018), 95-131.
  * [26] G.E. Dullerud and F. Paganini, A Course in Robust Control Theory: A Convex Approach, Texts in Applied Mathematics 36, Springer-Verlag, New York, 2000.
  * [27] E. Durszt, Contractions as shifts, Acta Sci. Math. (Szeged) 48, 129-134 (1985),
  * [28] S. Pal, The failure of rational dilation on the tetrablock, J. Functional Analysis 269 (2015), 1903–1924.
  * [29] S. Pal, Canonical decomposition of a tetrablock contraction and operator model, J. Math. Anal. Appl. 438 (2016), 274-284.
  * [30] S. Pal, Dilation, functional model and a complete unitary invariant for $C_{\cdot 0}$ $\Gamma_{n}$-contractions, arXiv:1708.06015v2 [math.FA] 7 Oct 2021.
  * [31] S. Parrott, Unitary dilations for commuting contractions, Pacific J. math.,34(1970), 481-490.
  * [32] H. Sau, A note on tetrablock contractions, New York J. Math. 21 (2015) 1347-1369.
  * [33] H. Sau, Andô dilations for a pair of commuting contractions: two explicit constructions and functional models, arXiv:1710.11368 [math.FA].
  * [34] B. Sz.-Nagy, C. Foias, H. Bercovici and L. Kerchy, Harmonic Analysis of Operators on Hilbert Space, Second edition, Revised and enlarged edition, Universitext, Springer, New York, 2010.
  * [35] B. Sz.-Nagy and C. Foias, On the structure of intertwining operators, Acta Sci. Math. (Szeged) 35 (1973), 225–254.
  * [36] J. von Neumann, Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann. 102 (1930), 49-131.
  * [37] N.Th. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operator theory,  J. Funct. Anal. 16 (1974) 83-100.
  * [38] H. Wold, A study in the analysis of stationary time series,, 2nd edition, Stockholm, 1954.