# Algebraic Properties of Toeplitz operators on the symmetrized polydisk

B. Krishna Das  Department of Mathematics, Indian Institute of Technology
Bombay, Powai, Mumbai, 400076, India.
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Haripada Sau  Department of
Mathematics, Indian Institute of Science Education and Research, Dr Homi
Bhabha Rd, Pashan, Pune, Maharashtra, 411008, India.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

This paper is an effort to continue the legacy of the classically successful
theory of Toeplitz operators on the Hardy space over the unit disk to a new
domain in $\mathbb{C}^{d}$ – the symmetrized polydisk. We obtain algebraic
characterizations of Toeplitz operators, analytic Toeplitz operators, compact
perturbation of Toeplitz operators and dual Toeplitz operators. We then
revisit the operator theory of this domain considered first in [10], to study
the generalized Toeplitz operators and find a commutant lifting type result.
This is a continuation of a previous work done in [8].

###### Key words and phrases:

Symmetrized Polydisk, Toeplitz Operator, Commutant Lifting Theorem

###### 2010 Mathematics Subject Classification:

47A13, 47A20, 47B35, 47B38, 46E20, 30H10

##  1\. Introduction

Let $\mathbb{D}$ be the open unit disk and $\mathbb{T}$ be the unit circle in
the complex plane $\mathbb{C}$, while $\mathbb{D}^{d}$ and $\mathbb{T}^{d}$
denote the corresponding cartesian products. The Hardy space over the unit
disk, denoted by $H^{2}$, is the Hilbert space of functions that are analytic
in $\mathbb{D}$ and the co-efficients of the Taylor series around the origin
are square summable (making it naturally isomorphic to $\ell^{2}$, the space
of square summable sequences in $\mathbb{C}$). A function $f$ analytic in
$\mathbb{D}$ is in $H^{2}$ if and only if

$\displaystyle\|f\|_{H^{2}}^{2}=\sup_{0<r<1}\int_{0}^{2\pi}|f(re^{i\theta})|^{2}d\theta<\infty.$

Multiplication by the coordinate function ‘$z$’, denoted by $T_{z}:f\mapsto
zf(z)$, is the first example of a bounded operator on $H^{2}$ that comes to
one’s mind. This is an example of a fairly large class of operators that has
remained a field of extensive research and development since mid 20th century,
viz., the Toeplitz operators. For a symbol $\varphi$ in $L^{\infty}$, the
algebra of essentially bounded function on $\mathbb{T}$ (the measures in this
paper will be the Lebesgue measure unless otherwise specified), the Toeplitz
operator $T_{\varphi}$ is defined as
$T_{\varphi}f(z)=P_{H^{2}}\varphi(z)f(z)$, where $P_{H^{2}}$ denotes the
orthogonal projection from $L^{2}$, the Hilbert space of square integrable
measurable functions on $\mathbb{T}$, onto $H^{2}$. Toeplitz operators often
serve as the elementary building blocks of the study of much complicated
operators. Ever since its invention, the Toeplitz operators have been studied
in many other Hilbert function spaces on several domains in $\mathbb{C}^{d}$,
notably the weighted Bergman spaces [4, 7, 20, 29, 33] and the $H^{p}$ spaces,
$1\leq p<\infty$ [12, 34]; also see [11] for a detailed discussion.

The goal of this paper is to extend the theory of Toeplitz operators in the
Hardy space setting to another domain in $\mathbb{C}^{d}$, $d\geq 2$, viz.,
the symmetrized polydisk. This polynomially convex but not convex domain has
attracted a considerable amount of attention for its complex geometry [19, 17,
24, 25, 23], function theory [22, 26, 31], and operator theory [10]. To
describe the domain, we consider the elementary symmetric functions
$s_{i}:\mathbb{C}^{d}\to\mathbb{C}$ of degree $i$ in $d$ variables defined as

$\displaystyle s_{0}:=1\text{ and }s_{i}(\bm{z}):=\sum_{1\leq
k_{1}<k_{2}<\cdots<k_{i}\leq d}z_{k_{1}}z_{k_{2}}\cdots z_{k_{i}}\text{ for
all $\bm{z}\in\mathbb{C}^{d}$ and $1\leq i\leq d$}$ (1.1)

and the symmetrization map $\bm{s}:\mathbb{C}^{d}\to\mathbb{C}^{d}$ is defined
by

$\displaystyle\bm{s}(\bm{z}):=(s_{1}(\bm{z}),s_{2}(\bm{z}),\dots,s_{d}(\bm{z}))\text{
for all $\bm{z}\in\mathbb{C}^{d}$}.$ (1.2)

The symmetrized polydisk, denoted by $\mathbb{G}_{d}$, is defined to be the
image of the polydisk $\mathbb{D}^{d}$ under the symmetrization map, i.e.,
$\mathbb{G}_{d}:=\bm{s}(\mathbb{D}^{d})$. A typical element of
$\mathbb{G}_{d}$ will be denoted by $(s_{1},s_{2},\dots,s_{d-1},p)$. Following
Ogle [27], the closure of $\mathbb{G}_{d}$ will be denoted by
$\Gamma_{d}:=\bm{s}(\overline{\mathbb{D}}^{d})$. The symmetrized polydisk came
as a natural multivariable generalization of the symmetrized bidisk
$\mathbb{G}_{2}=\\{(z_{1}+z_{2},z_{1}z_{2}):(z_{1},z_{2})\in\mathbb{D}^{2}\\}$,
which, with original motivation coming from the problem of $\mu$-synthesis in
robust control ([16]), has proved to possess rich holomorphic function theory,
complex geometry and operator theory; see [1, 2, 3, 9, 32, 13] and references
therein. The algebraic properties of Toeplitz operators on the symmetrized
bidisk are studied in [8].

As has always been the case in the history of extension of classical theory to
multivariable setting, adopting the Toeplitz operators to the symmetrized
polydisk has challenges. The main challenge is the absence of a matrix
structure of a Toeplitz operator unlike the classical case. We now lay out the
main results of this paper in the order they are proved.

  1. (1)

In §2, we introduce the Hardy space of the symmetrized polydisk
$H^{2}(\mathbb{G}_{d})$ and exhibit two isomorphic copies of it. These
isomorphic copies of the Hardy space are used to decipher algebraic properties
of the Toeplitz operators.

  2. (2)

Note that the Toeplitz operators on $H^{2}$ are precisely those that satisfy
$T_{z}^{*}TT_{z}=T$. This is due to Brown and Halmos [12]. We show that a
bounded operator on $H^{2}(\mathbb{G}_{d})$ is a Toeplitz operator if and only
if

$\displaystyle T_{s_{i}}^{*}TT_{p}=TT_{s_{d-i}}\text{ for each }1\leq i\leq
d-1\text{ and }T_{p}^{*}TT_{p}=T,$ (1.3)

where $(T_{s_{1}},T_{s_{2}},\dots,T_{s_{d-1}},T_{p})$ on
$H^{2}(\mathbb{G}_{d})$ is the tuple of the multiplication operator by the
coordinate functions. See Theorem 3.4 for a proof of the Brown–Halmos type
characterization. While the part that proves necessity is somewhat obvious,
the proof of the sufficiency part needs substantial analysis. We shall explain
in §5 that the role of the tuple
$(T_{s_{1}},T_{s_{2}},\dots,T_{s_{d-1}},T_{p})$ is a direct analogue of that
of $T_{z}$ on $H^{2}$.

  3. (3)

Perhaps the simplest class of Toeplitz operators are the analytic Toeplitz
operators. The analytic Toeplitz operators are those that come from an
analytic symbol $\varphi$, i.e., $\varphi$ has no negative Fourier coefficient
in $L^{2}$. Several characterizations of analytic Toeplitz operators on
$H^{2}(\mathbb{G}_{d})$ are given in Theorem 3.12.

  4. (4)

Compact operators on $H^{2}(\mathbb{G}_{d})$ are characterized in terms of
certain natural shifts. We then use this to study compact perturbation of
Toeplitz operators – the so called asymptotic Toeplitz operators. This is the
content of §4

  5. (5)

In §5, we consider an analogue of a single isometric operator in the
symmetrized polydisk case, called a $\Gamma_{d}$-isometry. These are
essentially those tuples $\underline{S}=(S_{1},S_{2},\dots,S_{d-1},P)$ of
commuting bounded Hilbert space operators that have an extension to what is
called a $\Gamma_{d}$-unitary, that is a tuple
$\underline{R}=(R_{1},R_{2},\dots,R_{d-1},U)$ of commuting normal operators
with the Taylor joint spectrum lying in the distinguished boundary of
$\Gamma_{d}$. It turns out that
$(T_{s_{1}},T_{s_{2}},\dots,T_{s_{d-1}},T_{p})$ is a prototype example of a
$\Gamma_{d}$-isometry. This motivates us to study those bounded operators that
satisfy the Brown–Halmos type relations (1.3) with respect to a general
$\Gamma_{d}$-isometry $\underline{S}$ instead of the particular one
$(T_{s_{1}},T_{s_{2}},\dots,T_{s_{d-1}},T_{p})$. We call these the
$\underline{S}$-Toeplitz operators or the generalized Toeplitz operators.

For a symbol $\varphi\in L^{\infty}$, let $M_{\varphi}$ denote the
multiplication operator on $L^{2}$. Apart from the concrete algebraic
characterization by Brown and Halmos, another way of viewing the Toeplitz
operators on $H^{2}$ is that a Toeplitz operator is precisely the compression
of a commutant of $M_{\zeta}$ on $L^{2}$ with the same norm. We observe a
similar phenomenon in the case of an $\underline{S}$-Toeplitz operator.
Indeed, given a $\Gamma_{d}$-isometry $\underline{S}$ on ${\mathcal{H}}$, we
construct a special minimal $\Gamma_{d}$-unitary extension $\underline{R}$ on
${\mathcal{K}}$ of $\underline{S}$ such that $X$ is an
$\underline{S}$-Toeplitz operator if and only if there is a $Y$ in the
commutant of $\underline{R}$ whose compression to ${\mathcal{H}}$ is $X$ and
$\|Y\|=\|X\|$, see part (2) of Theorem 5.6.

  6. (6)

For an $L^{\infty}$ function $\varphi$, the multiplication operator
$M_{\varphi}$ on $L^{2}$ has the following $2\times 2$ matrix form with
respect to the decomposition $L^{2}=H^{2}\oplus{H^{2}}^{\perp}$:

$\displaystyle\begin{bmatrix}T_{\varphi}&H_{\overline{\varphi}}^{*}\\\
H_{\varphi}&DT_{\varphi}\\\ \end{bmatrix},$ (1.4)

where $H_{\varphi}$ is the Hankel operator and $DT_{\varphi}$ is called the
dual Toeplitz operator associated to $\varphi$. We characterize the dual
Toeplitz operators on $H^{2}(\mathbb{G}_{d})$ in §6.

Finally, the authors would like to express their sincere gratitude to
Professor Tirthankar Bhattacharyya for introducing them to this flourishing
area of operator theory.

##  2\. The Hardy space

###  2.1. Basics of the Hardy Space

Let $\bm{s}$ be the symmetrization map as defined in (1.2) and $J_{\bm{s}}$
denote its complex Jacobian. For
$\bm{\theta}=(\theta_{1},\theta_{2},\ldots,\theta_{d})$ with
$\theta_{i}\in[0,2\pi)$ and $0<r\leq 1$, let $re^{i\bm{\theta}}$ denote the
$d$-tori element
$(\,re^{i\theta_{1}},\,re^{i\theta_{2}},\dots,\,re^{i\theta_{d}})$, while
$d\bm{\theta}$ denotes the normalized product measure on $\mathbb{T}^{d}$.

######  Definition 2.1.

The Hardy space of the symmetrized polydisk, denoted by
$H^{2}(\mathbb{G}_{d})$, is the vector space of those holomorphic functions
$f$ on $\mathbb{G}_{d}$ which satisfy

$\sup_{\,0<r<1}\int_{\mathbb{T}^{d}}|f\circ\bm{s}(re^{i\bm{\theta}})|^{2}|J_{\bm{s}}(re^{i\bm{\theta}})|^{2}d\bm{\theta}<\infty.$

The norm of $f\in H^{2}(\mathbb{G}_{d})$ is defined to be

$\|f\|_{H^{2}(\mathbb{G}_{d})}=\|J_{\bm{s}}\|^{-1}\Big{\\{}\sup_{0<r<1}\int_{\mathbb{T}^{d}}|f\circ\bm{s}(re^{i\bm{\theta}})|^{2}|J_{\bm{s}}(re^{i\bm{\theta}})|^{2}d\bm{\theta}\Big{\\}}^{1/2},$

where
$\|J_{\bm{s}}\|^{2}={{\int_{\mathbb{T}^{d}}}}|J_{\bm{s}}(e^{i\bm{\theta}})|^{2}d\bm{\theta}.$

We divide by $\|J_{\bm{s}}\|$ in the definition of the norm in
$H^{2}(\mathbb{G}_{d})$ to make sure that the norm of the constant function
$\bm{1}$ (sending every element of $\mathbb{G}_{d}$ to the constant $1$) is
$1$.

The space $H^{2}(\mathbb{G}_{d})$ as defined above is actually a reproducing
kernel Hilbert space and its reproducing kernel was explicitly computed in
[22, Theorem 3.1].

The distinguished boundary of a compact set $K$ in $\mathbb{C}^{d}$ is the
$\check{\text{S}}$ilov boundary with respect to the algebra $\mathcal{A}(K)$
of all complex-valued functions continuous on $K$ and holomorphic in the
interior of $K$; see Chapter 9 of [5] for a detailed discussion regarding
$\check{\text{S}}$ilov boundary. The distinguished boundary of the symmetrized
polydisk, denote by $b\Gamma_{d}$ is just the symmetrization of the $d$-tori
$\mathbb{T}^{d}$, i.e., $b\Gamma_{d}=\bm{s}(\mathbb{T}^{d}).$ See [10, Theorem
2.4] for characterizations of $b\Gamma_{d}$.

Consider the measure $\mu$ defined on a Borel subset $E$ of $\mathbb{T}^{d}$
induced by the Jacobian $J_{\bm{s}}$ of the symmetrization map $\bm{s}$ as

$\mu(E)=\int_{E}|J_{\bm{s}}(e^{i\bm{\theta}})|^{2}d\bm{\theta}.$

Let us denote by $\nu$ the push forward measure on $b\Gamma_{d}$ of $\mu$ and
consider the $L^{2}$ space of $b\Gamma_{d}$ with respect to it, i.e.,

$L^{2}(b\Gamma_{d}):=L^{2}(b\Gamma_{d},\nu)=\\{f:b\Gamma_{d}\to\mathbb{C}:\int_{\mathbb{T}^{d}}|f\circ\bm{s}(e^{i\bm{\theta}})|^{2}|J_{\bm{s}}(e^{i\bm{\theta}})|^{2}d\bm{\theta}<\infty\\}.$

Just like in the case of the classical Hardy space, Theorem 2.2 below shows
that $H^{2}(\mathbb{G}_{d})$ is a subspace of $L^{2}(b\Gamma_{d})$ with
respect to a natural identification.

We shall first set up some standard notations which will be used in what
follows.

  * •

A partition $\bm{p}$ of $\mathbb{Z}$ of size $d$ is a $d$-tuple
$(p_{1},p_{2},\dots,p_{d})$ of integers such that $p_{1}>p_{2}>\cdots>p_{d}$.
Let

$\llbracket
z\rrbracket:=\\{(p_{1},p_{2},\dots,p_{d}):p_{1}>p_{2}>\cdots>p_{d}\text{ and
}p_{j}\in\mathbb{Z}\\}$

and

$\llbracket
n\rrbracket:=\\{(p_{1},p_{2},\dots,p_{d}):p_{1}>p_{2}>\cdots>p_{d}\geq 0\\};$

  * •

$\Sigma_{d}$ denotes the permutation group of the set $\\{1,2,\dots,d\\}$.

  * •

For $\sigma\in\Sigma_{d}$, $\bm{m}=(m_{1},m_{2},\dots,m_{d})\in\llbracket
z\rrbracket$ and $\bm{z}=(z_{1},z_{2},\dots z_{d})\in\mathbb{C}^{d}$, let
$\bm{z}_{\sigma}:=(z_{\sigma(1)},z_{\sigma(2)},\dots z_{\sigma(d)})$,
$\bm{z^{m}}:=z_{1}^{m_{1}}z_{2}^{m_{2}}\cdots z_{d}^{m_{d}}$.

Therefore for a partition $\bm{m}$ in $\llbracket z\rrbracket$, $\sigma$ in
$\Sigma_{n}$ and $\bm{z}=(z_{1},z_{2},\dots z_{d})$ in $\mathbb{C}^{d}$,

$\bm{z}^{\bm{m}_{\sigma}}=z_{1}^{m_{\sigma(1)}}z_{2}^{m_{\sigma(2)}}\cdots
z_{d}^{m_{\sigma(d)}}.$

For $\bm{m}\in\llbracket n\rrbracket$, consider the function $\bm{a_{m}(z)}$
defined by

$\bm{a_{m}(z)}:=\sum_{\sigma\in\Sigma_{d}}\text{sgn}(\sigma)\bm{z}^{\bm{m}_{\sigma}}=\det\left(((z_{i}^{m_{j}}))_{i,j=1}^{d}\right),$

where $\text{sgn}(\sigma)$ denotes the sign of the permutation $\sigma$. The
above representation of $\bm{a_{m}}$ shows that it is zero if and only if
there exist $1\leq i<j\leq d$ such that $m_{i}=m_{j}$. Note that the functions
$\bm{a_{m}}$ continue to be defined on $\mathbb{T}^{d}$ for
$\bm{m}\in\llbracket z\rrbracket$ and that
$\bm{a_{m}}(\bm{z}_{\sigma})=\operatorname{sgn}(\sigma)\bm{a_{m}}(\bm{z})$ for
every $\sigma$ in $\Sigma_{n}$. This shows that each $\bm{a_{m}}$ is a member
of the following subspace of $L^{2}(\mathbb{T}^{d})$ consisting of anti-
symmetric functions

$L^{2}_{\rm anti}(\mathbb{T}^{d}):=\\{f\in
L^{2}(\mathbb{T}^{d}):f(e^{i\bm{\theta}_{\sigma}})=\text{sgn}(\sigma)f(e^{i\bm{\theta}})\text{
a.e. w.r.t. the product measure}\\}.$

Moreover, the set

$\\{\bm{a_{m}}:\bm{m}\in\llbracket z\rrbracket\\}$

actually forms an orthogonal basis for $L^{2}_{\rm anti}(\mathbb{T}^{d})$,
which follows from the facts that the operator
$\mathbb{P}:L^{2}(\mathbb{T}^{d})\to L^{2}_{\text{anti}}(\mathbb{T}^{d})$
defined by

$\mathbb{P}(f)(e^{i\bm{\theta}})=\frac{1}{d!}\sum_{\sigma\in\Sigma_{d}}\text{sgn}(\sigma)f(e^{i\bm{\theta}_{\sigma}})$

is an orthogonal projection onto $L^{2}_{\text{anti}}(\mathbb{T}^{d})$ and
that the set $\\{e^{i\bm{m}\bm{\cdot}\bm{\theta}}:\bm{m}\in\mathbb{Z}^{n}\\}$
is an orthonormal basis for $L^{2}(\mathbb{T}^{d})$. Here
$\bm{m}\cdot\bm{\theta}:=(m_{1}\theta_{1},m_{2}\theta_{2},\dots,m_{d}\theta_{d})$
for $\bm{m}=(m_{1},m_{2},\dots,m_{d})$ and
$\bm{\theta}=(\theta_{1},\theta_{2},\dots,\theta_{d})$.

Consider the following subspace of the Hardy space $H^{2}(\mathbb{D}^{d})$
over the $d$-disk

$H^{2}_{\rm anti}(\mathbb{D}^{d}):=\\{f\in
H^{2}(\mathbb{D}^{d}):f(\bm{z}_{\sigma})=\text{sgn}(\sigma)f(\bm{z}),\text{
for all }\bm{z}\in\mathbb{D}^{d}\text{ and }\sigma\in\Sigma_{d}\\}.$

It was observed in [22, Sec. 3] that the set

$\mathcal{B}=\\{\bm{a_{p}}:\bm{p}\in\llbracket n\rrbracket\\}$

is an orthogonal basis for $H^{2}_{\rm anti}(\mathbb{D}^{d})$. The following
theorem immediately allows us to consider boundary values of the Hardy space
functions.

######  Theorem 2.2.

There is an isometric embedding of the space $H^{2}(\mathbb{G}_{d})$ inside
$L^{2}(b\Gamma_{d})$.

###### Proof.

Define $U_{h}:H^{2}(\mathbb{G}_{d})\to H^{2}_{\rm anti}(\mathbb{D}^{d})$ by

$\displaystyle
U_{h}(f)=\frac{1}{\|J_{\bm{s}}\|}J_{\bm{s}}(\cdot)f\circ\bm{s}(\cdot),\text{
for all }f\in H^{2}(\mathbb{G}_{d})$ (2.1)

and ${U_{l}}:L^{2}(b\Gamma_{d})\to L^{2}_{\rm anti}(\mathbb{D}^{d})$ by

$\displaystyle
U_{l}f=\frac{1}{\|J_{\bm{s}}\|}J_{\bm{s}}(\cdot)f\circ\bm{s}(\cdot),\text{ for
all }f\in L^{2}(b\Gamma_{d}).$ (2.2)

It follows from the definitions of the norms in the respective spaces that
both $U_{l}$ and $U_{h}$ are isometries. Moreover, since they are easily seen
to be surjective, they are unitary. Also note that there is an isometry
$W:H^{2}_{\rm anti}(\mathbb{D}^{d})\to L^{2}_{\rm anti}(\mathbb{T}^{d})$ which
sends a function to its radial limit. Therefore we have the following
commutative diagram:

$\begin{CD}H^{2}(\mathbb{G}_{d})@>{U_{l}^{-1}\circ W\circ
U_{h}}>{}>L^{2}(b\Gamma_{d})\\\ @V{U_{h}}V{}V@V{}V{U_{l}}V\\\ H^{2}_{\rm
anti}(\mathbb{D}^{d})@>{}>{W}>L^{2}_{\rm anti}(\mathbb{T}^{d})\end{CD}.$

Consequently, the embedding of $H^{2}(\mathbb{G}_{d})$ into
$L^{2}(b\Gamma_{d})$ is carried out by the isometry $U_{l}^{-1}\circ W\circ
U_{h}$. ∎

Let $L^{\infty}(b\Gamma_{d})$ denote the algebra of all functions on
$b\Gamma_{d}$ which are bounded almost everywhere on $b\Gamma_{d}$ with
respect to the measure $\nu$. For a function $\varphi$ in
$L^{\infty}(b\Gamma_{d})$, let $M_{\varphi}$ be the operator on
$L^{2}(b\Gamma_{d})$ defined by

$M_{\varphi}f(s_{1},\dots,s_{d-1},p)=\varphi(s_{1},\dots,s_{d-1},p)f(s_{1},\dots,s_{d-1},p),$

for all $f$ in $L^{2}(b\Gamma_{d})$. We note that the tuple
$(M_{s_{1}},\dots,M_{s_{d-1}},M_{p})$ of multiplication by coordinate
functions on $L^{2}(b\Gamma_{d})$ is a commuting tuple of normal operators.

######  Definition 2.3.

For a function $\varphi$ in $L^{\infty}(b\Gamma_{d})$, the multiplication
operator $M_{\varphi}$ is called the Laurent operator with symbol $\varphi$.
The compression of $M_{\varphi}$ to $H^{2}(\mathbb{G}_{d})$ is called the
Toeplitz operator and denoted by $T_{\varphi}$. Therefore

$T_{\varphi}f=PrM_{\varphi}f\;\text{ for all $f$ in $H^{2}(\mathbb{G}_{d})$,}$

where $Pr$ denotes the orthogonal projection of $L^{2}(b\Gamma_{d})$ onto
$H^{2}(\mathbb{G}_{d})$.

Let $L^{\infty}_{\text{sym}}(\mathbb{T}^{d})$ denote the sub-algebra of
$L^{\infty}(\mathbb{T}^{d})$ consisting of symmetric functions, i,e.,
$f(e^{i\bm{\theta}_{\sigma}})=f(e^{i\bm{\theta}})$ a.e., and
$\Pi_{\bm{s}}:L^{\infty}(b\Gamma_{d})\to
L^{\infty}_{\text{sym}}(\mathbb{T}^{d})$ be the $*$-isomorphism defined by

$\varphi\mapsto\varphi\circ\bm{s}$

where $\bm{s}$ is the symmetrization map as defined in (1.2). Induced by the
unitary ${U_{l}}$ as in (2.2), let
$\Pi_{l}:\mathcal{B}(L^{2}(b\Gamma_{d}))\to\mathcal{B}(L^{2}_{\text{anti}}(\mathbb{T}^{d}))$
be the map

$\Pi_{l}:X\mapsto{U_{l}}X{U_{l}}^{*}.$

######  Theorem 2.4.

Let $\Pi_{1}$ and $\Pi_{2}$ be the above $*$-isomorphisms. Then the following
diagram is commutative:

$\begin{CD}L^{\infty}(b\Gamma_{d})@>{\Pi_{\bm{s}}}>{}>L^{\infty}_{\operatorname{sym}}(\mathbb{T}^{d})\\\
@V{i_{1}}V{}V@V{}V{i_{2}}V\\\
\mathcal{B}(L^{2}(b\Gamma_{d}))@>{}>{\Pi_{l}}>B(L^{2}_{\operatorname{anti}}(\mathbb{T}^{d}))\end{CD},$

where $i_{1}$ and $i_{2}$ are the canonical inclusion maps. Equivalently, for
$\varphi\in L^{\infty}(b\Gamma_{d})$, the operators $M_{\varphi}$ on
$L^{2}(b\Gamma_{d})$ and $M_{\varphi\circ\bm{s}}$ on
$L^{2}_{\operatorname{anti}}(\mathbb{T}^{d})$ are unitarily equivalent via the
unitary $U_{l}$ as in (2.2).

###### Proof.

To show that the above diagram commutes all we need to show is that
$U_{l}M_{\varphi}U_{l}^{*}=M_{\varphi\circ\bm{s}}$, for every $\varphi$ in
$L^{\infty}(b\Gamma_{d})$. This follows from the following computation: for
every $\varphi$ in $L^{\infty}(b\Gamma_{d})$ and $f\in
L^{2}_{\text{anti}}(\mathbb{T}^{d})$,

$U_{l}M_{\varphi}U_{l}^{*}(f)=U_{l}(\varphi
U_{l}^{*}f)=(\varphi\circ\bm{s})\frac{1}{\|J_{\bm{s}}\|}J_{\bm{s}}(U_{l}^{*}f\circ\bm{s})=M_{\varphi\circ\bm{s}}(f).$

∎

As a consequence of the above theorem, we obtain that the Toeplitz operators
on the Hardy space of the symmetrized polydisk are unitarily equivalent to
those on $H^{2}_{\text{anti}}(\mathbb{D}^{d})$.

######  Corollary 2.5.

For a $\varphi\in L^{\infty}(b\Gamma_{d})$, $T_{\varphi}$ is unitarily
equivalent to
$T_{\varphi\circ\bm{s}}:=P_{a}M_{\varphi\circ\bm{s}}|_{H^{2}_{\operatorname{anti}}(\mathbb{D}^{d})}$,
where $P_{a}$ stands for the projection of
$L^{2}_{\operatorname{anti}}(\mathbb{T}^{d})$ onto
$H^{2}_{\operatorname{anti}}(\mathbb{D}^{d})$.

###### Proof.

This follows from the fact that the operators $M_{\varphi}$ and
$M_{\varphi\circ\bm{s}}$ are unitarily equivalent via the unitary $U_{h}$ as
in (2.1). Indeed for every $f\in H^{2}(\mathbb{G}_{d})$

$U_{h}M_{\varphi}f=\frac{1}{\|J_{\bm{s}}\|}\varphi\circ\bm{s}(\cdot)f\circ\bm{s}(\cdot)J_{\bm{s}}(\cdot)=M_{\varphi\circ\bm{s}}U_{h}f.$

∎

######  Notation 2.6.

It is trivial to see that the coordinate functions $s_{1},\dots,s_{d-1},p$ are
in $L^{\infty}(b\Gamma_{d})$. Just like in the classical case, the Toeplitz
operators $T_{s_{1}},T_{s_{2}},\dots,T_{s_{d-1}},T_{p}$ will play a
significant role in describing a general Toeplitz operator on
$H^{2}(\mathbb{G}_{d})$. For each $j=1,2,\dots,d-1$, we shall denote by
$T_{s_{j}(\bm{z})}$ and $T_{s_{d}(\bm{z})}$ on
$H^{2}_{\operatorname{anti}}(\mathbb{D}^{d})$ the unitary copies of
$T_{s_{j}}$ and $T_{p}$, respectively. Similarly, we use the notations
$M_{s_{1}(\bm{z})},M_{s_{2}(\bm{z})},\dots,M_{s_{d}(\bm{z})}$ to denote the
unitary copies of $M_{s_{1}},M_{s_{2}},\dots,M_{p}$, respectively.

We end the subsection with one more isomorphic copy of the Hardy space
$H^{2}(\mathbb{G}_{d})$. If $\mathcal{E}$ is a Hilbert space, let
$\mathcal{O}(\mathbb{D},\mathcal{E})$ be the class of all $\mathcal{E}$-valued
holomorphic functions on $\mathbb{D}$. Let

$H^{2}_{\mathcal{E}}(\mathbb{D})=\\{f(z)=\sum
a_{k}z^{k}\in\mathcal{O}(\mathbb{D},\mathcal{E}):a_{k}\in\mathcal{E}\mbox{
with }\|f\|^{2}=\sum\|a_{k}\|^{2}<\infty\\}.$

######  Lemma 2.7.

There is a Hilbert space isomorphism $U_{1}$ from
$H^{2}_{\operatorname{anti}}(\mathbb{D}^{d})$ onto the vector-valued Hardy
space $H^{2}_{\mathcal{E}}(\mathbb{D})$, where

$\mathcal{E}=\overline{\operatorname{span}}\\{\bm{a_{p}(z)}:\bm{p}\in\llbracket
n\rrbracket\text{ such that }\bm{p}=(p_{1},p_{2},\dots,p_{d-1},0)\\}\subset
H^{2}_{\operatorname{anti}}(\mathbb{D}^{d}).$

Moreover, this unitary $U_{1}$ intertwines $T_{s_{d}(\bm{z})}$ on
$H^{2}_{\operatorname{anti}}(\mathbb{D}^{d})$ with the unilateral shift of
infinite multiplicity $T_{z}$ on $H^{2}_{\mathcal{E}}(\mathbb{D})$.

###### Proof.

As observed above, the set $\\{\bm{a_{p}(z)}:\bm{p}\in\llbracket
n\rrbracket\text{ such that }\bm{p}=(p_{1},p_{2},\dots,p_{d-1},0)\\}$ forms an
orthogonal basis in $\mathcal{E}$. Thus

$\\{z^{q}\bm{a}_{\bm{p}}:q\geq 0\text{ and }\bm{p}\in\llbracket
n\rrbracket\text{ such that }\bm{p}=(p_{1},p_{2},\dots,p_{d-1},0)\\}$

is an orthogonal basis for $H^{2}_{\mathcal{E}}(\mathbb{D})$. On the other
hand, the space $H^{2}_{\text{anti}}(\mathbb{D}^{d})$ is spanned by the
orthogonal set $\\{\bm{a_{p}(z)}:\bm{p}\in\llbracket n\rrbracket\\}$ and if
$\bm{p}=(p_{1},p_{2},\dots,p_{d-1},p_{d})$, then

$\bm{a_{p}(z)}=(z_{1}z_{2}\cdots
z_{d})^{p_{d}}\bm{a_{\tilde{p}}(z)}={T^{p_{d}}_{s_{d}(\bm{z})}}\bm{a_{\tilde{p}}(z)},$

where $\bm{\tilde{p}}=(p_{1}-p_{d},p_{2}-p_{d},\dots,p_{d-1}-p_{d},0)$. Define
the unitary operator from $H^{2}_{\text{anti}}(\mathbb{D}^{d})$ onto
$H^{2}_{\mathcal{E}}(\mathbb{D})$ by

$U_{1}:\bm{a_{p}(z)}\mapsto z^{p_{d}}\bm{a_{\tilde{p}}(z)},$

and then extending linearly. This preserves norms because $T_{s_{d}(\bm{z})}$
is an isometry on $H^{2}_{\text{anti}}(\mathbb{D}^{d})$ and $T_{z}$ is an
isometry on $H^{2}_{\mathcal{E}}(\mathbb{D})$. It is surjective and obviously
intertwines $T_{s_{d}(\bm{z})}$ and $T_{z}$. ∎

By virtue of the isomorphisms $U_{h}$ and $U_{1}$ described above, we have the
following commutative diagram:

${(H^{2}(\mathbb{G}_{d}),T_{p})}$${(H^{2}_{\text{anti}}(\mathbb{D}^{d}),T_{s_{d}(\bm{z})})}$${(H^{2}_{\mathcal{E}}(\mathbb{D}),T_{z})}$$\scriptstyle{U_{h}}$$\scriptstyle{U_{2}}$$\scriptstyle{U_{1}}$

i.e., the operator $T_{p}$ on $H^{2}(\mathbb{G}_{d})$ is unitarily equivalent
to the unilateral shift $T_{z}$ on the vector valued Hardy space
$H^{2}_{\mathcal{E}}(\mathbb{D})$ via the unitary $U_{2}$. We call
$\mathcal{E}$ the coefficient space of the symmetrized polydisk. It is a
subspace of $H^{2}_{\mathcal{E}}(\mathbb{D})$ which is naturally identifiable
with the subspace of constant functions in $H^{2}_{\mathcal{E}}(\mathbb{D})$.

##  3\. An algebraic characterization of Toeplitz operators and their basic
properties

Although a Toeplitz operator is defined in terms of an $L^{\infty}$ function,
it is a difficult question of how to recognize a given bounded operator $T$ on
the relevant Hilbert space as a Toeplitz operator. This question was answered
for the Hardy space of the unit disc by Brown and Halmos in Theorem 6 of [12]
where they showed that $T$ has to be invariant under conjugation by the
unilateral shift. We show that in our context one needs not only $T_{p}$ but
all of $T_{s_{i}}$, $i=1,2,\dots,d-1$ to give such a characterization.

We first note the following preliminary results each of which will play a
significant role in obtaining a Brown–Halmos type relations for
$H^{2}(\mathbb{G}_{d})$.

######  Lemma 3.1.

Let $M$ be a bounded operator on $L^{2}(b\Gamma_{d})$ that commutes with each
entry of the tuple $(M_{s_{1}},M_{s_{2}},\dots,M_{p})$. Then $M$ is of the
form $M_{\varphi}$, for some function $\varphi$ in $L^{\infty}(b\Gamma_{d})$.

###### Proof.

Since $(M_{s_{1}},M_{s_{2}},\dots,M_{p})$ is a commuting tuple of normal
operators, the Taylor joint spectrum
$\sigma_{T}(M_{s_{1}},M_{s_{2}},\dots,M_{p})=b\Gamma_{d}$. Now it follows from
the spectral theorem for commuting normal operators that the von Neumann
algebra generated by $\\{M_{s_{1}},M_{s_{2}},\dots,M_{p}\\}$ is
$L^{\infty}(b\Gamma_{d})$, which is a maximal abelian von Neumann algebra,
i.e., $\\{M_{s_{1}},M_{s_{2}},\dots,M_{p}\\}^{\prime}$ is $*$-isomorphic to
$L^{\infty}(b\Gamma_{d})$. ∎

In view of identification results Theorem 2.2 and Theorem 2.4, the following
is then an obvious consequence of the above lemma.

######  Corollary 3.2.

Any bounded operator $M$ on $L^{2}_{\operatorname{anti}}(\mathbb{T}^{d})$ that
commutes with each entry of the tuple
$(M_{s_{1}(\bm{z})},M_{s_{2}(\bm{z})},\dots,M_{s_{d}(\bm{z})})$ is of the form
$M_{\varphi}$, for some function $\varphi$ in
$L^{\infty}_{\operatorname{sym}}(\mathbb{T}^{d})$.

The following algebraic relations of the multipliers
$M_{s_{1}},M_{s_{2}},\dots,M_{p}$ on $L^{2}(b\Gamma_{d})$ and their
restrictions to $H^{2}(\mathbb{G}_{d})$ give us a major lead to the
Brown–Halmos relations.

######  Lemma 3.3.

For every $j=1,2,\dots d-1$, $M_{s_{j}}^{*}M_{p}=M_{s_{d-j}}$ and
$T_{s_{j}}^{*}T_{p}=T_{s_{d-j}}$.

###### Proof.

From the definition of the elementary symmetric functions $s_{j}$, it follows
that if $\bm{z}\in\mathbb{T}^{d}$, then one has

$s_{d-j}(\bm{z})=\overline{s_{j}(\bm{z})}s_{d}(\bm{z})$

and hence follows the first set of equations. For the second part we choose
$f,g$ in $H^{2}(\mathbb{G}_{d})$ and compute using the first part

$\langle T_{s_{j}}^{*}T_{p}f,g\rangle=\langle T_{p}f,T_{s_{j}}g\rangle=\langle
M_{s_{j}}^{*}M_{p}f,g\rangle=\langle M_{s_{d-j}}f,g\rangle=\langle
T_{s_{d-j}}f,g\rangle.$

This proves the lemma. ∎

######  Theorem 3.4.

Let $T$ be a bounded operator on $H^{2}(\mathbb{G}_{d})$. Then $T$ is a
Toeplitz operator if and only if for each $i=1,2,\dots,d-1$,

$\displaystyle T_{s_{i}}^{*}TT_{p}=TT_{s_{d-i}}\text{ and }T_{p}^{*}TT_{p}=T.$
(3.1)

###### Proof.

Let us first prove the easier direction – the ‘only if’ part. Let $T$ be a
Toeplitz operator with symbol $\varphi$. Then for $f,g\in
H^{2}(\mathbb{G}_{d})$,

$\displaystyle\langle T_{p}^{*}T_{\varphi}T_{p}f,g\rangle=\langle
T_{\varphi}T_{p}f,T_{p}g\rangle=\langle PrM_{\varphi}T_{p}f,T_{p}g\rangle$
$\displaystyle=$ $\displaystyle\langle M_{\varphi}M_{p}f,M_{p}g\rangle=\langle
M_{\varphi}f,g\rangle$ $\displaystyle=$ $\displaystyle\langle
PrM_{\varphi}f,g\rangle=\langle T_{\varphi}f,g\rangle.$

Also for $i=1,2,\dots,d-1$, using Lemma 3.3 we compute

$\displaystyle\langle T_{s_{i}}^{*}T_{\varphi}T_{p}f,g\rangle_{H^{2}}$
$\displaystyle=$ $\displaystyle\langle
PrM_{\varphi}T_{p}f,T_{s_{i}}g\rangle_{H^{2}}=\langle
M_{\varphi}M_{p}f,M_{s_{i}}g\rangle_{L^{2}}=\langle
M_{s_{i}}^{*}M_{p}M_{\varphi}f,g\rangle_{L^{2}}$ $\displaystyle=$
$\displaystyle\langle M_{\varphi}M_{s_{d-i}}f,g\rangle_{L^{2}}=\langle
PrM_{\varphi}M_{s_{d-i}}f,g\rangle_{H^{2}}=\langle
T_{\varphi}T_{s_{d-i}}f,g\rangle_{H^{2}}.$

For the sufficient part, we work on $H^{2}_{\rm anti}(\mathbb{D}^{d})$ and
invoke Corollary 2.5 to draw the conclusion. So, let $T$ be a bounded operator
on $H^{2}_{\rm anti}(\mathbb{D}^{d})$ that satisfies
$T_{s_{i}(\bm{z})}^{*}TT_{s_{d}(\bm{z})}=TT_{s_{d-i}(\bm{z})}$ and
$T_{s_{d}(\bm{z})}^{*}TT_{s_{d}(\bm{z})}=T$, where note that $s_{i}$ is the
$i$-th symmetric function defined in (1.1). For a partition
$\bm{p}=(p_{1},p_{2},\dots,p_{d})$ in $\llbracket z\rrbracket$, note that for
$r\geq 0$, $M_{s_{d}(\bm{z})}^{r}\bm{a_{p}}=\bm{a_{p+r}}$, where $\bm{r}$ is
the $d$-tuple $(r,r,\dots,r)$ and
$\bm{p}+\bm{r}=(p_{1}+r,p_{2}+r,\dots,p_{d}+r)$. Therefore for every partition
$\bm{p}$ in $\llbracket z\rrbracket$, there exists a sufficiently large $r$
(depending on $\bm{p}$) such that $M_{s_{d}(\bm{z})}^{n}\bm{a_{p}}\in
H^{2}_{\rm anti}(\mathbb{D}^{d})$. For each non-negative integer $r$, define
an operator $T_{r}$ on $L^{2}_{\text{anti}}(\mathbb{T}^{d})$ by

$T_{r}:={M^{*r}_{s_{d}(\bm{z})}}TP_{a}M_{s_{d}(\bm{z})}^{r},$

where $P_{a}$ is the orthogonal projection of
$L^{2}_{\text{anti}}(\mathbb{T}^{d})$ onto
$H^{2}_{\text{anti}}(\mathbb{D}^{d})$. Let $\bm{p}$ and $\bm{q}$ be two
partitions in $\llbracket z\rrbracket$, then for sufficiently large $r$, we
have

$\displaystyle\langle T_{r}\bm{a_{p}},\bm{a_{q}}\rangle=\langle
TM_{s_{d}(\bm{z})}^{r}\bm{a_{p}},M_{s_{d}(\bm{z})}^{r}\bm{a_{q}}\rangle=\langle
T\bm{a_{p+r}},\bm{a_{q+r}}\rangle.$ (3.2)

Since $T_{s_{d}(\bm{z})}^{*}TT_{s_{d}(\bm{z})}=T$, we have for every $r\geq
0$, $T_{s_{d}(\bm{z})}^{*r}TT_{s_{d}(\bm{z})}^{r}=T$. Let $\bm{p}$ and
$\bm{q}$ be two partitions in $\llbracket n\rrbracket$, then for every $r\geq
0$,

$\displaystyle\langle T\bm{a_{p}},\bm{a_{q}}\rangle=\langle
TT_{s_{d}(\bm{z})}^{r}\bm{a_{p}},T_{s_{d}(\bm{z})}^{r}\bm{a_{q}}\rangle=\langle
T\bm{a_{p+r}},\bm{a_{q+r}}\rangle.$ (3.3)

Since $\\{\bm{a_{p}}:\bm{p}\in\llbracket z\rrbracket\\}$ is an orthogonal
basis for $L^{2}_{\rm anti}(\mathbb{T}^{d})$ and the sequence of operators
$\\{T_{r}\\}$ on $L^{2}_{\rm anti}(\mathbb{T}^{d})$ is uniformly bounded by
$\|T\|$, by (3.2) and $(\ref{auxtoepcharctwo})$ the sequence $\\{T_{r}\\}$
converges weakly to some operator $T_{\infty}$ (say) acting on $L^{2}_{\rm
anti}(\mathbb{T}^{d})$.

We now use Corollary 3.2 to conclude that $T_{\infty}=M_{\varphi}$, for some
$\varphi\in L^{\infty}_{\text{sym}}(\mathbb{T}^{d})$. Therefore we have to
show that $T_{\infty}$ commutes with each entry of
$(M_{s_{1}(\bm{z})},M_{s_{2}(\bm{z})},\dots,M_{s_{d}(\bm{z})})$. That
$T_{\infty}$ commutes with $M_{s_{d}(\bm{z})}$ is clear from the definition of
$T_{\infty}$. The following computation shows that $T_{\infty}$ commutes with
the other entries also. Let $\bm{p}$ and $\bm{q}$ be two partitions in
$\llbracket z\rrbracket$ and $1\leq i\leq d-1$. Then

$\displaystyle\langle
M_{s_{i}(\bm{z})}^{*}T_{\infty}^{*}\bm{a_{p}},\bm{a_{q}}\rangle$
$\displaystyle=$ $\displaystyle\lim_{r}\langle
M_{s_{i}(\bm{z})}^{*}M_{s_{d}(\bm{z})}^{*r}T^{*}P_{a}M_{s_{d}(\bm{z})}^{r}\bm{a_{p}},\bm{a_{q}}\rangle$
$\displaystyle=$ $\displaystyle\lim_{r}\langle
T_{s_{i}(\bm{z})}^{*}T^{*}M_{s_{d}(\bm{z})}^{r}\bm{a_{p}},M_{s_{d}(\bm{z})}^{r}\bm{a_{q}}\rangle\;\;(\text{for
sufficiently large $r$})$ $\displaystyle=$ $\displaystyle\lim_{r}\langle
T_{s_{d}(\bm{z})}^{*}T^{*}T_{s_{d-i}(\bm{z})}M_{s_{d}(\bm{z})}^{r}\bm{a_{p}},M_{s_{d}(\bm{z})}^{r}\bm{a_{q}}\rangle\;\;(\text{applying
(\ref{Toeplitzcharc})})$ $\displaystyle=$ $\displaystyle\lim_{r}\langle
M_{s_{d}(\bm{z})}^{*r+1}T^{*}P_{a}M_{s_{d}(\bm{z})}^{r+1}M^{*}_{s_{d}(\bm{z})}M_{s_{d-i}(\bm{z})}\bm{a_{p}},\bm{a_{q}}\rangle$
$\displaystyle=$ $\displaystyle\lim_{r}\langle
M_{s_{d}(\bm{z})}^{*r+1}P_{a}T^{*}P_{a}M_{s_{d}(\bm{z})}^{r+1}M_{s_{i}(\bm{z})}^{*}\bm{a_{p}},\bm{a_{q}}\rangle\;\;(\text{by
Lemma \ref{L:GammaUni}})$ $\displaystyle=$ $\displaystyle\langle
T_{\infty}^{*}M_{s_{i}(\bm{z})}^{*}\bm{a_{p}},\bm{a_{q}}\rangle.$

Therefore there exists a $\varphi\in L^{\infty}_{\text{sym}}(\mathbb{T}^{d})$
such that $T_{\infty}=M_{\varphi}$. Now for $f$ and $g$ in
$H^{2}_{\text{anti}}(\mathbb{D}^{d})$, we have

$\displaystyle\langle P_{a}M_{\varphi}f,g\rangle=\langle
M_{\varphi}f,g\rangle=\langle T_{\infty}f,g\rangle$ $\displaystyle=$
$\displaystyle\lim_{r}\langle T_{r}f,g\rangle=\lim_{r}\langle
TT_{s_{d}(\bm{z})}^{r}f,T_{s_{d}(\bm{z})}^{r}g\rangle=\langle Tf,g\rangle.$

Hence $T$ is the Toeplitz operator with symbol $\varphi$. ∎

The following is a straightforward consequence of Lemma 3.3 and the
characterization of Toeplitz operators obtained above.

######  Corollary 3.5.

If $T$ is a bounded operator on $H^{2}(\mathbb{G}_{d})$ that commutes with
each entry of $(T_{s_{1}},\dots,T_{s_{d-1}},T_{p})$, then $T$ is a Toeplitz
operator.

######  Definition 3.6.

The relations in (3.1) will be referred to as the Brown–Halmos relations.

We progress with basic properties of Toeplitz operators. It is a natural
question whether any of the Brown–Halmos relations implies the other. We give
here an example of an operator $Y$ which satisfies none other than the last
one.

######  Example 3.7.

We actually construct an example on the space $H^{2}_{\rm
anti}(\mathbb{D}^{d})$ and invoke the unitary equivalences we established in
Section 3 to draw the conclusion. For each $1\leq j\leq d-1$, we define
operators $Y_{j}$ on $H^{2}_{\text{anti}}(\mathbb{D}^{d})$ by defining its
action on the basis elements $\\{\bm{a_{p}}:\bm{p}\in\llbracket
n\rrbracket\\}$ as

$Y_{j}\bm{a_{p}}=\bm{a_{p+f_{j}}},\text{ where
}\bm{f_{j}}=(\underbrace{1,\dots,1}_{j\text{-times}},0\dots,0).$ (3.4)

Note that $M_{s_{d}(\bm{z})}\bm{a_{p}}=\bm{a_{p+f_{d}}}$ for every
$\bm{p}\in\llbracket n\rrbracket$. Therefore each $Y_{j}$ commutes with
$M_{s_{d}(\bm{z})}|_{H^{2}_{\rm anti}(\mathbb{D}^{d})}$. Thus by the algebraic
relations in Lemma 3.3, it follows that for $Y_{j}$ to satisfy other
Brown–Halmos relations, it is necessary for it to commute with each
$T_{s_{i}(\bm{z})}$. But we show below that $Y_{j}$ commutes with none of the
$T_{s_{i}(\bm{z})}$.

To that end let us fix some $j$ between $1$ and $d-1$. Choose the following
partition

$\bm{p}=(p_{1},\dots,p_{j-1},p_{j+1}+1,p_{j+1},\dots,p_{d})$

in a way so that the difference between each entry with its next entry is
strictly greater than one (except the $j$-th entry). For this $\bm{p}$, note
that the difference between each adjacent entries of $\bm{p}+\bm{f}_{j}$ is
greater than $2$. Hence we conclude that $T_{s_{i}(\bm{z})}Y_{j}\bm{a_{p}}$
has exactly $n!$ many non-zero terms. On the other hand, note that the $i$-the
elementary symmetrization function corresponds to the partition

$\bm{m}^{i}=(\underbrace{1,\dots,1}_{i-\text{times}},0,\dots,0)$

in the sense that $s_{i}=\bm{s}_{\bm{m}^{i}}(\bm{z})$. Therefore
$T_{s_{i}(\bm{z})}\bm{a_{p}}=s_{i}\bm{a_{p}}=\bm{s}_{\bm{m}^{i}}\bm{a_{p}}=\sum_{\sigma\in\Sigma_{d}}\bm{a_{p+\bm{m}^{i}_{\sigma}}}$.
Note that since $i<n$, we can choose a $\sigma$ form $\Sigma_{d}$ so that
$\bm{m}^{i}_{\sigma}$ has $0$ and $1$ in its $j$-th and $(j+1)$-th entry,
respectively. For this $\sigma$, it follows that
$\bm{a}_{\bm{p}+\bm{m}^{i}_{\sigma}}=0$. Hence $T_{s_{i}(\bm{z})}\bm{a_{p}}$
has at most $n!-1$ many non-zero terms and hence so does
$Y_{j}T_{s_{i}(\bm{z})}\bm{a_{p}}$. Therefore $Y_{j}$ and $T_{s_{i}(\bm{z})}$
can not commute.

######  Lemma 3.8.

For $\varphi\in L^{\infty}(b\Gamma_{d})$ if $T_{\varphi}$ is the zero
operator, then $\varphi=0$, a.e. In other words, the map $\varphi\mapsto
T_{\varphi}$ from $L^{\infty}(b\Gamma_{d})$ into the set of all Toeplitz
operators on the symmetrized polydisk, is injective.

###### Proof.

Let
$\varphi\circ\bm{s}(\bm{z})=\sum_{\bm{m}\in\mathbb{Z}^{d}}\alpha_{\bm{m}}\bm{z}^{\bm{m}}\in
L^{\infty}_{\text{sym}}(\mathbb{T}^{d})$. Since $\varphi\circ\bm{s}$ is
symmetric, the coefficients $\alpha_{\bm{m}}$ are such that
$\alpha_{\bm{m}_{\sigma}}=\alpha_{\bm{m}}$ for every $\sigma\in\Sigma_{d}$.
Also, $\varphi\circ\bm{s}$ in $L^{2}(\mathbb{T}^{d})$ implies the coefficient
sequence $\\{\alpha_{\bm{m}}\\}$ is square-summable. Suppose that
$T_{\varphi\circ\bm{s}}$ on $H^{2}_{\text{anti}}(\mathbb{D}^{d})$ is the zero
operator. Then for every $\bm{p}$ and $\bm{q}$ in $\llbracket n\rrbracket$ we
have

$\displaystyle 0=\langle T_{\varphi\circ\bm{s}}\bm{a_{p}},\bm{a_{q}}\rangle$
$\displaystyle=$
$\displaystyle\langle\sum_{\bm{m}\in\mathbb{Z}^{d}}\alpha_{\bm{m}}\bm{z^{m}}\sum_{\sigma\in\Sigma_{d}}\operatorname{sgn}(\sigma)\bm{z}^{\bm{p}_{\sigma}},\sum_{\delta\in\Sigma_{d}}\operatorname{sgn}(\delta)\bm{z}^{\bm{q}_{\delta}}\rangle$
(3.5) $\displaystyle=$
$\displaystyle\langle\sum_{\bm{m}\in\mathbb{Z}^{d},\sigma\in\Sigma_{d}}\operatorname{sgn}(\sigma)\alpha_{\bm{m}}\bm{z}^{\bm{m}+\bm{p}_{\sigma}},\sum_{\delta\in\Sigma_{d}}\operatorname{sgn}(\delta)\bm{z}^{\bm{q}_{\delta}}\rangle$
$\displaystyle=$
$\displaystyle\sum_{\bm{m}\in\mathbb{Z}^{d},\sigma,\delta\in\Sigma_{d}}\operatorname{sgn}(\sigma)\operatorname{sgn}(\delta)\alpha_{\bm{m}}\bm{\langle}z^{\bm{m}+\bm{p}_{\sigma}},\bm{z}^{\bm{q}_{\delta}}\rangle$
$\displaystyle=$
$\displaystyle\sum_{\sigma,\delta\in\Sigma_{d}}\operatorname{sgn}(\sigma)\operatorname{sgn}(\delta)\alpha_{\bm{q}_{\delta}-\bm{p}_{\sigma}}$
$\displaystyle=$
$\displaystyle\sum_{\sigma,\delta\in\Sigma_{d}}\operatorname{sgn}(\sigma)\operatorname{sgn}(\delta)\alpha_{(\bm{q}_{\sigma^{-1}\delta}-\bm{p})_{\sigma}}$
$\displaystyle=$
$\displaystyle\sum_{\sigma,\delta\in\Sigma_{d}}\operatorname{sgn}(\sigma)\operatorname{sgn}(\delta)\alpha_{\bm{q}_{\sigma^{-1}\delta}-\bm{p}}=d!\sum_{\sigma\in\Sigma_{d}}\operatorname{sgn}(\sigma)\alpha_{\bm{q}_{\sigma}-\bm{p}}.$

Our claim is that the above sum being zero for every $\bm{p}$ and $\bm{q}$ in
$\llbracket n\rrbracket$ implies $\alpha_{\bm{m}}=0$ for every
$\bm{m}\in\mathbb{Z}^{d}$. Since $\alpha_{\bm{m}}$ remains the same for
different permutation of $\bm{m}$, without loss of generality, we choose an
$\bm{m}=(m_{1},m_{2},\dots,m_{d})$ such that

$m_{1}\leq m_{2}\leq\cdots\leq m_{d}\leq r\;(\text{say}).$

By choice of $r$, one can assume it to be bigger or equal to $-1$. Choose
$\bm{q}=(r+d,\dots,r+1)$ and $\bm{p}=\bm{q}-\bm{m}$. Then both $\bm{p}$ and
$\bm{q}$ are in $\llbracket n\rrbracket$. With this choice we have from the
above inner product computation that

$\alpha_{\bm{m}}=-\sum_{e\neq\sigma\in\Sigma_{d}}\operatorname{sgn}(\sigma)\alpha_{\bm{q}_{\sigma}-\bm{p}},$

where $e$ is the identity permutation. Fix a positive integer $n$ and consider
the following element in $\llbracket n\rrbracket$

$\bm{n}=((d-1)n,\dots,n,0).$

Applying (3.5) to the choices $\bm{p}_{n}=\bm{p}+\bm{n}$ and
$\bm{q}_{n}=\bm{q}+\bm{n}$, we obtain

$\displaystyle\alpha_{m}=-\sum_{e\neq\sigma\in\Sigma_{d}}\operatorname{sgn}(\sigma)\alpha_{(\bm{q}_{\sigma}-\bm{p})+(\bm{n}_{\sigma}-\bm{n})}.$
(3.6)

Note that if $\sigma\neq e$, then at least one entry of
$\bm{n}_{\sigma}-\bm{n}$ is non-zero and therefore for a fixed $\sigma\neq e$,

$\beta(\sigma,n):=\alpha_{(\bm{q}_{\sigma}-\bm{p})+(\bm{n}_{\sigma}-\bm{n})}$

is a square-summable sequence. And therefore the linear combination

$\gamma_{n}:=-\sum_{e\neq\sigma\in\Sigma_{d}}\operatorname{sgn}(\sigma)\beta(\sigma,n)$

must also be square-summable. But from (3.6) we see that for every $n\geq 1$,
$\gamma_{n}=\alpha_{\bm{m}}$. This implies that $\alpha_{\bm{m}}=0$ which was
to prove.

∎

It is easy to see that the space $H^{\infty}(\mathbb{G}_{d})$ consisting of
all bounded analytic functions on $\mathbb{G}_{d}$ is contained in
$H^{2}(\mathbb{G}_{d})$. We identify $H^{\infty}(\mathbb{G}_{d})$ with its
boundary functions. In other words,

$H^{\infty}(\mathbb{G}_{d})=\\{\varphi\in
L^{\infty}(b\Gamma_{d}):\varphi\circ\bm{s}\text{ has no negative Fourier
coefficients}\\}$

######  Definition 3.9.

A Toeplitz operator with symbol $\varphi$ is called an analytic Toeplitz
operator if $\varphi$ is in $H^{\infty}(\mathbb{G}_{d})$. A Toeplitz operator
with symbol $\varphi$ is called a co-analytic Toeplitz operator if
$T_{\varphi}^{*}$ is an analytic Toeplitz operator.

Our next goal is to characterize analytic Toeplitz operators. But to be able
to do that we need to define the following notion and the proposition
following it.

######  Definition 3.10.

Let $\varphi$ be in $L^{\infty}(b\Gamma_{d})$. The operator
$H_{\varphi}:H^{2}(\mathbb{G}_{d})\to L^{2}(b\Gamma_{d})\ominus
H^{2}(\mathbb{G}_{d})$ defined by

$H_{\varphi}f=(I-Pr)M_{\varphi}f$

for all $f\in H^{2}(\mathbb{G}_{d})$, is called a Hankel operator.

We write down a few observations about Toeplitz operators some of which will
be used in the theorem following it. The proofs are not written because they
go along the same line as in the case of the unit disk.

######  Proposition 3.11.

Let $\varphi\in L^{\infty}(b\Gamma_{d})$. Then

  1. (1)

$T_{\varphi}^{*}=T_{\overline{\varphi}}$.

  2. (2)

If $\psi\in L^{\infty}(b\Gamma_{d})$ is another function, then the product
$T_{\varphi}T_{\psi}$ is another Toeplitz operator if $\overline{\varphi}$ or
$\psi$ is analytic. In each case, $T_{\varphi}T_{\psi}=T_{\varphi\psi}$.

  3. (3)

If $\psi\in L^{\infty}(b\Gamma_{d})$, then
$T_{\varphi}T_{\psi}-T_{\varphi\psi}=-H_{\overline{\varphi}}^{*}H_{\psi}.$

  4. (4)

If $T_{\varphi}$ is compact, then $\varphi=0$.

###  3.1. Characterizations of analytic Toeplitz operators

Now we are ready to characterize Toeplitz operators with analytic symbols. In
the proof of this result, we make use of the following notation:

$[z]=\\{\bm{m}=(m_{1},m_{2},\dots,m_{d}):m_{1}\geq m_{2}\geq\cdots\geq
m_{d}\\}.$

Note the difference between $[z]$ and $\llbracket z\rrbracket$, defined in §2.
For an $\bm{m}\in[z]$, we denote by $\bm{s}_{\bm{m}}$, the symmetrization of
the monomial $\bm{z}^{\bm{m}}$, i.e.,

$\bm{s}_{\bm{m}}(\bm{z})=\sum_{\sigma\in\Sigma_{d}}\bm{z}^{\bm{m}_{\sigma}}.$

It is easy to check that the set $\\{\bm{s}_{\bm{m}}:\bm{m}\in[z]\\}$ is an
orthogonal basis of $L^{2}_{sym}(\mathbb{T}^{d})$, the space consisting of the
symmetric functions in $L^{2}(\mathbb{T}^{d})$.

######  Theorem 3.12.

Let $T_{\varphi}$ be a Toeplitz operator. Then the following are equivalent:

  1. (i)

$T_{\varphi}$ is an analytic Toeplitz operator;

  2. (ii)

$T_{\varphi}$ commutes with $T_{p}$;

  3. (iii)

$T_{\varphi}(RanT_{p})\subseteq RanT_{p}$;

  4. (iv)

$T_{p}T_{\varphi}$ is a Toeplitz operator;

  5. (v)

For each $1\leq i\leq d-1$, $T_{\varphi}$ commutes with $T_{s_{i}}$;

  6. (vi)

For each $1\leq i\leq d-1$, $T_{s_{i}}T_{\varphi}$ is a Toeplitz operator.

###### Proof.

$(i)\Leftrightarrow(ii)$: The implication $(i)\Rightarrow(ii)$ is easy. To
prove the other direction, we note that if $T_{\varphi}$ commutes with
$T_{p}$, then by part $(3)$ of Proposition 3.11 we have
$H_{\overline{p}}^{*}H_{\varphi}=0$. This shows that the corresponding product
of Hankel operators on $H^{2}_{\text{anti}}(\mathbb{D}^{d})$ is also zero,
i.e., $H_{\overline{p}}^{*}H_{\varphi\circ\bm{s}}=0$. Let the power series
expansion of $\varphi\circ\bm{s}\in L^{\infty}_{\text{sym}}(\mathbb{T}^{d})$
be

$\varphi\circ\bm{s}(\bm{z})=\sum_{\bm{m}\in\mathbb{Z}^{d}}\alpha_{\bm{m}}\bm{z}^{m}$

with the convention that $\alpha_{\bm{m}_{\sigma}}=\alpha_{\bm{m}}$ for every
$\sigma\in\Sigma_{d}$. Let $\bm{p}=(p_{1},p_{2},\dots,p_{d})$ and
$\bm{q}=(q_{1},\dots,q_{d-1},0)$ be in $\llbracket n\rrbracket$. Then by a
similar computation as done in (3.5) we have

$\displaystyle 0=\langle
H_{\varphi\circ\bm{s}}\bm{a_{p}},H_{\overline{p}}\bm{a_{q}}\rangle_{L^{2}(\mathbb{T}^{d})}$
$\displaystyle=$
$\displaystyle\langle\sum_{\bm{m}\in[z]}\alpha_{\bm{m}}\bm{z}^{\bm{m}}\bm{a_{p}},\bm{a}_{\bm{q}^{\prime}}\rangle_{L^{2}(\mathbb{T}^{d})}\;\;\;[\bm{q}^{\prime}:=\bm{q}-(1,1,\dots,1)]$
(3.7) $\displaystyle=$ $\displaystyle
d!\sum_{\sigma\in\Sigma_{d}}\operatorname{sgn}(\sigma)\alpha_{\bm{q}^{\prime}_{\sigma}-\bm{p}}.$

Since the last entry of $\bm{q}^{\prime}$ is $-1$, at least one entry of
$\bm{q}^{\prime}_{\sigma}-\bm{p}$ is negative for every $\bm{p}\in\llbracket
n\rrbracket$ and $\sigma\in\Sigma_{d}$. Now following a similar line of
analysis done in the proof of Lemma 3.8, one can conclude from 3.7 that
$\alpha_{\bm{m}}=0$ whenever $\bm{m}\in[z]$ with the last entry negative.
Hence by the symmetric property of $\varphi\circ\bm{s}$, $\alpha_{\bm{m}}=0$,
if any coordinates of $\bm{m}$ is negative, in other words, $\varphi$ is
analytic in $\mathbb{G}_{d}$.

$(ii)\Leftrightarrow(iii)$: The part $(ii)\Rightarrow(iii)$ is easy.
Conversely, suppose that $RanT_{p}$ is invariant under $T_{\varphi}$. Since
$RanT_{p}$ is closed, we have for every $f\in H^{2}(\mathbb{G}_{d})$,

$\displaystyle T_{\varphi}T_{p}f=T_{p}g_{f}\text{ for some $g_{f}$ in
$H^{2}(\mathbb{G}_{d})$.}$ $\displaystyle\Rightarrow$ $\displaystyle
T_{p}^{*}T_{\varphi}T_{p}f=g_{f}\Rightarrow T_{\varphi}f=g_{f}\;(\text{by the
Brown--Halmos relations \eqref{Toeplitzcharc}}).$

Hence $T_{\varphi}T_{p}=T_{p}T_{\varphi}$.

$(ii)\Leftrightarrow(iv)$: If $T_{\varphi}$ commutes with $T_{p}$, then
$T_{p}T_{\varphi}$ is same as $T_{\varphi}T_{p}$, which is a Toeplitz operator
by Proposition 3.11. Conversely, if $T_{p}T_{\varphi}$ is a Toeplitz operator,
then it satisfies Brown–Halmos relations, the second one of which implies that
$T_{\varphi}$ commutes with $T_{p}$.

$(i)\Leftrightarrow(v)$: For an analytic symbol $\varphi$, $T_{\varphi}$
obviously commutes with each $T_{s_{i}}$. The proof of the converse direction
is done by the same technique as in the proof of $(ii)\Rightarrow(i)$. If
$T_{\varphi}$ commutes with $T_{s_{i}}$, then by part $(3)$ of Proposition
3.11 we have $H_{\overline{s_{i}}}^{*}H_{\varphi}=0$. Suppose
$\varphi\circ\bm{s}\in L^{\infty}_{\text{sym}}(\mathbb{T}^{d})$ has the
following power series expansion

$\varphi\circ\bm{s}(\bm{z})=\sum_{\bm{m}\in\mathbb{Z}^{d}}\alpha_{\bm{m}}\bm{z}^{\bm{m}}.$

Let $\bm{q}=(q_{1},\dots,q_{d-1},0)$ and $\bm{p}=(p_{1},p_{2},\dots,p_{d})$ be
in $\llbracket n\rrbracket$ such that $q_{j}-q_{j-1}=1$ for every $2\leq j\leq
d-1$. Note that for this choice of $\bm{q}$, the only non analytic term in the
expression of $\overline{s_{i}}\bm{a_{q}}(\bm{z})$ is $\bm{a_{q^{\prime}}}$,
where

$\bm{q}^{\prime}=\bm{q}-(0,\dots,0,\underbrace{1,\dots,1}_{i-\text{times}}).$

A similar computation as in (3.5) yields

$\displaystyle 0=\langle
H_{\varphi\circ\bm{s}}\bm{a_{p}},H_{\overline{s_{i}}}\bm{a_{q}}\rangle_{L^{2}(\mathbb{T}^{d})}=\langle
H_{\varphi\circ\bm{s}}\bm{a_{p}},\bm{a_{q^{\prime}}}\rangle_{L^{2}(\mathbb{T}^{d})}=d!\sum_{\sigma\in\Sigma_{d}}\operatorname{sgn}(\sigma)\alpha_{\bm{q}^{\prime}_{\sigma}-\bm{p}}.$

By a similar analysis as in the proof of $(ii)\Rightarrow(i)$ above, one can
conclude that $\alpha_{\bm{m}}=0$ whenever $\bm{m}$ has at least one negative
entry. This proves that $\varphi$ is analytic.
$(v)\Leftrightarrow(vi)$: The implication $(v)\Rightarrow(vii)$ follows from
Proposition 3.11. Conversely suppose that $T_{s_{i}}T_{\varphi}$ is a Toeplitz
operator. Therefore applying Theorem 3.4 and Lemma 3.3, we get
$T_{\varphi}T_{s_{i}}={T^{*}_{s_{d-i}}}T_{\varphi}T_{p}={T_{p}}^{*}T_{s_{i}}T_{\varphi}T_{p}=T_{s_{i}}T_{\varphi}.$
∎

######  Remark 3.13.

Since adjoint of an analytic Toeplitz operator is a co-analytic Toeplitz
operator and vice versa, Theorem 3.12 in turn characterizes co-analytic
Toeplitz operators as well.

##  4\. Compact perturbation of Toeplitz operators

In this section we first find a characterization of compact operators on
$H^{2}(\mathbb{G}_{d})$ and then use it to characterize compact perturbation
of Toeplitz operators – the so called asymptotic Toeplitz operators.

Note that for a bounded operator $B$ on $H^{2}(\mathbb{D})$, $B$ is a Toeplitz
operator if and only if there exists $T$ on $H^{2}(\mathbb{D})$ such that
${T_{z}^{*}}^{n}TT_{z}^{n}\to B$ weakly on $H^{2}(\mathbb{D})$. Although
$T_{p}$ of $H^{2}(\mathbb{G}_{d})$ is unitarily equivalent to $T_{z}$ on
$H^{2}_{\mathcal{E}}(\mathbb{D})$, a similar convergence like above does not
guarantee that the limit is a Toeplitz operator, see Example 3.7. A necessary
and sufficient condition for the limit to be Toeplitz is given below.

######  Lemma 4.1.

Let $T$ and $B$ be bounded operators on $H^{2}(\mathbb{G}_{d})$ such that
${T_{p}^{*}}^{n}TT_{p}^{n}\to B$ weakly. Then $B$ is Toeplitz operator if and
only if for each $1\leq i\leq d-1$

${T_{p}^{*}}^{n}[T,T_{s_{i}}]T_{p}^{n}\to 0\text{ weakly,}$

where $[T,T_{s_{i}}]$ denotes the commutator of $T$ and $T_{s_{i}}$.

###### Proof.

Note that if $T$ and $B$ are bounded operators on $H^{2}(\mathbb{G}_{d})$ such
that ${T_{p}^{*}}^{n}TT_{p}^{n}\to B$ weakly, then $T_{p}^{*}BT_{p}=B$.
Suppose ${T_{p}^{*}}^{n}[T,T_{s}]T_{p}^{n}\to 0$ weakly. To prove that $B$ is
Toeplitz, it remains to show that $B$ satisfies the other Brown–Halmos
relations with respect to the $\Gamma_{d}$-isometry
$(T_{s_{1}},\dots,T_{s_{d-1}},T_{p})$. For any $i$ with $1\leq i\leq d-1$

$\displaystyle T_{s_{i}}^{*}BT_{p}=\text{w-}\\!\lim
T_{s_{i}}^{*}({T_{p}^{*}}^{n}TT_{p}^{n})T_{p}=\text{w-}\\!\lim{T_{p}^{*}}^{n}(T_{s_{i}}^{*}TT_{p})T_{p}^{n}=\text{w-}\\!\lim{T_{p}^{*}}^{n+1}T_{s_{d-i}}TT_{p}^{n+1}$
$\displaystyle=$
$\displaystyle\text{w-}\\!\lim{T_{p}^{*}}^{n+1}(T_{s_{d-i}}T-TT_{s_{d-i}}+TT_{s_{d-i}})T_{p}^{n+1}=\text{w-}\\!\lim{T_{p}^{*}}^{n+1}TT_{p}^{n+1}T_{s_{d-i}}=BT_{s_{d-i}}.$

Conversely, suppose that the weak limit $B$ of ${T_{p}^{*}}^{n}TT_{p}^{n}$ is
a Toeplitz operator and hence satisfies the Brown–Halmos relations. Thus,

$\displaystyle\text{w-}\\!\lim{T_{p}^{*}}^{n}(TT_{s_{i}}-T_{s_{i}}T)T_{p}^{n}=\text{w-}\\!\lim({T_{p}^{*}}^{n}TT_{p}^{n}T_{s_{i}}-{T_{p}^{*}}^{n}T_{s_{d-i}}^{*}T_{p}TT_{p}^{n})$
$\displaystyle=$
$\displaystyle\text{w-}\\!\lim({T_{p}^{*}}^{n}TT_{p}^{n}T_{s_{i}}-T_{s_{d-i}}^{*}{T_{p}^{*}}^{d-1}TT_{p}^{d-1}T_{p})=BT_{s_{i}}-T_{s_{d-i}}^{*}BT_{p}=0,$

for all $1\leq i\leq d-1$. ∎

Before we characterize the compact operators on $H^{2}(\mathbb{G}_{d})$, we
recall the analogous result for the polydisk, discovered in [21]. The one
dimensional case was proved by Feintuch [18].

For $m\geq 1$, define a completely positive map
$\eta_{m}:\mathcal{B}(H^{2}(\mathbb{D}^{d})\rightarrow\mathcal{B}(\oplus_{i=1}^{n}H^{2}(\mathbb{D}^{d}))$
by

$\eta_{m}(T)=\begin{bmatrix}T_{z_{1}}^{*m}\\\ T_{z_{2}}^{*m}\\\ \vdots\\\
T_{z_{d}}^{*m}\end{bmatrix}T\begin{bmatrix}T_{z_{1}}^{m},&T_{z_{2}}^{m},&\cdots,&T_{z_{d}}^{m}\\\
\end{bmatrix}.$

######  Theorem 4.2.

A bounded operator $T$ on $H^{2}(\mathbb{D}^{d})$ is compact if and only if
$\eta_{m}(T)\to 0$ in norm as $m\to\infty$.

This shows the importance of the forward shifts in characterizing the compact
operators.

For $1\leq j\leq d-1$, let $Y_{j}$ be the operator on
$H^{2}_{\text{anti}}(\mathbb{D}^{d})$ as defined in (3.4). Let $X_{j}$ be the
restriction of $Y_{j}$ to $\mathcal{E}$, the coefficient space of the
symmetrized polydisk, i.e., for every $\bm{p}\in\llbracket n\rrbracket$ such
that $\bm{p}=(p_{1},\dots,p_{d-1},0)$,

$X_{j}\bm{a_{p}}=\bm{{a_{p+f_{j}}}},\text{ where
}\bm{f_{j}}=(\underbrace{1,\dots,1}_{j\text{-times}},0\dots,0).$

Then each $Y_{j}$ on $H^{2}_{\text{anti}}(\mathbb{D}^{d})$ has the following
expression:

$Y_{j}=\sum_{r=0}^{\infty}T_{p}^{r}X_{j}T_{p}^{*r}.$

It should be noted that each $Y_{j}$ is a pure isometry and the set of
operators $\\{Y_{1},Y_{2},\dots,Y_{d-1}\\}$ is doubly commuting. Consider the
following operator on $\mathcal{E}$

$E_{l}:=(P_{\mathcal{E}}-X_{1}^{l}X_{1}^{*l})(P_{\mathcal{E}}-X_{2}^{l}X_{2}^{*l})\cdots(P_{\mathcal{E}}-X_{d-1}^{l}X_{d-1}^{*l}).$

Since the operators $X_{1},X_{2},\cdots,X_{d-1}$ are doubly commuting, the
operator $E_{l}$ is an orthogonal projection of $\mathcal{E}$ onto the $l$
dimensional space

$\bigcap_{j=1}^{d-1}\ker
X_{j}^{*l}=\text{span}\\{\bm{a}_{\bm{p}}:\bm{p}=\big{(}k+(d-2)l,k+(d-3)l,\dots,k,0\big{)},\;1\leq
k\leq l\\}.$

Also note that

$\displaystyle E_{l}=$ $\displaystyle
P_{\mathcal{E}}-\sum_{j=1}^{d-1}X_{j}^{l}X_{j}^{*l}+\sum_{1\leq
j_{1}<j_{2}\leq d-1}X_{j_{1}}^{l}X_{j_{2}}^{l}X_{j_{1}}^{*l}X_{j_{2}}^{*l}+$
$\displaystyle\cdots+(-1)^{k}\sum_{1\leq j_{1}<\cdots<j_{k}\leq
d-1}X_{j_{1}}^{l}\cdots X_{j_{k}}^{l}X_{j_{1}}^{*l}\cdots
X_{j_{k}}^{*l}+\cdots+(-1)^{d-1}X_{1}^{l}\cdots X_{n}^{l}X_{1}^{*l}\cdots
X_{n}^{*l}.$

We have the following characterization of compact operators on
$H^{2}(\mathbb{G}_{d})$ using the set of pure isometries
$\\{Y_{1},\dots,Y_{d-1},T_{p}\\}$.

######  Theorem 4.3.

For $j\geq 1$, define completely positive maps

$\eta_{j}:\mathcal{B}(H^{2}(\mathbb{G}_{d}))\to\mathcal{B}(\oplus_{i=1}^{n}H^{2}(\mathbb{G}_{d}))$

by

$\eta_{j}(T)=\begin{bmatrix}Z_{1}^{*j}\\\ \vdots\\\ Z_{d-1}^{*j}\\\
T_{p}^{*j}\\\ \end{bmatrix}T\begin{bmatrix}Z_{1}^{j}\cdots
Z_{d-1}^{j}T_{p}^{n}\\\ \end{bmatrix},$

where $Z_{l}=U^{*}Y_{l}U$. A bounded operator $T$ on $H^{2}(\mathbb{G}_{d})$
is compact if and only if $\eta_{j}(T)\to 0$ in norm as $j\to\infty$.

###### Proof.

The necessity follows from a straightforward application of Lemma 3.1 of [21],
which states that if $P$ is a pure contraction, $T$ is a contraction and $K$
is a compact operator on a Hilbert space, then $P^{*l}KT\to 0$ in the operator
norm as $l\to\infty$.

Conversely, suppose that $T$ is a bounded operator on $H^{2}(\mathbb{G}_{d})$
satisfying the convergence condition. We shall conclude by finding a finite
rank operators approximation of $T$. To that end note that
$U^{*}P_{\mathcal{E}}U=U^{*}(I-T_{p}T_{p}^{*})U=I-T_{p}T_{p}^{*}$ and

$\displaystyle
U^{*}E_{l}U=(I-T_{p}T_{p}^{*})-\sum_{k=1}^{d-1}(-1)^{k}\sum_{1\leq
j_{1}<\cdots<j_{k}\leq d-1}W_{j_{1}}^{l}\cdots
W_{j_{k}}^{l}W_{j_{1}}^{*l}\cdots W_{j_{k}}^{*l},$ (4.1)

where $W_{j}=U^{*}X_{j}U$. For each positive integer $l$, consider the
following finite rank operators on $H^{2}(\mathbb{G}_{d})$:

$F_{l}=U^{*}\left(E_{l}+T_{p}E_{l}T_{p}^{*}+\cdots+T_{p}^{l-1}E_{l}T_{p}^{*l-1}\right)U.$

Using (4.1) we get $F_{l}$ to be the same as

$\displaystyle(I-T_{p}^{l}T_{p}^{*l})-\sum_{r=0}^{l-1}T_{p}^{r}\left(\sum_{j=1}^{d-1}W_{j}^{l}W_{j}^{*l}\right)T_{p}^{*r}+\cdots$
$\displaystyle+(-1)^{k}\sum_{r=0}^{l-1}T_{p}^{r}\left(\sum_{1\leq
j_{1}<\cdots<j_{k}\leq d-1}W_{j_{1}}^{l}\cdots
W_{j_{k}}^{l}W_{j_{1}}^{*l}\cdots W_{j_{k}}^{*l}\right)T_{p}^{*r}+\cdots$
$\displaystyle+(-1)^{d-1}\sum_{r=0}^{l-1}T_{p}^{r}\left(W_{1}^{l}\cdots
W_{n}^{l}W_{1}^{*l}\cdots W_{n}^{*l}\right)T_{p}^{*r}.$

Let $P_{l}$ be the projection of $H^{2}(\mathbb{G}_{d})$ onto the space $\ker
T_{p}^{*l-1}$. Since $Z_{j}=\sum_{r=0}^{\infty}T_{p}^{r}W_{j}T_{p}^{*r}$,

$\displaystyle I-F_{l}$ $\displaystyle=$
$\displaystyle{T_{p}}^{l}{T_{p}}^{*l}+\sum_{j=1}^{d-1}P_{l}Z_{j}^{l}Z_{j}^{*l}P_{l}+\cdots+(-1)^{k}\sum_{1\leq
j_{1}<\cdots<j_{k}\leq d-1}P_{l}Z_{j_{1}}^{l}\cdots
Z_{j_{k}}^{l}Z_{j_{1}}^{*l}\cdots Z_{j_{k}}^{*l}P_{l}$
$\displaystyle+(-1)^{n}P_{l}Z_{1}^{l}\cdots Z_{d-1}^{l}Z_{1}^{*l}\cdots
Z_{d-1}^{*l}P_{l}.$

Then the operator $\tilde{F}_{l}=TF_{l}+F_{l}T-F_{l}TF_{l}$ is also a finite
rank operator and note that $T-\tilde{F}_{l}=(I-F_{l})T(I-F_{l})$. By the
above form of $(I-F_{l})$ and from the hypotheses it follows that
$\|T-\tilde{F}_{l}\|\to 0$. Hence, $T$ is compact. ∎

######  Definition 4.4.

A bounded operator $T$ on $H^{2}(\mathbb{G}_{d})$ is called an asymptotic
Toeplitz operator if $T=T_{\varphi}+K$ for some $\varphi\in
L^{\infty}(b\Gamma)$ and compact operator $K$ on $H^{2}(\mathbb{G}_{d})$.

We end this section with the following characterization of asymptotic Toeplitz
operators.

######  Theorem 4.5.

A bounded operator $T$ on $H^{2}(\mathbb{G}_{d})$ is an asymptotic Toeplitz
operator if and only if ${T_{p}^{*}}^{n}[T,T_{s}]T_{p}^{n}\to 0$,
${T_{p}^{*}}^{n}TT_{p}^{n}\to B$ and $\eta_{d}(T-B)\to 0$.

###### Proof.

If $T$ satisfies the convergence conditions as in the statement, then it
follows from Lemma 4.1 that $B$ is a Toeplitz operator because
${T_{p}^{*}}^{n}[T,T_{s}]T_{p}^{n}\to 0$. Also, since $\eta_{d}(T-B)\to 0$, by
Theorem 4.3, $T-B$ is a compact operator. Hence $T$ is the sum of a compact
operator and a Toeplitz operator.

Conversely, suppose $T$ is an asymptotic Toeplitz operator, i.e.,
$T=K+T_{\varphi}$, where $K$ is some compact operator. Then by Theorem 4.3,
${T_{p}^{*}}^{n}TT_{p}^{n}\to T_{\varphi}$. Since $T_{\varphi}$ is Toeplitz,
by Lemma 4.1, ${T_{p}^{*}}^{n}[T,T_{s}]T_{p}^{n}\to 0$. And finally, since $K$
is compact, by Theorem 4.3, $\eta_{d}(T-T_{\varphi})\to 0$. ∎

##  5\. Generalized Toeplitz operators

In this section we briefly describe the operator theory of the symmetrized
polydisk. This was introduced in [10]. We then extend the notion of Toeplitz
operators to what we call an $\underline{S}$-Toeplitz operator, where a
commuting $d$-tuple $\underline{S}$ is an ‘isometry’ associated to the
symmetrized polydisk, to be formally defined below.

######  Definition 5.1.

A $d$-tuple $\underline{T}=(T_{1},\dots,T_{d-1},P)$ of commuting bounded
operators on a Hilbert space $\mathcal{H}$ is called a
$\Gamma_{d}$-contraction, if $\Gamma_{d}$ is a spectral set for
$\underline{T}$, i.e., (since $\Gamma_{d}$ is polynomially convex) for every
polynomial $f$ in $d$ commuting variables

$\|f(T_{1},\dots,T_{d-1},P)\|\leq\sup_{\bm{z}\in\Gamma_{d}}|f(\bm{z})|=:\|f\|_{\infty,\Gamma_{d}}.$

It is clear from the definition that $(T_{1},\dots,T_{d-1},P)$ is a
$\Gamma_{d}$-contraction if and only if $(T_{1}^{*},\dots,T_{d-1}^{*},P^{*})$
is so. It is also clear that the last component $P$ of a
$\Gamma_{d}$-contraction is a contraction, which follows by applying the
definition of a $\Gamma_{d}$-contraction to the particular function that
projects $\mathbb{C}^{d}$ to the last component. A profound study of
$\Gamma_{2}$-contraction – initiated by Agler and Young – is what prompted the
study of such a tuple. There are natural $\Gamma_{d}$-analogues of unitaries
and isometries.

######  Definition 5.2.

A $d$-tuple $\underline{R}=(R_{1},\dots,R_{d-1},U)$ of commuting bounded
normal operators on a Hilbert space is said to be a $\Gamma_{d}$-unitary, if
the Taylor joint spectrum of $\underline{R}$, $\sigma(\underline{R})$ is
contained in the distinguished boundary of $\Gamma_{d}$.

A $\Gamma_{d}$-isometry is a $d$-tuple $\underline{S}=(S_{1},\dots,S_{d-1},V)$
of commuting bounded operators that has a $\Gamma_{d}$-unitary extension.

Let us recall that the projection of the distinguished boundary of
$\Gamma_{d}$ to the last component is $\mathbb{T}$. Therefore by definition,
the last entry of a $\Gamma_{d}$-isometry and a $\Gamma_{d}$-unitary are an
isometry and a unitary, respectively. The following characterizations of
$\Gamma_{d}$-unitaries and $\Gamma_{d}$-isometries will be used.

######  Theorem 5.3 (Theorem 4.2 [10]).

Let $(R_{1},\dots,R_{d-1},U)$ be a pair of commuting operators defined on a
Hilbert space $\mathcal{H}.$ Then the following are equivalent:

  1. (1)

$(R_{1},\dots,R_{d-1},U)$ is a $\Gamma_{d}$-unitary;

  2. (2)

there exist commuting unitary operators $U_{1},U_{2}\dots,U_{d}$ on
$\mathcal{H}$ such that for $1\leq i\leq d$,

$R_{i}=\sum_{1\leq k_{1}<\cdots<k_{i}\leq d}U_{k_{1}}U_{k_{2}}\cdots
U_{k_{i}};$

  3. (3)

$U$ is unitary, $R_{d-i}=R_{i}^{*}U,$ and
$(\gamma_{1}R_{1},\gamma_{2}R_{2},\dots,\gamma_{d-1}R_{d-1})$ is a
$\Gamma_{d-1}$-contraction where $\gamma_{i}=\frac{d-i}{d}$ for
$i=1,2,\dots,d-1$.

######  Theorem 5.4 (Theorem 4.12 [10]).

Let $(S_{1},\dots,S_{d-1},V)$ be a pair of commuting operators defined on a
Hilbert space $\mathcal{H}.$ Then the following are equivalent:

  1. (1)

$(S_{1},\dots,S_{d-1},V)$ is a $\Gamma_{d}$-isometry;

  2. (2)

$V$ is isometry, $S_{d-i}=S_{i}^{*}V,$ and
$(\gamma_{1}S_{1},\gamma_{2}S_{2},\dots,\gamma_{d-1}S_{d-1})$ is a
$\Gamma_{d-1}$-contraction where $\gamma_{i}=\frac{d-i}{d}$ for
$i=1,2,\dots,d-1$.

The tuple $(T_{s_{1}},\dots,T_{s_{d-1}},T_{p})$ of multiplication by
coordinate functions on $H^{2}(\mathbb{G}_{d})$ is a prototype example of a
$\Gamma_{d}$-isometry. By Theorem 3.4, Toeplitz operators on
$H^{2}(\mathbb{G})$ are precisely those that satisfy the Brown–Halmos
relations

$T_{s_{i}}^{*}TT_{p}=TT_{s_{d-i}}\text{ and }T_{p}^{*}TT_{p}=T\text{ for each
}i=1,2\dots,d-1.$

This motivates us to consider the following more general situation.

######  Definition 5.5.

Given a $\Gamma_{d}$-isometry $\underline{S}=(S_{1},\cdots,S_{d-1},V)$ acting
on a Hilbert space ${\mathcal{H}}$, a bounded operator $X$ on ${\mathcal{H}}$
is called an $\underline{S}$-Toeplitz operator if it satisfies the
Brown–Halmos relation with respect to the $\Gamma_{d}$-isometry
$\underline{S}$, i.e.,

${S_{i}}^{*}XV=XS_{d-i}\text{ and }V^{*}XV=X\text{ for each }i=1,2\dots,d-1.$

The following is the main result of this section.

######  Theorem 5.6.

Let $\underline{S}=(S_{1},\dots,S_{d-1},V)$ be a $\Gamma_{d}$-isometry on a
Hilbert space $\mathcal{H}$. Then

  1. (1)

Extension: There exists a $\Gamma_{d}$-unitary
$\underline{R}=(R_{1},\dots,R_{d-1},U)$ acting on a Hilbert space
$\mathcal{K}$ containing $\mathcal{H}$ such that $\underline{R}$ is the
minimal extension of $\underline{S}$. In fact,

$\displaystyle\mathcal{K}=\overline{\operatorname{span}}\\{U^{m}h:h\in\mathcal{H},\text{
and }m\in\mathbb{Z}\\}.$ (5.1)

Moreover, any operator $X$ acting on $\mathcal{H}$ commutes with
$\underline{S}$ if and only if $X$ has a unique norm preserving extension $Y$
acting on $\mathcal{K}$ commuting with $\underline{R}$;

  2. (2)

Halmos dilation: An operator $X$ is an $\underline{S}$-Toeplitz operator if
and only if there exists a unique norm-preserving Halmos dilation of $X$ in
$\\{R_{1},\dots,R_{d-1},U\\}^{\prime}$, i.e., there exists a unique operator
$Y$ in $\\{R_{1},\dots,R_{d-1},U\\}^{\prime}$ such that

$\displaystyle\|X\|=\|Y\|\text{ and }X=P_{\mathcal{H}}Y|_{\mathcal{H}};$ (5.2)

  3. (3)

Extension of $C^{*}(\underline{R})$: Let $\mathcal{C}^{*}(\underline{S})$ and
$\mathcal{C}^{*}(\underline{R})$ denote the unital $\mathcal{C}^{*}$-algebras
generated by $\\{S_{1},\dots,S_{d-1},V\\}$ and $\\{R_{1},\dots,R_{d-1},U\\}$,
respectively and $\mathcal{I}(\underline{S})$ denote the closed ideal of
$\mathcal{C}^{*}(\underline{S})$ generated by all the commutators $XY-YX$ for
$X,Y\in\mathcal{C}^{*}(\underline{S})\cap\mathcal{T}(\underline{S})$. Then
there exists a short exact sequence

$0\rightarrow\mathcal{I}(\underline{S})\hookrightarrow\mathcal{C}^{*}(\underline{S})\xrightarrow{\pi_{0}}\mathcal{C}^{*}(\underline{R})\rightarrow
0$

with a completely isometric cross section, where
$\pi_{0}:\mathcal{C}^{*}(\underline{S})\to\mathcal{C}^{*}(\underline{R})$ is
the canonical unital $*$-homomorphism which sends the generating set
$\underline{S}$ to the corresponding generating set $\underline{R}$, i.e.,
$\pi_{0}(V)=U$ and $\pi_{0}(S_{i})=R_{i}$ for all $1\leq i\leq d-1$.

###### Proof.

The main idea is to apply Theorem 1.2 of Prunaru [28] to the isometry part $V$
of the $\Gamma_{d}$-isometry $\underline{S}=(S_{1},\dots,S_{d-1},V)$ and then
to progress with a similar line of analysis as in Prunaru’s proof of his
theorem for spherical isometries to conclude the proof of the theorem.

The starting point is the completely positive and unital map
$\Phi_{0}:{\mathcal{B}}({\mathcal{H}})\to{\mathcal{B}}({\mathcal{H}})$ defined
as

$\Phi_{0}:X\to V^{*}XV.$

We note that $\Phi_{0}$ is a normal mapping, i.e.,
$\Phi_{0}(A_{i})\to\Phi_{0}(A)$ in the strong operator topology for every net
$\\{A_{i}\\}_{i\in I}$ of increasing operators that goes to $A$ in the strong
operator topology. Thus, the map $\Phi_{0}$ satisfies all the hypotheses of
Lemma 2.3 of [28], a simpler version of which states that if $\Phi_{0}$ is a
completely positive, unital and normal mapping on
${\mathcal{B}}({\mathcal{H}})$, then there exists an idempotent completely
positive map $\Phi$ on ${\mathcal{B}}({\mathcal{H}})$ such that

${\mathcal{T}}(V):=\\{X:\Phi_{0}(X)=X\\}=\operatorname{Ran}\Phi.$

Now using a well-known result of Choi–Effros [14] for the idempotent unital
completely positive map $\Phi$, one has

$\Phi(\Phi(X)Y)=\Phi(X\Phi(Y))=\Phi(\Phi(X)\Phi(Y))\quad(X,Y\in\mathcal{B}({\mathcal{H}})).$
(5.3)

Let us denote by $\Phi_{\nu}$ the restriction of $\Phi$ to
$C^{*}({\mathcal{T}}(V))$, the $C^{*}$-algebra generated by
${\mathcal{T}}(V)$. Then it readily follows from (5.3) that
$\operatorname{Ker}\Phi_{\nu}$ is an ideal. Let $(\Pi,\pi,{\mathcal{K}})$
denote the minimal Stinespring dilation of $\Phi_{\nu}$, i.e.,
$\Pi:{\mathcal{H}}\to{\mathcal{K}}$ is an isometry (since $\Phi_{\nu}$ is
unital) and

$\displaystyle\Phi_{\nu}(X)=\Pi^{*}\pi(X)\Pi\quad\text{ for all $X\in
C^{*}({\mathcal{T}}(V))$}.$ (5.4)

As a consequence of (5.3) and the minimality of the Stinespring dilation
$(\Pi,\pi,{\mathcal{K}})$, we get

$\operatorname{Ker}\Phi_{\nu}=\operatorname{Ker}\pi.$

Indeed, from (5.4) the inclusion
$\operatorname{Ker}\Phi_{\nu}\supset\operatorname{Ker}\pi$ is clear, while for
the other inclusion we note using (5.4) that for any $X,Y,Z$ in
$C^{*}({\mathcal{T}}(V))$ and $h,k$ in ${\mathcal{H}}$

$\displaystyle\langle\Phi_{\nu}(X^{*}YZ)h,k\rangle=\langle\Pi^{*}\pi(X^{*}YZ)\Pi
h,k\rangle=\langle\pi(Y)\pi(Z)\Pi(h),\pi(X)\Pi(k)\rangle.$

This implies that if $Y$ in $C^{*}({\mathcal{T}}(V))$ is such that
$\Phi_{\nu}(Y)=0$, then since the kernel of $\Phi_{\nu}$ is an ideal,
$\Phi_{\nu}(XYZ)=0$ and hence $\pi(Y)=0$ because by minimality of the dilation

$\overline{\operatorname{span}}\\{\pi(X)\Pi h:X\in
C^{*}({\mathcal{T}}(V)),\;h\in{\mathcal{H}}\\}={\mathcal{K}}.$

Let us now define the mapping
$\rho:\pi(C^{*}({\mathcal{T}}(V)))\to{\mathcal{B}}({\mathcal{H}})$ as

$\displaystyle\rho(\pi(X))=\Pi^{*}\pi(X)\Pi=\Phi_{\nu}(X).$ (5.5)

Since $\Phi_{\nu}$ is idempotent, it follows that for every $X$ in
$C^{*}({\mathcal{T}}(V))$, $\Phi_{\nu}$ kills $X-\Phi_{\nu}(X)$ and therefore
so does $\pi$. Using this fact, one easily deduces that

$\pi\circ\rho=I_{\operatorname{Ran}\pi}$

and the mapping $\rho$ is a complete isometry. We now note down two further
crucial properties of the Stinespring dilation $(\Pi,\pi,{\mathcal{K}})$ which
can be derived from what we discussed above. This can also be seen from the
proof of Theorem 1.2 in [28] when applied to the case of a single isometry
$V$.

  1. ($\textbf{P}_{1}$)

The operator

$U:=\pi(V),$

is the minimal unitary extension of $V$, i.e., ${\mathcal{K}}$ is as in (5.1);

  2. ($\textbf{P}_{2}$)

if $X$ commutes with $V$, then $\pi(X)$ is the unique operator that commutes
with $U$, $\|\pi(X)\|=\|X\|$ and $\pi(X)|_{\Pi({\mathcal{H}})}=\Pi
X\Pi^{*}|_{\Pi({\mathcal{H}})}$;

We shall identify ${\mathcal{H}}$ with $\Pi{\mathcal{H}}$ and therefore view
${\mathcal{H}}$ as a subspace of ${\mathcal{K}}$. To prove item (1) of the
theorem, we proceed by defining

$R_{j}=\pi(S_{j})\quad\text{for each }j=1,2,\dots,d-1.$

Commutativity of the tuple $(R_{1},\dots,R_{d-1},U)$ follows from that of
$(S_{1},\dots,S_{d-1},V)$. We shall use item (3) of Theorem 5.3 to prove that
$\underline{R}=(R_{1},\dots,R_{d-1},U)$ is a $\Gamma_{d}$-unitary. By
$({\bf{P_{1}}})$ we know that $U$ is a unitary. We also see that for each
$j=1,2,\dots,d-1$

$R^{*}_{j}U=\pi(S_{j}^{*})\pi(V)=\pi(S_{j}^{*}V)=\pi(S_{d-j})=R_{d-j},$

where we used item (2) of Theorem 5.4. Now to see that

$(\frac{d-1}{d}R_{1},\frac{d-2}{d}R_{2},\ldots,\frac{1}{d}R_{d-1})$

is a $\Gamma_{d-1}$-contraction, we choose any polynomial $f$ in $d-1$
variables, and compute

$\displaystyle\|f(\frac{d-1}{d}R_{1},\frac{d-2}{d}R_{2},\ldots,\frac{1}{d}R_{d-1})\|$
$\displaystyle=$
$\displaystyle\|f(\frac{d-1}{d}\pi(S_{1}),\frac{d-2}{d}\pi(S_{2}),\ldots,\frac{1}{d}\pi(S_{d-1}))\|$
$\displaystyle=$
$\displaystyle\|\pi(f(\frac{d-1}{d}S_{1},\frac{d-2}{d}S_{2},\ldots,\frac{1}{d}S_{d-1}))\|\quad[\mbox{because
}\pi\mbox{ is multiplicative}]$ $\displaystyle\leq$
$\displaystyle\|f(\frac{d-1}{d}S_{1},\frac{d-2}{d}S_{2},\ldots,\frac{1}{d}S_{d-1})\|\quad[\mbox{because
}\pi\mbox{ is contractive}].$

Now we use Lemma 2.7 of [10] which states that
$(\frac{d-1}{d}S_{1},\frac{d-2}{d}S_{2},\ldots,\frac{1}{d}S_{d-1})$ is a
$\Gamma_{d-1}$-contraction whenever $(S_{1},\ldots,S_{d-1},P)$ is a
$\Gamma_{d}$-contraction to conclude that the last line of the displayed
equations above is dominated by the supremum of $|f|$ over the set
$\Gamma_{d-1}$. Consequently, we have shown that
$(\frac{d-1}{d}R_{1},\frac{d-2}{d}R_{2},\ldots,\frac{1}{d}R_{d-1})$ is a
$\Gamma_{d-1}$-contraction.

The minimality of the extension $\underline{R}=(R_{1},\dots,R_{d-1},U)$
follows from $({\bf{P_{1}}})$.

For the moreover part in item (1), let $X$ be an operator on $\mathcal{H}$
which commutes with $\underline{S}$. Set $Y:=\pi(X)$. By $({\bf{P_{2}}})$
above, we know that $Y$ is the unique operator that commutes with $U$ and is a
norm preserving extension of $X$. Furthermore, $Y$ commutes with each of
$R_{j}$ as $X$ commutes with each of $S_{j}$. Conversely, if $X$ in
${\mathcal{B}}({\mathcal{H}})$ has an extension that commutes with
$(R_{1},\dots,R_{d-1},U)$, then clearly $X$ would commute with
$(S_{1},\dots,S_{d-1},V)$ because $(R_{1},\dots,R_{d-1},U)$ is an extension of
$(S_{1},\dots,S_{d-1},V)$. This completes the proof of part (1) of this
theorem.

For part (2), let $X$ be an $\underline{S}$-Toeplitz operator and as before
set $Y:=\pi(X)$. By definition we have

${S_{i}}^{*}XV=XS_{d-i}\text{ and }V^{*}XV=X\text{ for each }i=1,2\dots,d-1,$

applying $\pi$ on both sides of which we get

${R_{i}}^{*}YU=YR_{d-i}\text{ and }U^{*}YU=Y\text{ for each }i=1,2\dots,d-1.$

The last equation is equivalent to $YU=UY$, which, in view of the other
equations, further implies that $YR_{j}=R_{j}Y$ for each $i=1,2\dots,d-1$. For
the norm equality we simply observe that $X$ is the compression of $Y$ to
${\mathcal{H}}$ and therefore $\|X\|\leq\|Y\|=\|\pi(X)\|\leq\|X\|.$ For the
uniqueness part we use $({\bf{P_{3}}})$. Suppose that there are two possibly
distinct operators $Y_{1}$ and $Y_{2}$ that satisfy (5.2). Then

$0=P_{{\mathcal{H}}}(Y_{1}-Y_{2})|_{{\mathcal{H}}}=\rho(Y_{1}-Y_{2}).$

Since $\rho$ is an isometry, $Y_{1}=Y_{2}$.

For the converse direction of (2), let us first note that commutativity is
much stronger than being a generalized Toeplitz operator, i.e., if $X$
commutes with $\underline{S}$, then $X$ trivially becomes an
$\underline{S}$-Toeplitz operator. Therefore if $Y$ commutes with
$(R_{1},\dots,R_{d-1},U)$, then in particular we have

$\displaystyle R_{j}^{*}YU=YR_{d-j}\text{ and }U^{*}YU=U\text{ for each
}j=1,2\dots,d-1.$ (5.6)

Moreover we have the following $2\times 2$ matrix structure of the concerned
operators:

$R_{j}=\left[\begin{array}[]{cc}S_{j}&*\\\ 0&*\\\
\end{array}\right],U=\left[\begin{array}[]{cc}V&*\\\ 0&*\\\
\end{array}\right]\text{ and }Y=\left[\begin{array}[]{cc}X&*\\\ &*\\\
\end{array}\right].$

Using these matrix structures and (5.6), one readily obtains that $X$ is an
$\underline{S}$-Toeplitz operator. This completes the proof of part (2).

To prove part (3), we first note that the representation $\pi_{0}$ in the
statement of the theorem is actually the restriction of $\pi$ to
$\mathcal{C}^{*}(\underline{S})$ as the representation $\pi$ also maps the
generating set $\underline{S}$ of $\mathcal{C}^{*}(\underline{S})$ to the
generating set $\underline{R}$ of $\mathcal{C}^{*}(\underline{R})$. Since
$\pi_{0}(\underline{S})=\underline{R}$, range of $\pi_{0}$ is
$\mathcal{C}^{*}(\underline{R})$. Therefore to prove that the following
sequence

$0\rightarrow\mathcal{I}(\underline{S})\hookrightarrow\mathcal{C}^{*}(\underline{S})\xrightarrow{\pi_{0}}\mathcal{C}^{*}(\underline{R})\rightarrow
0$

is a short exact sequence, all we need to show is that
ker$\pi_{0}=\mathcal{I}(\underline{S})$.

We do that now. Since $\pi_{0}(\mathcal{C}^{*}(\underline{S}))$ is
commutative, we have $XY-YX$ in the kernel of $\pi_{0}$, for any
$X,Y\in\mathcal{C}^{*}(\underline{S})\cap\mathcal{T}(\underline{S})$. Hence
$\mathcal{I}(\underline{S})\subseteq$ ker$\pi_{0}$. To prove the other
inclusion, consider a finite product $Z_{1}$ of members of
$\underline{S}^{*}=(S_{1}^{*},\dots,S_{d-1}^{*},V^{*})$ and a finite product
$Z_{2}$ of members of $\underline{S}$. Let $Z=Z_{1}Z_{2}$. Since
$Z\in\mathcal{T}(\underline{S})\subseteq\mathcal{T}(V)$, we have
$\Phi_{0}(Z)=Z$. Note that
$\Phi_{0}(Z)=P_{\mathcal{H}}\pi_{0}(Z)|_{\mathcal{H}}$, for every
$Z\in\mathcal{C}^{*}(\underline{S})$. Now let $Z$ be any arbitrary finite
product of members from $\underline{S}$ and $\underline{S}^{*}$. Since
$\pi_{0}(\underline{S})=\underline{R}$, which is a family of normal operators,
we obtain, by Fuglede-Putnam theorem that, action of $\Phi_{0}$ on $Z$ has all
the members from $\underline{S}^{*}$ at the left and all the members from
$\underline{S}$ at the right. It follows from
$\text{ker}\pi=\text{ker}\Phi_{0}$ and $\Phi_{0}$ is idempotent that
ker$\pi_{0}=\\{Z-\Phi_{0}(Z):Z\in\mathcal{C}^{*}(\underline{S})\\}$. Also,
because of the above action of $\Phi_{0}$, if $Z$ is a finite product of
elements from $\underline{S}$ and $\underline{S}^{*}$ then a simple commutator
manipulation shows that $Z-\Phi_{0}(Z)$ belongs to the ideal generated by all
the commutators $XY-YX$, where
$X,Y\in\mathcal{C}^{*}(\underline{S})\cap\mathcal{T}(\underline{S})$. This
shows that ker$\pi_{0}=\mathcal{I}(\underline{S})$.

In order to find a completely isometric cross section, recall the completely
isometric map
$\rho:\pi(\mathcal{C}^{*}(\mathcal{T}(V)))\to\mathcal{B}(\mathcal{H})$ defined
by $Y\mapsto V^{*}YV$ such that
$\pi\circ\rho=id_{\pi(\mathcal{C}^{*}(\mathcal{T}(V)))}$. Set
$\rho_{0}:=\rho|_{\pi(\mathcal{C}^{*}(\underline{S}))}$. Then by the
definition of $\rho$ and the action of $\Phi_{0}$, it follows that
$\rho_{0}(\pi(X))=V^{*}\pi(X)V=\Phi_{0}(X)\in\mathcal{C}^{*}(\underline{S})$
for all $X\in\mathcal{C}^{*}(\underline{S})$. Thus
$\text{Ran}\rho_{0}\subseteq\mathcal{C}^{*}(\underline{S})$ and therefore is a
completely isometric cross section. This completes the proof of the theorem. ∎

A remark on a possible strengthening of Theorem 5.6 is in order.

######  Remark 5.7.

One can work with a commuting family of $\Gamma_{d}$-isometries instead of
just one and obtain similar results as in Thereom 5.6. This was done in the
pair case in [8].

##  6\. Dual Toeplitz operators

Dual Toeplitz operators have been studied for a while on the Bergman space of
the unit disc $\mathbb{D}$ in [30] and on the Hardy space of the Euclidean
ball $\mathbb{B}_{d}$ in [15]. In our setting, consider the space

$H^{2}(\mathbb{G}_{d})^{\perp}=L^{2}(b\Gamma_{d})\ominus
H^{2}(\mathbb{G}_{d}).$

Let $(I-Pr)$ be the projection of $L^{2}(b\Gamma_{d})$ onto
$H^{2}(\mathbb{G}_{d})^{\perp}$. If $\varphi\in L^{\infty}(b\Gamma_{d})$,
define the dual Toeplitz operator on $H^{2}(\mathbb{G}_{d})^{\perp}$ by
$DT_{\varphi}=(I-Pr)M_{\varphi}|_{H^{2}(\mathbb{G}_{d})^{\perp}}$. With
respect to the decomposition above,

$\displaystyle
M_{\varphi}=\left(\begin{array}[]{cc}T_{\varphi}&H_{\overline{\varphi}}^{*}\\\
H_{\varphi}&DT_{\varphi}\\\ \end{array}\right).$ (6.3)

######  Lemma 6.1.

The $d$-tuple
$\underline{D}=(DT_{\bar{s}_{1}},\dots,DT_{\bar{s}_{d-1}},DT_{\overline{p}})$
is a $\Gamma_{d}$-isometry with the $d$-tuple
$(M_{\bar{s}_{1}},\dots,M_{\bar{s}_{d-1}},M_{\overline{p}})$ as its minimal
$\Gamma_{d}$-unitary extension.

###### Proof.

It is a $\Gamma_{d}$-isometry because it is the restriction of the
$\Gamma_{d}$-unitary
$(M_{\bar{s}_{1}},\dots,M_{\bar{s}_{d-1}},M_{\overline{p}})$ to the space
$H^{2}(\mathbb{G}_{d})^{\perp}$. And this extension is minimal because
$M_{\overline{p}}$ is the minimal unitary extension of $DT_{\overline{p}}$. ∎

######  Theorem 6.2.

A bounded operator $T$ on $H^{2}(\mathbb{G}_{d})^{\perp}$ is a dual Toeplitz
operator if and only if it satisfies the Brown–Halmos relations with respect
to $\underline{D}$.

###### Proof.

The easier part is showing that every dual Toeplitz operator on
$H^{2}(\mathbb{G}_{d})^{\perp}$ satisfies the Brown–Halmos relations with
respect to $\underline{D}$. It follows from the following identities

${M_{\bar{s}_{i}}}^{*}M_{\varphi}M_{\overline{p}}=M_{\varphi}M_{\bar{s}_{d-i}}\text{
and }{M_{\overline{p}}}^{*}M_{\varphi}M_{\overline{p}}=M_{\varphi}\text{ for
every }\varphi\in L^{\infty}(b\Gamma_{d})$

and from the $2\times 2$ matrix representations of the operators in concern.
For the converse, let $T$ on $H^{2}(\mathbb{G}_{d})^{\perp}$ satisfy the
Brown–Halmos relations with respect to the $\Gamma_{d}$-isometry
$\underline{D}$. By part (2) of Theorem 5.6 and the fact that any operator
that commutes with each of $M_{s_{1}},\dots,M_{s_{d-1}}$ and $M_{p}$ is of the
form $M_{\varphi}$, for some $\varphi\in L^{\infty}(b\Gamma_{d})$, there is a
$\varphi\in L^{\infty}(b\Gamma_{d})$ such that $T$ is the compression of
$M_{\varphi}$ to $H^{2}(\mathbb{G}_{d})^{\perp}$. ∎

Acknowledgement: The research works of the first and second named authors are
supported by DST-INSPIRE Faculty Fellowships DST/INSPIRE/04/2015/001094 and
DST/INSPIRE/04/2018/002458 respectively. The second named author thanks Indian
Institute of Technology, Bombay for a post-doctoral fellowship under which
most of this work was done.

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