# Reducing subspaces of $C_{00}$ contractions

Chafiq Benhida  Laboratoire Paul Painlevé, Université Lille 1, 59 655
Villeneuve d’Ascq Cédex<EMAIL_ADDRESS>,  Emmanuel Fricain
Laboratoire Paul Painlevé, Université Lille 1, 59 655 Villeneuve d’Ascq Cédex
<EMAIL_ADDRESS>and  Dan Timotin  Simion Stoilov Institute of
Mathematics of the Romanian Academy, PO Box 1–764, Bucharest 014700, Romania
<EMAIL_ADDRESS>
###### Abstract.

Using the Sz.-Nagy–Foias theory of contractions, we obtain general results
about reducibility for a class of completely nonunitary contractions. These
are applied to certain truncated Toeplitz operators, previously considered by
Li–Yang–Lu and Gu. In particular, a negative answer is given to a conjecture
stated by the latter.

###### Key words and phrases:

Truncated Toeplitz operators, reducibility, model space

###### 2010 Mathematics Subject Classification:

47B35,47A15,47B38

This work was partially supported by Labex CEMPI (ANR-11-LABX-0007).

##  1\. Introduction

We will denote by $L^{2}$ the Lebesgue space $L^{2}(\mathbb{T},dm)$, where
$dm$ is normalized Lebesgue measure. The subspace of functions whose negative
Fourier coefficients are zero is denoted by $H^{2}$; it is identified with the
space of analytic functions in the unit disc with square summable Taylor
coefficients. An inner function is an element of $H^{2}$ whose values have
modulus 1 almost everywhere on $\mathbb{T}$.

If $\theta$ is an inner function, then the space
$K_{\theta}=H^{2}\ominus\theta H^{2}$ is usually called a model space; it has
been the focus of much research, in function theory in the unit disc as well
as in operator theory (see, for instance,[8, 6]; or [2] for a more recent
account). In particular, in the last two decades several papers discuss the
so-called _truncated Toeplitz operators_ , introduced in [9], which are
compressions to $K_{\theta}$ of multiplication operators on $L^{2}$.

Originating with work in [1], the question of reducibility of a certain class
of truncated Toeplitz operators has been recently investigated in papers by
Yi, Yang, and Lu [4, 5] and Gu [3]. Besides certain remarkable results, they
also contain intriguing questions that have not yet found their solution.

The current paper has several purposes. First, we put the problem of
reducibility of the truncated Toeplitz operators in a larger context, that of
the Sz.-Nagy–Foias theory of completely nonunitary contractions [7], and show
that some results in the above quoted papers may be generalized or given more
transparent proofs. Secondly, we answer in the negative a conjecture stated in
[3] and prove a statement that replaces it.

The plan of the paper is the following. After presenting in the next section
the elements of Sz.-Nagy theory that interest us, we obtain in Section 3 some
general results about reducibility for completely nonunitary contractions.
These results are applied in Section 4 to a certain class of truncated
Toeplitz operators. The connection to [5] is achieved in Section 5, while the
relation to [3] is the content of the last section.

##  2\. Sz.-Nagy–Foias dilation theory

The general reference for this section is the monograph [7], in particular
chapters I, II, and VI.

###  2.1. Minimal isometric dilation

If ${\mathcal{H}}$ is a Hilbert space and ${\mathcal{H}}_{1}$ is a closed
subspace, we will denote by $P_{{\mathcal{H}}_{1}}$ the orthogonal projection
onto ${\mathcal{H}}_{1}$.

A closed subspace $M$ of ${\mathcal{H}}$ is said to be _reducing_ for an
operator $T$ if both $M$ and $M^{\perp}$ are invariant with respect to $T$. A
_completely nonunitary contraction_ $T\in{\mathcal{L}}({\mathcal{H}})$ is a
linear operator that satisfies $\|T\|\leq 1$, and there is no reducing
subspace of $T$ on which it is unitary. The defect of $T$ is the operator
$D_{T}=(I-T^{*}T)^{1/2}$, and the defect space is
${\mathcal{D}}_{T}=\overline{D_{T}{\mathcal{H}}}$.

We write $T\in C_{\cdot 0}$ if $T^{*}{}^{n}$ tends strongly to $0$, and $T\in
C_{00}$ if $T$ and $T^{*}$ are in $C_{\cdot 0}$, that is $T^{n}$ and
$T^{*}{}^{n}$ both tend strongly to $0$. If $T\in C_{00}$, then it can be
shown that $\dim{\mathcal{D}}_{T}=\dim{\mathcal{D}}_{T^{*}}$. The subclass of
$C_{00}$ for which this dimension is finite and equal to $N$ is denoted by
$C_{0}(N)$. We will mostly be interested by contractions in the class
$C_{00}$.

An isometric dilation of $T$ is an isometric operator
$V\in{\mathcal{L}}({\mathcal{K}})$, with ${\mathcal{K}}\supset{\mathcal{H}}$,
such that $P_{\mathcal{H}}V^{n}|{\mathcal{H}}=T^{n}$ for any $n\in\mathbb{N}$.
Note that if $T=P_{\mathcal{H}}V|{\mathcal{H}}$ and $VH^{\perp}\subset
H^{\perp}$, then $V$ is a dilation. An isometric dilation
$V\in{\mathcal{L}}({\mathcal{K}})$ is called minimal if
${\mathcal{K}}=\bigvee_{n=0}^{\infty}V^{n}{\mathcal{H}}$. This is uniquely
defined, modulo a unitary isomorphism commuting with the dilations; in [7]
there is a precise description of its geometric structure. This becomes
simpler for contractions in $C_{\cdot 0}$; since this is the only case we are
interested in, we will describe the minimal isometric dilation in this case.

We will say that a subspace $X\subset{\mathcal{K}}$ is _wandering_ for $V$ if
$V^{n}X\perp V^{m}X$ for any $n\not=m$, and in this case we will denote
$M_{+}(X):=\bigoplus_{n=0}^{\infty}V^{n}X$. Note that $M_{+}(X)$ is invariant
with respect to $V$.

######  Lemma 2.1.

If $T$ is a completely nonunitary contraction and $V$ is its minimal isometric
dilation, then $T\in C_{\cdot 0}$ if and only if there exist wandering
subspaces $L,L_{*}\subset{\mathcal{K}}$ for $V$, with $\dim
L=\dim{\mathcal{D}}_{T}$ and $\dim L_{*}=\dim{\mathcal{D}}_{T^{*}}$, such that

(2.1) ${\mathcal{K}}=M_{+}(L_{*})={\mathcal{H}}\oplus M_{+}(L).$

In this case, the operators

(2.2) $\phi:D_{T}x\mapsto(V-T)x,\quad\phi_{*}:D_{T^{*}}x\mapsto x-VT^{*}x$

extend to unitary operators ${\mathcal{D}}_{T}\to L$ and
${\mathcal{D}}_{T^{*}}\to L_{*}$.

###  2.2. Analytic vector valued functions

If ${\mathcal{E}}$ is a Hilbert space, then $H^{2}({\mathcal{E}})$ is the
Hilbert space of ${\mathcal{E}}$-valued analytic functions in $\mathbb{D}$
with the norms of the Taylor coefficients square summable. As in the scalar
case, these functions have strong radial limits almost everywhere on
$\mathbb{T}$, and so may be identified with their boundary values, defined on
$\mathbb{T}$.

Denote by $\bm{T}^{\mathcal{E}}_{z}$ multiplication by $z$ acting on
$H^{2}({\mathcal{E}})$; it is an isometric operator. If
$\omega:{\mathcal{E}}\to{\mathcal{E}}^{\prime}$ is unitary, then the notation
$\tilde{\omega}$ will indicate the unique unitary extension
$\tilde{\omega}:H^{2}({\mathcal{E}})\to H^{2}({\mathcal{E}}^{\prime})$ such
that
$\tilde{\omega}\bm{T}^{\mathcal{E}}_{z}=\bm{T}^{{\mathcal{E}}^{\prime}}_{z}\tilde{\omega}$.

Suppose $X\subset{\mathcal{K}}$ is wandering for the isometry
$V\in{\mathcal{L}}({\mathcal{K}})$. Then the map $\mathfrak{F}_{X}$, defined
by

(2.3)
$\mathfrak{F}_{X}(\sum_{n=0}^{\infty}V^{n}x_{n})=\sum_{n=0}^{\infty}\lambda^{n}x_{n},$

is unitary from $M_{+}(X)$ to $H^{2}(X)$.

Another class of functions that we have to consider take as values operators
between two Hilbert spaces ${\mathcal{E}},{\mathcal{E}}_{*}$. More precisely,
we will be interested in _contractive analytic functions_ ; that is, functions
$\Theta:\mathbb{D}\to{\mathcal{L}}({\mathcal{E}},{\mathcal{E}}^{*})$, which
satisfy $\|\Theta(z)\|\leq 1$ for all $z\in\mathbb{D}$. As in the scalar case,
$\Theta$ has boundary values $\Theta(e^{it})$ almost everywhere on
$\mathbb{T}$.

A contractive analytic function is called _pure_ if $\|\Theta(0)x\|<\|x\|$ for
any $x\in{\mathcal{E}}$, $x\neq 0$. Any contractive analytic function admits a
decomposition in a direct sum $\Theta=\Theta_{p}\oplus\Theta_{u}$, where
$\Theta_{p}$ is pure and $\Theta_{u}$ is a constant unitary operator; then
$\Theta_{p}$ is called the _pure part_ of $\Theta$. A contractive analytic
function will be called _bi-inner_ if $\Theta(e^{it})$ is almost everywhere
unitary. (We prefer this shorter word rather than call them inner and
*-inner).

The appropriate equivalence relation for contractive analytic functions is
that of coincidence: two analytic functions
$\Theta:\mathbb{D}\to{\mathcal{L}}({\mathcal{E}},{\mathcal{E}}_{*})$,
$\Theta^{\prime}:\mathbb{D}\to{\mathcal{L}}({\mathcal{E}}^{\prime},{\mathcal{E}}^{\prime}_{*})$
are said to _coincide_ if there exist unitary operators
$\omega:{\mathcal{E}}\to{\mathcal{E}}^{\prime}$,
$\omega_{*}:{\mathcal{E}}_{*}^{\prime}\to{\mathcal{E}}_{*}^{\prime}$, such
that $\Theta^{\prime}(\lambda)\omega=\omega_{*}\Theta(\lambda)$ for all
$\lambda\in\mathbb{D}$.

###  2.3. Functional model and characteristic function

The model theory of Sz.-Nagy and Foias associates to any completely nonunitary
contraction $T$ a pure contractive analytic function $\Theta_{T}(z)$, with
values in ${\mathcal{L}}({\mathcal{D}}_{T},{\mathcal{D}}_{T^{*}})$, defined by
the formula

(2.4) $\Theta_{T}(z)=-T+zD_{T^{*}}(I-zT^{*})^{-1}D_{T}|{\mathcal{D}}_{T}.$

A functional model space and an associated model operator are constructed by
means of $\Theta_{T}$, and one can prove that $T$ is unitarily equivalent to
this model operator.

As we will be interested only in $C_{00}$ contractions, we will describe the
model only in this case, in which it takes a significantly simpler form. The
reason is that $T\in C_{00}$ is equivalent to $\Theta_{T}$ bi-inner. The
functional model associated to a bi-inner contractive analytic function
$\Theta:\mathbb{D}\to{\mathcal{L}}({\mathcal{E}},{\mathcal{E}}^{*})$ is
defined as follows: the _model space_ is

(2.5) ${\mathcal{H}}_{\Theta}=H^{2}({\mathcal{E}}_{*})\ominus\Theta
H^{2}({\mathcal{E}}),$

while the _model operator_ $\bm{S}_{\Theta}$ is the compression to
${\mathcal{H}}_{\Theta}$ of $\bm{T}_{z}^{{\mathcal{E}}_{*}}$. If $\Theta$ is
pure, then $\bm{T}_{z}^{{\mathcal{E}}_{*}}$ is precisely a minimal unitary
dilation of $\bm{S}_{\Theta}$.

Note that (2.5) shows that $\bm{S}_{\Theta}$ satisfies the assumptions of
Lemma 2.1 with $L=\Theta{\mathcal{E}}$, $L_{*}={\mathcal{E}}_{*}$. In
particular,

(2.6)
$\dim{\mathcal{D}}_{\bm{S}_{\Theta}}=\dim{\mathcal{E}},\qquad\dim{\mathcal{D}}_{\bm{S}^{*}_{\Theta}}=\dim{\mathcal{E}}_{*}.$

Suppose $\Theta:\mathbb{D}\to{\mathcal{L}}({\mathcal{E}},{\mathcal{E}}_{*})$
and
$\Theta^{\prime}:\mathbb{D}\to{\mathcal{L}}({\mathcal{E}}^{\prime},{\mathcal{E}}^{\prime}_{*})$
coincide, by means of the operators
$\omega:{\mathcal{E}}\to{\mathcal{E}}^{\prime}$,
$\omega_{*}:{\mathcal{E}}_{*}^{\prime}\to{\mathcal{E}}_{*}^{\prime}$. Then the
unitary $\tilde{\omega}_{*}:H^{2}({\mathcal{E}}_{*})\to
H^{2}({\mathcal{E}}^{\prime}_{*})$ satisfies
$\tilde{\omega}({\mathcal{H}}_{\Theta})={\mathcal{H}}_{\Theta^{\prime}}$ and

$\tilde{\omega}_{*}\bm{S}_{\Theta}=\bm{S}_{\Theta^{\prime}}\tilde{\omega}_{*}.$

Returning now to the contraction $T$ and its characteristic function, the next
lemma is a particular case of one of the basic results in [7, Chapter VI].

######  Lemma 2.2.

If $T\in C_{00}$, then the formula (2.4) defines a bi-inner pure analytic
function with values in
${\mathcal{L}}({\mathcal{D}}_{T},{\mathcal{D}}_{T^{*}})$, and $T$ is unitarily
equivalent to $\bm{S}_{\Theta_{T}}$. $\bm{S}_{\Theta_{T}}$ is called the
_functional model_ of $T$.

There is a relation between the functional model and the geometrical structure
of a minimal unitary dilation given by (2.1), as shown by the next result.

######  Lemma 2.3.

Suppose $T\in C_{00}$, $V\in{\mathcal{L}}({\mathcal{K}})$ is a minimal
isometric dilation of $T$, and $L,L_{*}$ are wandering subspaces for $V$
satisfying (2.1). Extend $\phi,\phi_{*}$ in (2.2) to unitary operators
$\tilde{\phi}:H^{2}({\mathcal{D}}_{T})\to H^{2}(L)$,
$\tilde{\phi}_{*}:H^{2}({\mathcal{D}}_{T^{*}})\to H^{2}(L_{*})$, and define
$\Omega=\mathfrak{F}_{L_{*}}^{*}\Phi_{*}$.

  * (i)

The map $\mathfrak{F}_{L_{*}}^{*}\Phi_{*}\Theta_{T}\Phi^{*}\mathfrak{F}_{L}$
is the inclusion of $M_{+}(L)$ into $M_{+}(L_{*})$.

  * (ii)

We have

(2.7)
$\begin{split}\Omega{\mathcal{H}}_{\Theta_{T}}&={\mathcal{H}},\quad\Omega({\mathcal{D}}_{T^{*}})=L_{*},\quad\Omega\Theta_{T}({\mathcal{D}}_{T})=L,\\\
\Omega\bm{T}_{z}^{{\mathcal{D}}_{T^{*}}}&=V\Omega,\quad\Omega\bm{S}_{\Theta_{T}}=T\Omega.\end{split}$

  * (iii)

If $\Theta=\phi_{*}\Theta_{T}\phi^{*}$ is written
$\Theta(\lambda)=\sum_{n=0}^{\infty}\lambda^{n}\Theta_{n}$ (with
$\Theta_{n}:L\to L_{*}$), then

(2.8) $\Theta_{n}=P_{L_{*}}(V^{*})^{n}J,$

where $J$ denotes the embedding of $L$ into $M_{+}(L_{*})$.

##  3\. Reducibility

In the sequel of the paper we will be interested by reducibility of certain
contractions. Fortunately, this can be easily characterized through
characteristic functions.

######  Lemma 3.1.

Suppose $T\in C_{00}$ has characteristic function
$\Theta_{T}:{\mathcal{D}}_{T}\to{\mathcal{D}}_{T^{*}}$. Then the following are
equivalent.

  * (i)

$T=T_{1}\oplus T_{2}$.

  * (ii)

There exist nontrivial orthogonal decompositions
${\mathcal{D}}_{T}=E^{1}\oplus E^{2}$, ${\mathcal{D}}_{T^{*}}=E^{1}_{*}\oplus
E^{2}_{*}$ which diagonalize $\Theta_{T}(\lambda)$ for all
$\lambda\in\mathbb{D}$; that is,

(3.1) $\Theta_{T}(\lambda)=\begin{pmatrix}\Theta_{1}(\lambda)&0\\\
0&\Theta_{2}(\lambda)\end{pmatrix}.$

In this case $\dim{\mathcal{D}}_{T_{i}}=\dim
E^{i}=\dim{\mathcal{D}}_{T_{i}^{*}}=\dim E^{i}_{*}$, and $\Theta_{T_{i}}$
coincides with $\Theta_{i}$.

###### Proof.

If $T=T_{1}\oplus T_{2}$, then
${\mathcal{D}}_{T}={\mathcal{D}}_{T_{1}}\oplus{\mathcal{D}}_{T_{2}}$,
${\mathcal{D}}_{T^{*}}={\mathcal{D}}_{T_{1}^{*}}\oplus{\mathcal{D}}_{T_{2}^{*}}$,
and formula (2.4) splits according to these decompositions into
$\Theta_{T}(\lambda)=\Theta_{T_{1}}(\lambda)\oplus\Theta_{T_{2}}(\lambda)$. So
(3.1) is valid, taking $E^{i}={\mathcal{D}}_{T_{i}}$,
$E^{i}_{*}={\mathcal{D}}_{T_{i}^{*}}$.

Conversely, if
$\Theta(\lambda):=\Theta_{T}(\lambda)=\Theta_{1}(\lambda)\oplus\Theta_{2}(\lambda)$,
then, according to (2.5),
${\mathcal{H}}_{\Theta}={\mathcal{H}}_{\Theta_{1}}\oplus{\mathcal{H}}_{\Theta_{2}}$,
and ${\mathcal{H}}_{\Theta_{1}},{\mathcal{H}}_{\Theta_{2}}$ are invariant with
respect to $\bm{S}_{\Theta}$. Since this last operator is unitarily equivalent
to $T$, $T$ is also reducible. Moreover,
$\bm{S}_{\Theta}|{\mathcal{H}}_{\Theta_{i}}$ is unitarily equivalent to
$\bm{S}_{\Theta_{i}}$, and the equality of the dimensions follows from (2.6).
∎

######  Corollary 3.2.

Suppose $T\in C_{00}$. Then $T$ is reducible if and only if there exist
nontrivial subspaces $E\subset{\mathcal{D}}_{T}$,
$E_{*}\subset{\mathcal{D}}_{T^{*}}$, such that $\Theta_{T}(e^{it})E=E_{*}$ for
almost all $t$.

###### Proof.

If nontrivial subspaces as assumed exist, then, since $\Theta_{T}(e^{it})$ is
unitary almost everywhere, we also have
$\Theta_{T}(e^{it})E^{\perp}=E_{*}^{\perp}$ for almost all $t$. The
decompositions ${\mathcal{D}}_{T}=E\oplus E^{\perp}$,
${\mathcal{D}}_{T^{*}}=E_{*}\oplus E_{*}^{\perp}$ satisfy then (3.1). ∎

The following is a geometrical reformulation of Corollary 3.2 in terms of the
spaces $L,L_{*}$ appearing in an arbitrary minimal isometric dilation of $T$.

######  Corollary 3.3.

Suppose $T\in C_{00}$ and $V\in{\mathcal{L}}({\mathcal{K}})$ is a minimal
dilation of $T$, such that (2.1) is valid for $L,L_{*}$ wandering subspaces
for $V$. Let $d$ be a finite positive integer or $\infty$. Then:

  * (i)

If $T$ has a nontrivial reducing subspace such that the restriction has
$d$-dimensional defects, then there exist nontrivial subspaces $L^{1}\subset
L$, $L^{1}_{*}\subset L_{*}$, both of dimension $d$, such that

(3.2) $L^{1}\subset M_{+}(L^{1}_{*}).$

  * (ii)

The converse also holds if $d<\infty$.

###### Proof.

(i). Suppose $T$ has a reducing subspace with defect of dimension $d$. We
apply Lemma 3.1, which gives decomposition (3.1), where
$\Theta_{i}(\lambda):E^{i}\to E^{i}_{*}$, and $\dim E^{1}=\dim E^{1}_{*}=d$.
So $\Theta_{1}H^{2}(E^{1})\subset H^{2}(E^{1}_{*})$; in particular, if we look
at $E^{1}$ as the constant functions in $H^{2}(E^{1})$, we have

(3.3) $\Theta_{1}E^{1}\subset H^{2}(E^{1}_{*}).$

Denote then $L^{1}=\phi E^{1}$ and $L^{1}_{*}=\phi_{*}E^{1}_{*}$
($\phi,\phi_{*}$ in (2.2)). We consider the unitary operator $\Omega$ from
Lemma 2.3. Formulas (2.7) yield also $\Omega E^{1}_{*}=L^{1}_{*}$,
$\Omega\Theta_{1}(E^{1})=L^{1}$, and
$\Omega(H^{2}(E^{1}_{*}))=M_{+}(L^{1}_{*})$. Therefore (3.3) implies (3.2).

(ii) Conversely, suppose we have the required spaces satisfying (3.2);
therefore $M_{+}(L^{1})\subset M_{+}(L^{1}_{*})$. Define
$\Theta^{\prime}(\lambda)=\phi_{*}\Theta_{T}(\lambda)\phi^{*}:L\to L_{*}$. By
using $\mathfrak{F}_{L_{*}}$, we obtain $\Theta^{\prime}H^{2}(L^{1})\subset
H^{2}(L^{1}_{*})$, which means that $\Theta^{\prime}(e^{it})L^{1}\subset
L^{1}_{*}$ almost everywhere. Since $\dim L^{1}=\dim L^{1}_{*}=d<\infty$, we
have in fact $\Theta^{\prime}(e^{it})L^{1}=L^{1}_{*}$ almost everywhere. As in
the proof of Corollary 3.2, it follows that
$\Theta^{\prime}(e^{it})L^{1}{}^{\perp}\subset L^{1}_{*}{}^{\perp}$ almost
everywhere, whence we may obtain a decomposition similar to (3.1). This
implies the reducibility of $\Theta^{\prime}$, and thus the reducibility of
$\Theta_{T}$ and of $T$. ∎

In particular, we obtain a nice result if we consider reducing subspaces with
defects of dimension 1.

######  Corollary 3.4.

An operator $T\in C_{00}$ has a reducing subspace with defects of dimension 1
if and only if there exists $y\in L$, $y_{*}\in L_{*}$, $y,y^{\prime}\neq 0$,
and a scalar inner function $u$, such that $y=u(V)y_{*}$. In this case the
characteristic function of the reduced operator is precisely $u$.

###### Proof.

By Corollary 3.3 applied to $d=1$, the existence of a reducing subspace with
defects of dimension 1 is equivalent to the existence of elements of norm 1
$y\in L,y_{*}\in L_{*}$, such that $y\in M_{+}(y_{*})$. The Fourier
representation $\mathfrak{F}_{y_{*}}$ maps $M_{+}(y_{*})$ onto $H^{2}$; more
precisely, from (2.3) it follows that $\mathfrak{F}_{y_{*}}(f(V)y_{*})=f$. In
particular, $y$ is a wandering vector for $V$, which implies that
$u:=\mathfrak{F}_{y_{*}}y$ is an inner function.

If we denote by ${\mathcal{H}}_{1}$ the reducing subspace of dimension 1
obtained, then have ${\mathcal{H}}_{1}=M^{+}(y_{*})\ominus M^{+}(y)$. Through
the Fourier representation $\mathfrak{F}_{y_{*}}$, this becomes $H^{2}\ominus
uH^{2}$. By comparing with the general formula for the functional model, we
see that the characteristic function of the reduced operator is $u$. ∎

######  Remark 3.5.

Part of the results in this section may be extended to more general
contractions. Thus Lemma 3.1 is true for a general completely nonunitary
contraction; we have then to use in the proof the more complicated general
form of the functional model associated to $T$. Appropriately modified
versions of Corollaries 3.2 and 3.3 can also be stated. However, since the
statements are less neat, we have preferred to restrict ourselves to the case
$T\in C_{00}$, which will be used in the applications in the sequel of the
paper.

##  4\. A class of contractions

In the rest of the paper we will work in the Hardy space $H^{2}$, applying the
above results to a particular class of contractions. By $T_{\varphi}$ we will
denote the usual Toeplitz operator on $H^{2}$, that is, the compression of the
operator of multiplication by $\varphi$ on $H^{2}$. Recall here that, for a
scalar inner function $K_{\theta}=H^{2}\ominus\theta
H^{2}={\mathcal{H}}_{\theta}$ (see (2.5) with
${\mathcal{E}}={\mathcal{E}}_{*}=\mathbb{C}$).

Let then $\theta,B$ be two scalar inner functions that satisfy the basic
assumption

(4.1) $\ker T_{\theta\overline{B}}=\\{0\\}.$

Note that $f\in\ker T_{\theta\overline{B}}$ if and only if $\theta f\in\ker
T_{\overline{B}}=K_{B}$, whence (4.1) is equivalent to $\theta H^{2}\cap
K_{B}=\\{0\\}$.

We will consider the operator $A^{\theta}_{B}\in{\mathcal{L}}(K_{\theta})$,
defined by

(4.2) $A^{\theta}_{B}=P_{K_{\theta}}T_{B}|K_{\theta}.$

The operator $A^{\theta}_{B}$ is usually called the _truncated Toeplitz
operator_ on $K_{\theta}$ with symbol $B$. It is known [9] that truncated
Toeplitz operators are _complex symmetric_ ; that is, there exist a complex
conjugation $C_{\theta}$ on $K_{\theta}$ such that

(4.3) $(A^{\theta}_{B})^{*}=C_{\theta}A^{\theta}_{B}C_{\theta}.$

The next theorem identifies concretely a minimal isometric dilation of
$A^{\theta}_{B}$; it is a generalization of [5, Lemma 3.1].

######  Theorem 4.1.

Let $B$ and $\theta$ two inner functions satisfying (4.1). The operator
$T_{B}\in{\mathcal{L}}(H^{2})$ is a minimal isometric dilation of
$A^{\theta}_{B}$. For this minimal isometric dilation we have

(4.4) $L=\theta K_{B},\quad L_{*}=K_{B}.$

###### Proof.

$T_{B}$ is an isometry on $H^{2}$, and $T_{B}(K_{\theta}^{\perp})=T_{B}(\theta
H^{2})\subset\theta H^{2}=K_{\theta}^{\perp}$. Thus it follows from (4.2) that
$T_{B}$ is a dilation of $A^{\theta}_{B}$.

We show now by induction according to $n$ that

(4.5) $K_{\theta}+BK_{\theta}+\dots+B^{n}K_{\theta}=K_{B^{n}\theta}.$

Equality (4.5) is obviously true for $n=0$. Suppose that it is true up to
$n-1$. We are left then to prove that

(4.6) $K_{B^{n-1}\theta}+B^{n}K_{\theta}=K_{B^{n}\theta}.$

It is immediate from the definitions that the left hand side is contained in
the right hand side. On the other hand, we have

$K_{B^{n}\theta}=K_{B^{n-1}\theta}\oplus B^{n-1}\theta
K_{B}=B^{n}K_{\theta}\oplus K_{B^{n}}.$

If $f\in K_{B^{n}\theta}$ is orthogonal to $K_{B^{n-1}\theta}$ as well as to
$B^{n}K_{\theta}$, it follows that

$f\in(\theta B^{n-1}K_{B})\cap K_{B^{n}}.$

So $f=\theta B^{n-1}g$ with $g\in K_{B}$; and also $f\perp B^{n}H^{2}$, which
means $\theta g\perp BH^{2}$, or $\theta g\in K_{B}$. It follows that
$0=T_{\overline{B}}(\theta g)=T_{\theta\overline{B}}g$. By (4.1), this implies
$g=0$, whence $f=0$.

Since

$\big{(}\bigvee_{n}K_{B^{n}\theta}\big{)}^{\perp}=\bigcap_{n}B^{n}\theta
H^{2}=\\{0\\},$

it follows that

(4.7) $H^{2}=\bigvee_{n=0}^{\infty}T_{B}^{n}K_{\theta}.$

Therefore $T_{B}$ is a _minimal_ isometric dilation of $A_{B}^{\theta}$.

Then

(4.8)
$H^{2}=\bigoplus_{n=0}^{\infty}B^{n}K_{B}=\bigoplus_{n=0}^{\infty}T_{B}^{n}K_{B}=M_{+}(K_{B}),$

whence $L_{*}=K_{B}$.

On the other hand, we have

(4.9) $K_{B\theta}=K_{\theta}\oplus\theta K_{B}=K_{B}\oplus BK_{\theta},$

Therefore

(4.10) $H^{2}=\bigoplus_{n=0}^{\infty}B^{n}K_{B}=K_{\theta}\oplus\theta
H^{2}=K_{\theta}\oplus\bigoplus_{n=0}^{\infty}T_{B}^{n}\theta
K_{B}=K_{\theta}\oplus M_{+}(\theta K_{B}),$

whence $L=\theta K_{B}$. ∎

######  Corollary 4.2.

With the above assumptions, $A_{B}^{\theta}$ is in $C_{00}$.

###### Proof.

In view of equation (4.10), it follows from Lemma 2.1 that $A_{B}^{\theta}$ is
in $C_{\cdot 0}$. On the other hand, it follows from (4.3) that

$((A^{\theta}_{B})^{*})^{n}=C_{\theta}(A^{\theta}_{B})^{n}C_{\theta},$

whence $A_{B}^{\theta}$ is also in $C_{0\cdot}$. ∎

Using the identification of a minimal unitary dilation in Theorem 4.1 we may
compute the characteristic function of $A^{\theta}_{B}$. The next theorem
generalizes [3, Theorem 2.4] (see Section 6 below).

######  Theorem 4.3.

Let $B$ and $\theta$ two inner functions satisfying (4.1). The characteristic
function of $A^{\theta}_{B}$ is $\Phi:\mathbb{D}\to{\mathcal{L}}(K_{B})$
defined by

(4.11) $\Phi(\lambda)=A^{B}_{\frac{\theta}{1-\lambda\overline{B}}}.$

###### Proof.

We have identified in Theorem 4.1 $L,L_{*}$ with $\theta K_{B},K_{B}$
respectively. We intend to apply Lemma 2.3 (iii). Since we want to consider
the characteristic function of $A$ as an analytic function with values in
${\mathcal{L}}(K_{B})$, the embedding $J$ is precisely multiplication by
$\theta$. Then, if $\Phi(\lambda)=\sum_{n=0}^{\infty}\lambda^{n}\Phi_{n}$,
(2.8) yields

$\Phi_{n}f=P_{K_{B}}\overline{B}^{n}\theta f$

for $f\in K_{B}$. Thus $\Phi_{n}=A^{B}_{\theta\overline{B}^{n}}$. Therefore

$\Phi(\lambda)=\sum_{n=0}^{\infty}\lambda^{n}A^{B}_{\theta\overline{B}^{n}}=A^{B}_{\theta\sum_{n=0}^{\infty}\lambda^{n}\overline{B}^{n}}=A^{B}_{\frac{\theta}{1-\lambda\overline{B}}}.\qed$

We may also obtain a more precise form of Corollary 3.4.

######  Corollary 4.4.

Let $B$ and $\theta$ two inner functions satisfying (4.1). Then the following
assertions are equivalent:

  * (i)

The operator $A^{\theta}_{B}$ has a reducing subspace such that the
restriction has one-dimensional defects.

  * (ii)

There exist $u$ inner and $h_{1},h_{2}\in K_{B}$, $h_{1},h_{2}\neq 0$, such
that

(4.12) $\theta=\frac{h_{2}}{h_{1}}(u\circ B),$

  * (iii)

There exist $u,v_{1},v_{2}$ inner, with

$\ker T_{v_{1}\bar{B}}\cap\ker T_{v_{2}\bar{B}}\not=\\{0\\},$

such that

(4.13) $\theta=\frac{v_{2}}{v_{1}}(u\circ B).$

###### Proof.

The equivalence of (i) and (ii) follows by applying in this case Corollary
3.4. We have $L_{*}=K_{B}$, $L=\theta K_{B}$, and so the existence of the
required reducing subspace is equivalent to the existence of $h_{1},h_{2}\in
K_{B}$, $h_{1},h_{2}\neq 0$ and an inner function $u$, such that $\theta
h_{1}=u(V)h_{2}$. Since $V=T_{B}$, $u(V)$ is multiplication by $u\circ B$, and
we have

(4.14) $\theta h_{1}=h_{2}(u\circ B).$

If (ii) is true, then we must have $h_{i}=v_{i}g$ for some inner functions
$v_{1},v_{2}$ and $g$ outer, so (4.13) is true. Note that, if $v$ is an
analytic and bounded function in $\mathbb{D}$, then

(4.15) $vh\in K_{B}\Leftrightarrow h\in\ker T_{v\bar{B}}.$

So $v_{1}g,v_{2}g\in K_{B}$ is equivalent to $g\in\ker
T_{v_{1}\bar{B}}\cap\ker T_{v_{2}\bar{B}}$.

The implication (iii)$\implies$(ii) follows easily by reversing the steps. ∎

Note that the function $u$ in (ii) and (iii) of the previous corollary is non
constant because otherwise $\theta h_{1}\in K_{B}$, and thus $h_{1}\in\ker
T_{\theta\overline{B}}$ which contradicts hypothesis (4.1).

##  5\. A particular case

Let us consider now the particular case when $B$ is a finite Blaschke product.
Denote $\phi_{\alpha}(z)=(z-\alpha)/(1-\bar{\alpha}z)$. If $B$ has roots
(counting with multiplicities) $w_{1},\dots,w_{k}$, it is known that

(5.1) $K_{B}=\Big{\\{}\frac{p(z)}{\prod_{i=1}^{k}(1-\bar{w_{i}}z)}:p\text{
polynomial of degree }\leq k-1\Big{\\}}.$

In this case condition (4.1) has a simple equivalent form.

######  Lemma 5.1.

If $B$ is a finite Blaschke product, then (4.1) is satisfied if and only if

(5.2) $\dim K_{B}\leq\dim K_{\theta}.$

###### Proof.

Indeed, first assume that (5.2) is satisfied, and let $f\in\ker
T_{\theta\overline{B}}$. Then $\theta f\in\ker T_{\overline{B}}=K_{B}$, whence
$f=T_{\overline{\theta}}(\theta f)\in T_{\overline{\theta}}K_{B}\subset
K_{B}$. If $f\not\equiv 0$, then $\theta=\frac{\theta f}{f}$ is a quotient of
two polynomials of degree at most $\mathrm{deg}B-1$, which contradicts
assumption (5.2).

Suppose now that $\dim K_{B}>\dim K_{\theta}$. Then $\theta H^{2}$ has finite
codimension in $H^{2}$ strictly smaller than $\dim K_{B}$, whence $\theta
H^{2}\cap K_{B}\not=\\{0\\}$. Applying (4.15) in case $v=\theta$, it follows
that $\ker T_{\theta\overline{B}}\not=\\{0\\}$, contradicting (4.1). ∎

Condition (5.2) is precisely the one considered in [3] and [5]. To discuss
this case, we need one more elementary lemma.

######  Lemma 5.2.

Suppose $h_{1},h_{2}$ are two polynomials of degree at most $k-1$ and

(5.3) $|h_{1}|=|h_{2}|\text{ a.e. on }\mathbb{T}.$

Then,

$\frac{h_{2}}{h_{1}}=\frac{B_{2}}{B_{1}},$

where $B_{i}$ are Blaschke products with $\deg B_{1}+\deg B_{2}\leq k-1$.

###### Proof.

First, a general remark. Suppose that $h$ is a polynomial and write
$h(z)=z^{p}g(z)$, with $p\in\mathbb{N}\cup\\{0\\}$ and $g(0)\not=0$. Denote
the roots (counting with multiplicities) of $g$ by
$\alpha_{1},\dots,\alpha_{\ell}$. Then, $h^{o}$, the outer part of $h$, is a
polynomial of degree $\deg g$, which has roots $Z_{o}(h)\cup Z_{i}(h)$, where
$Z_{o}(h):=\\{\alpha_{i}:|\alpha_{i}|\geq 1\\}$ and
$Z_{i}(h):=\\{1/\bar{\alpha}_{i}:0<|\alpha_{i}|<1\\}$.

We may assume that $h_{1},h_{2}$ have no common roots (otherwise we cancel
them). It also follows then that $h_{1}$ and $h_{2}$ have no roots on
$\mathbb{T}$ (since this would be a common root by (5.3) ). Also, only one of
them may have the root 0; suppose it is $h_{1}$, and write, as above,
$h_{1}(z)=z^{p}g_{1}(z)$, with $g_{1}(0)\not=0$.

Assumption (5.3) implies that the outer parts of $g_{1}$ and $h_{2}$ coincide.
Since $g_{1}$ and $h_{2}$ have no common roots, but $g_{1}{}^{o}=h_{2}^{o}$,
we must have $Z_{o}(g_{1})=Z_{i}(h_{2})$ and $Z_{i}(g_{1})=Z_{o}(h_{2})$. Then
we can write $h_{2}/h_{1}=B_{2}/B_{1}$, with

$B_{1}=z^{p}\prod_{\alpha_{i}\in Z^{\sharp}_{i}(h_{1})}\phi_{\alpha_{i}},\quad
B_{2}=\prod_{\alpha_{i}\in Z^{\sharp}_{i}(h_{2})}\phi_{\alpha_{i}},$

where
$Z^{\sharp}_{i}(p)=\\{\alpha_{i}:p(\alpha_{i})=0,0<|\alpha_{i}|<1\\}=\\{1/\bar{\alpha}_{i}:\alpha_{i}\in
Z_{i}(p)\\}$. Since we have

$\deg B_{1}+\deg
B_{2}=p+|Z_{i}(g_{1})|+|Z_{i}(h_{2})|=p+|Z_{i}(g_{1})|+|Z_{o}(g_{1})|\leq
k-1,$

the lemma is proved. ∎

The next theorem generalizes [5, Theorem 1.4].

######  Theorem 5.3.

Suppose $B$ is a finite Blaschke product, while $\theta$ is an inner function
with $\deg\theta\geq\deg B$. Then the operator $A^{\theta}_{B}$ has a reducing
subspace such that the restriction has one-dimensional defects if and only if

(5.4) $\theta=\frac{B_{2}}{B_{1}}(u\circ B),$

where $u$ is a non constant inner function, while $B_{1},B_{2}$ are finite
Blaschke products with $\deg B_{1}+\deg B_{2}\leq\deg B-1$.

###### Proof.

We apply to this case Corollary 4.4 (ii). The existence of the required
reducing subspace is then equivalent to the existence of $h_{1},h_{2}\in
K_{B}$ and an inner function $u$, such that

(5.5) $\theta h_{1}=h_{2}(u\circ B).$

By (5.1), it is equivalent to assume in this equality that $h_{i}$ are
polynomials of degree $\leq k-1$, where $k=\deg B$. Taking absolute values, we
obtain, since $\theta$ and $u\circ B$ are inner, that $|h_{1}|=|h_{2}|$ on
$\mathbb{T}$. We may then apply Lemma 5.2 to obtain the desired formula (5.4).

The converse is immediate, since (5.4) implies (5.5), with the degrees of
$h_{1}$ and $h_{2}$ at most $k-1$. If we further write
$g_{i}(z)=\frac{h_{i}(z)}{\prod_{i=1}^{k}(1-\bar{w_{i}}z)}$, we obtain

$\theta g_{1}=g_{2}(u\circ B).$

Since $g_{i}\in K_{B}$, this is equivalent, by Corollary 4.4, to the existence
of the required reducing subspace. ∎

The condition becomes simpler if $\theta$ is singular.

######  Theorem 5.4.

Let $\theta$ be a singular inner function and let $B$ be a finite Blaschke
product. Then the operator $A^{\theta}_{B}$ has a reducing subspace such that
the restriction has one-dimensional defects if and only if

(5.6) $\theta=S\circ B,$

for some singular inner function $S$.

###### Proof.

According to Theorem 5.3, it is sufficient to prove that (5.6) and (5.4) are
equivalent. The implication $\eqref{eq3:main one-
dim}\Longrightarrow\eqref{eq2:main one-dim}$ is clear. Assume now that (5.4)
is satisfied, that is we can write

$B_{1}\theta=B_{2}(u\circ B),$

where $B_{1}$ and $B_{2}$ are finite Blaschke products with $\deg B_{1}+\deg
B_{2}\leq N-1$ and $N=\deg B$.

Since $\theta$ is singular, $B_{2}$ must be a factor of $B_{1}$ and may be
canceled. So we may assume $B_{2}=1$, or $B_{1}\theta=u\circ B$, where $\deg
B_{1}\leq N-1$.

Write then $u=B_{3}S$, where $B_{3}$ is a Blaschke product and $S$ is the
singular part of $u$. Thus we have

$B_{1}\theta=(B_{3}\circ B)(S\circ B).$

We have $\deg(B_{3}\circ B)=\deg B_{3}\deg B$; so, if $B_{3}$ is not constant,
then

$\deg(B_{3}\circ B)\geq\deg B=N>\deg B_{1}.$

The contradiction obtained implies that $B_{3}$ is constant, and so

$B_{1}\theta=S\circ B.$

Since the right hand side is singular, it follows that $B_{1}$ is constant,
which proves the theorem. ∎

##  6\. The case $B(z)=z^{N}$

The case $B(z)=z^{N}$ is investigated at length in [3]. In particular, the
characteristic function of $A^{\theta}_{z^{N}}$ is computed; let us show how
Gu’s result follows from Theorem 4.3 above.

We use the canonical basis of $K_{B}$ formed by $1,z,\dots z^{N-1}$. To obtain
the matrix of $A^{B}_{\frac{\theta}{1-\lambda\overline{B}}}$, consider first
$A^{B}_{\frac{z^{n}}{1-\lambda\overline{B}}}$. We have

$\frac{z^{n}}{1-\lambda\overline{B}}=\sum_{j=0}^{\infty}\lambda^{j}z^{n-jN}=\sum_{j=0}^{\infty}\lambda^{j}z^{N(n^{\prime}-j)+m},$

where $n=Nn^{\prime}+m$, with $0\leq m\leq N-1$. Since $A^{B}_{z^{p}}$ is
nonzero only for $-(N-1)\leq p\leq N-1$, we have to consider in the above sum
only two terms, corresponding to $j=n^{\prime}$ and $j=n^{\prime}+1$. Thus

$A^{B}_{\frac{z^{n}}{1-\lambda\overline{B}}}=A^{B}_{\lambda^{n^{\prime}}z^{m}+\lambda^{n^{\prime}+1}z^{m-N}}.$

Its matrix with respect to the canonical basis is

(6.1)
$A^{B}_{\lambda^{n^{\prime}}z^{m}+\lambda^{n^{\prime}+1}z^{m-N}}=\begin{pmatrix}\ddots&\ddots&\lambda^{n^{\prime}+1}&\ddots&\ddots\\\
\ddots&\ddots&\ddots&\lambda^{n^{\prime}+1}&\ddots\\\
\lambda^{n^{\prime}}&\ddots&\ddots&\ddots&\ddots\\\
\ddots&\lambda^{n^{\prime}}&\ddots&\ddots&\ddots\\\
\ddots&\ddots&\ddots&\ddots&\ddots\end{pmatrix},$

with two nonzero constant diagonals (one in case $m=0$), corresponding to
entries $a_{ij}$ with $i-j=m$ or $i-j=m-N$.

Therefore, if we decompose

$\theta(z)=\theta_{0}(z^{N})+z\theta_{1}(z^{N})+\dots+z^{N-1}\theta_{N-1}(z^{N}),$

then

(6.2)
$A^{B}_{\frac{\theta}{1-\lambda\overline{B}}}=\begin{pmatrix}\theta_{0}(\lambda)&\lambda\theta_{N-1}(\lambda)&\lambda\theta_{N-2}(\lambda)&\dots&\lambda\theta_{1}(\lambda)\\\
\theta_{1}(\lambda)&\theta_{0}(\lambda)&\lambda\theta_{N-1}(\lambda)&\dots&\lambda\theta_{2}(\lambda)\\\
\theta_{2}(\lambda)&\theta_{1}(\lambda)&\theta_{0}(\lambda)&\dots&\lambda\theta_{3}(\lambda)\\\
\ddots&\ddots&\ddots&\ddots&\ddots\\\
\theta_{N-1}(\lambda)&\theta_{N-2}(\lambda)&\theta_{N-3}(\lambda)&\dots&\theta_{0}(\lambda)\end{pmatrix}.$

This is precisely the form given by [3, Theorem 2.4].

In the sequel we will solve a conjecture about $A^{\theta}_{z^{N}}$ left open
in [3]. This appears as Conjecture 3.5 therein, and has the following
statement.

######  Conjecture 6.1.

Suppose $B(z)=z^{N}$. Then the following are equivalent:

  * (i)

$A_{B}^{\theta}$ has a reducing subspace such that the restriction has one-
dimensional defects.

  * (ii)

$\theta(z)=b(z)u(z^{N})$ for some inner function $u$, while either $b\equiv 1$
or

(6.3) $b(z)=\prod_{i=1}^{l}\psi_{\alpha_{i},J_{i}},$

where $l\leq N-1$, $J_{i}\subset\\{0,\dots,N-1\\}$, and $\psi_{\alpha,J}$ is
defined by

(6.4) $\psi_{\alpha,J}(z)=\prod_{i\in J}\phi_{\omega^{i}\alpha}(z).$

[3, Theorem 3.4] shows that (i)$\implies$(ii), while (ii)$\implies$(i) is
proved only for $N=3$ in [3, Section 5].

######  Theorem 6.2.

Conjecture 6.1 is false for $N=4$.

###### Proof.

Take two different nonzero values $\alpha,\beta\in\mathbb{D}$, and define

$\theta(z)=\frac{(z^{2}-\alpha^{2})(z^{2}-\beta^{2})}{(1-\bar{\alpha}^{2}z^{2})(1-\bar{\beta}^{2}z^{2})}.$

We have then

$\theta(z)=\psi_{\alpha,J}\psi_{\beta,J}$

with $J=\\{0,2\\}\subset\\{0,1,2,3\\}$, so $\theta$ satisfies condition (ii)
of Conjecture 6.1.

On the other hand, if $\theta$ would satisfy condition (i), it would follow by
Theorem 5.3 that one should have

(6.5) $B_{2}(z)\theta(z)=B_{1}(z)u(z^{4}),$

with $u$ inner and $B_{1},B_{2}$ finite Blaschke products with $\deg
B_{1}+\deg B_{2}\leq 3$.

Obviously $u$ has also to be a finite Blaschke product. Equating the degrees
in both sides yields

$\deg B_{1}+4=\deg B_{2}+4\deg u.$

First, $\deg B_{1}=3$ would imply $\deg B_{2}=0$, so $7=4\deg u$: a
contradiction. So the degree of the left hand side of (6.5) is between 4 and
6, which implies $\deg u=1$. Again equating the degrees yields $\deg
B_{1}=\deg B_{2}=0\ \text{or}\ 1$.

Now $u(z^{4})$ has either the root 0 of multiplicity 4, or four distinct
roots. Both possibilities are incompatible with the fact that the left hand
side of (6.5) has either 2 or three roots. We have obtained the desired
contradiction, so $\theta$ does not satisfy (i) of Conjecture 6.1. ∎

In fact, we may replace Conjecture 6.1 with a precise result. We will need the
next lemma, also proved in [3, Theorem 3.4].

######  Lemma 6.3.

If $\alpha\in\mathbb{D}$, then

$\phi_{\alpha^{N}}(z^{N})=\prod_{i=0}^{N-1}\phi_{\omega^{i}\alpha}(z).$

######  Theorem 6.4.

Suppose $B(z)=z^{N}$. Then the following are equivalent:

  * (i)

$A_{B}^{\theta}$ has a reducing subspace such that the restriction has one-
dimensional defects.

  * (ii)

$\theta(z)=b(z)u(z^{N})$ for some inner function $u$, while $b$ is either 1 or
a finite Blaschke product given by (6.3), where $l\leq N-1$,
$J_{i}\subset\\{0,\dots,N-1\\}$, $\psi_{\alpha,J}$ are defined by (6.4), and,
moreover,

(6.6) $\sum\limits_{i=1}^{l}\min\\{|J_{i}|,N-|J_{i}|\\}\leq N-1.$

###### Proof.

(i)$\implies$(ii). From Theorem 5.3 we know that $\theta$ is given by (5.4),
where $B_{1}$ and $B_{2}$ have no common roots. We may denote the roots of
$B_{1}$ (counting multiplicities) by

$\\{\alpha_{1}^{1},\dots,\alpha^{1}_{s_{1}};\alpha^{2}_{1},\dots,\alpha^{2}_{s_{2}};\dots;\alpha^{p}_{1},\dots,\alpha^{p}_{s_{p}}\\},$

where, for each $i=1,\dots,p$, the values
$\alpha^{i}_{1},\dots,\alpha^{i}_{s_{i}}$ are all distinct, and

$(\alpha^{i}_{1})^{N}=\dots=(\alpha^{i}_{s_{i}})^{N}.$

Similarly, we denote the roots of $B_{2}$ by

$\\{\beta_{1}^{1},\dots,\beta^{1}_{r_{1}};\beta^{2}_{1},\dots,\beta^{2}_{r_{2}};\dots;\beta^{q}_{1},\dots,\beta^{q}_{r_{q}}\\},$

where, for each $i=1,\dots,q$, the values
$\beta^{i}_{1},\dots,\beta^{i}_{r_{i}}$ are all distinct, and

$(\beta^{i}_{1})^{N}=\dots=(\beta^{i}_{s_{i}})^{N}.$

Note that the condition $\deg B_{1}+\deg B_{2}\leq N-1$ is transcribed as

(6.7) $s_{1}+\dots+s_{p}+r_{1}+\dots+r_{q}\leq N-1.$

In particular, $p+q\leq N-1$.

Now, it is easy to see that, for each $i=1,\dots,q$, the Blaschke product

$\phi_{\beta^{i}_{1}}\dots\phi_{\beta^{i}_{r_{i}}}$

is equal to $\psi_{\beta^{i}_{1},J_{i}}$ for some
$J_{i}\subset\\{0,\dots,N-1\\}$. So

(6.8) $B_{2}=\prod_{i=1}^{q}\psi_{\beta^{i}_{1},J_{i}}.$

The matter is more subtle as concerns $B_{1}$: it appears at the denominator,
which we do not want. We have, similarly,

(6.9) $B_{1}=\prod_{i=1}^{p}\psi_{\alpha^{i}_{1},J^{\prime}_{i}}$

for some $J^{\prime}_{i}\subset\\{0,\dots,N-1\\}$.

The factor $\phi_{\alpha^{1}_{1}}(z)$ must be canceled by a factor in
$u(z^{N})$, so $\alpha^{1}_{1}$ must be a root of $u(z^{N})$. But then
$u(z^{N})$ must also have the roots $\omega^{j}\alpha^{1}_{1}$ for
$j=1,\dots,N-1$, and so

$u(z^{N})=\prod_{j=0}^{N-1}\phi_{\omega^{j}\alpha^{1}_{1}}(z)u_{1}(z^{N}).$

Since

$\frac{\phi_{\omega^{j}\alpha^{1}_{1}}(z)}{\psi_{\alpha^{1}_{1},J^{\prime}_{1}}}=\psi_{\alpha^{1}_{1},J^{\prime\prime}_{1}}$

with $J^{\prime\prime}_{1}=\\{0,\dots,N-1\\}\setminus J^{\prime}_{1}$, we have

$\frac{u(z^{N})}{\psi_{\alpha^{1}_{1},J^{\prime}_{1}}}=\psi_{\alpha^{1}_{1},J^{\prime\prime}_{1}}u_{1}(z^{N}).$

We may continue the argument (or use an appropriate induction) to obtain

(6.10)
$\frac{u(z^{N})}{B_{1}(z)}=\prod_{i=1}^{p}\psi_{\alpha^{i}_{1},J^{\prime\prime}_{i}}u^{\prime}(z^{N})$

for an inner function $u^{\prime}$, where
$J^{\prime\prime}_{i}=\\{0,\dots,N-1\\}\setminus J^{\prime}_{i}$. From (5.4),
(6.8), and (6.10) it follows that

$\theta(z)=\prod_{i=1}^{q}\psi_{\beta^{i}_{1},J_{i}}\prod_{i=1}^{p}\psi_{\alpha^{i}_{1},J^{\prime\prime}_{i}}\
u^{\prime}(z^{N}).$

This is exactly the form given by (6.3). Moreover
$\min\\{|J_{i}|,N-|J_{i}|\\}\leq r_{i}$ and
$\min\\{|J_{i}^{\prime\prime}|,N-|J_{i}^{\prime\prime}|\\}\leq s_{i}$, so
(6.7) implies (6.6).

(ii)$\implies$(i). Suppose $b(z)$ is given by (6.3), with (6.6) satisfied.
Define

$B_{2}=\prod_{\min\\{|J_{i}|,N-|J_{i}|\\}=|J_{i}|}\psi_{\alpha_{i},J_{i}}$

and

$B_{1}=\prod_{\min\\{|J_{i}|,N-|J_{i}|\\}=N-|J_{i}|}\psi_{\alpha_{i},N\setminus
J_{i}}.$

Then

$\theta(z)=\frac{B_{2}(z)}{B_{1}(z)}u_{1}(z^{N}),$

where

$u_{1}(z)=u(z)\prod_{\min\\{|J_{i}|,N-|J_{i}|\\}=N-|J_{i}|}\phi_{\alpha_{i}^{N}}(z);$

note that we have used Lemma 6.3. ∎

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