# Toeplitz operators on $\mathcal{L}^{p}$-spaces of a tree

Mingmei Huang  1 College of Mathematics and Statistics, Chongqing University,
Chongqing, 401331, P. R. China<EMAIL_ADDRESS>,  Xiaoyan Zhang  2
College of Mathematics and Statistics, Chongqing University, Chongqing,
401331, P. R. China<EMAIL_ADDRESS>and  Xianfeng Zhao  3 College of
Mathematics and Statistics, Chongqing University, Chongqing, 401331, P. R.
China<EMAIL_ADDRESS>
###### Abstract.

Let $T$ be a rooted, countable infinite tree without terminal vertices. In the
present paper, we characterize the spectra, self-adjointness and positivity of
Toeplitz operators on the spaces of $p$-summable functions on $T$. Moreover,
we obtain a necessary and sufficient condition for Toeplitz operators to have
finite rank on such function spaces.

###### Key words and phrases:

Toeplitz operator; spectrum; positivity; finite rank operator; tree

###### 2010 Mathematics Subject Classification:

47A30, 47B37, 05C05

##  1\. Introduction

Let us begin with the preliminary definitions, concepts and relevant notation
on trees. A _graph_ $G$ is a pair $(V,E)$ consisting of a countably-infinite
set of _vertices_ $V$ and a set of _edges_ $E\subset\big{\\{}(u,v):u,v\in
V,u\neq v\big{\\}}$ (which is a subset of $V\times V$). If $(u,v)\in E$, then
we say that $u$ and $v$ are _adjacent_ (or _neighbors_) and we denote this by
$u\thicksim v$. For each vertex $v$, the _degree_ of $v$ is the number of
vertices adjacent to $v$, which is denoted by $\mathrm{deg}(v)$. We say that
$G$ is _locally finite_ if the degree of every vertex is finite. We call that
a vertex $v$ is a _terminal vertex_ if $v$ has a unique neighbor. A _path of
length $n$_ joining two vertices $u$ and $v$ is a finite sequence of $(n+1)$
distinct vertices

$u=u_{0}\thicksim u_{1}\thicksim u_{2}\thicksim\cdots\thicksim u_{n}=v.$

By a _tree_ $T$ we mean a locally finite, connected and simply-connected
graph, i.e., for each pair of vertices there is one and only one path between
them. We shall identify $T$ with the collection of its vertices $V$. In the
present paper, the tree we consider has a distinguished vertex, which we call
the root of $T$ and we denote it by $o$. The length of a path joining the root
$o$ and a vertex $v$ is called the _length_ of $v$ and is denoted by $|v|$.

Given a tree $T$ rooted at $o$ and a vertex $v\in T$, a vertex $w$ is called a
_descendant_ of $v$ if $v$ lies in the unique path from $o$ to $w$. In this
case, the vertex $v$ is called an _ancestor_ of $w$. We denote by $v^{-}$ the
neighbor of a vertex $v$ which is an ancestor of $v$, and the vertex $v$ is
called a _child_ of $v^{-}$. For each $w\in T$, we denote the set of all
children of $w$ by $\mathrm{Chi}(w)$. Given a vertex $v$, the _sector
determined by $v$_ is the set $S_{v}$ consisting of $v$ and all its
descendants. In particular, $S_{o}=T$.

Let $1\leqslant p<\infty$. The function space $\mathcal{L}^{p}(T)$ of $T$ is
defined as the set of functions $f:T\rightarrow\mathbb{C}$ such that

$\sum_{v\in T}|f(v)|^{p}<\infty.$

It is easy to show that every $\mathcal{L}^{p}(T)$ is a Banach space when
endowed with the norm

$\|f\|_{p}:=\bigg{(}\sum_{v\in T}|f(v)|^{p}\bigg{)}^{\frac{1}{p}},$

since it is isomorphic to the $p$-summable sequence space $\ell^{p}(V)$.
Moreover, the dual space of $\mathcal{L}^{p}(T)$ with $1<p<\infty$ can be
identified with $\mathcal{L}^{q}(T)$ under the sesquilinear dual pairing

$\langle f,g\rangle:=\sum_{v\in T}f(v)\overline{g(v)},\ \ \
f\in\mathcal{L}^{p}(T),\ \ g\in\mathcal{L}^{q}(T),$

where $q=\frac{p}{p-1}$ is the conjugate exponent of $p$. In particular,
$\mathcal{L}^{2}(T)$ is a Hilbert space with the obvious inner product. When
$p=\infty$, the space $\mathcal{L}^{\infty}(T)$ is the collection of bounded
functions $f$ on $T$ equipped with the supremum norm
$\|f\|_{\infty}=\sup\big{\\{}|f(v)|:v\in T\big{\\}}$. In addition, we say that
$q=\infty$ is the conjugate exponent of $p=1$ and that $q=1$ is the conjugate
exponent of $p=\infty$.

The interest of the study of operators on infinite trees is motivated mainly
by the research in harmonic analysis dealing with the spectrum of the Laplace
operator on discrete structures, one can consult [9] and [10] for more
information. Linear operators on discrete structures other than Laplacian have
been studied by many authors. For instance:

  * •

_Toeplitz operators._ Pavone studied the properties of Toeplitz operators on a
discrete group and discussed the extent to which they paraller the properties
of Topelitz operators on the Hardy space of the unit circle, see [21, 22]; In
[14], Colonna and Martínez-Avendaño defined the Toeplitz operator on
$\mathcal{L}^{p}$-spaces of a tree, where $1\leqslant p\leqslant\infty$, and
then characterized the boundedness and the eigenvalues of such Toeplitz
operator; Motivated by the paper [14], Zhang and Zhao [23] established several
sufficient conditions for Toeplitz operators to be compact on
$\mathcal{L}^{p}$-spaces of a tree.

  * •

_Composition operators._ Pavone [20] characterized the class of hypercyclic
composition operators on the $\mathcal{L}^{p}$-space of the Poisson boundary
of a homogeneous tree, and showed that such operators are actually chaotic.
Allen, Colonna and Easley [5] investigated the boundedness, compactness and
spectra of the composition operators on the Lipschitz space of a tree; Later,
Allen and Pons ([7, 8]) obtained the characterizations for the (weighted)
composition operators with closed range, and to be bounded, compact, bounded
below, invertible and Fredholm on weighted Banach spaces of a tree; In
addition, the bounded, compact, invertible, isometric composition operators on
Hardy spaces of the homogenous rooted trees were investigated in the recent
paper [19].

  * •

_Shift operators._ The operator-norm estimate, spectral properties,
hyponormality, subnormality and hypercyclicity of (weighted) shift operators
on certain function spaces of a tree were well-studied in [16, 17, 18];
Moreover, the norm estimate, kernels and eigenvalues of the shift operator
(usually called the adjacency operator) on the $\mathcal{L}^{p}$-space of a
graph were systematically investigated in the paper [1].

  * •

_Multiplication operators._ The boundedness, compactness, invertibility,
hypercyclicity and essential norm of multiplication operators on Lipschitz-
type spaces of a tree were discussed in a series of papers of Allen, Colonna,
Easley and Prudhom [2, 3, 4, 6, 11, 12].

However, little is known about the spectral properties, self-adjointness and
positivity of classical operators on discrete structures. The purpose of this
paper is to study some fundamental properties about Toeplitz operators on
$\mathcal{L}^{p}$-spaces of a tree, such as the spectrum, self-adjointness,
positivity and finite rank. Before stating the definition of the Toeplitz
operator on $\mathcal{L}^{p}$-spaces of a tree, we need to introduce the
following two important linear operators on such function spaces. For any
function $f:T\rightarrow\mathbb{C}$, the operator $\nabla$ is defined by

$\displaystyle(\nabla f)(u):=f(u)-\sum_{v^{-}=u}f(v),\ \ \ \ u\in T.$

Recall that the operator $\Delta$ is the transformation with domain
$\mathcal{L}^{1}(T)$ defined by

$(\Delta f)(u):=\sum_{v\in S_{u}}f(v),\ \ \ \ f\in\mathcal{L}^{1}(T),\ \ \
u\in T,$

where $S_{u}$ is the sector determined by $u$. For more details concerning the
above two operators, one can consult [14]. Let $1\leqslant p\leqslant\infty$
and $\varphi$ be a complex-valued function on $T$. The _Toeplitz operator_ on
$\mathcal{L}^{p}(T)$ with symbol $\varphi$ is defined by

$T_{\varphi}f:=\Delta(\varphi\nabla f),\ \ \ \ f\in\mathcal{L}^{p}(T).$

Some useful properties and important conclusions about $\nabla$, $\Delta$ and
Toeplitz operators on $\mathcal{L}^{p}(T)$ will be included in the next
section.

The main part of this paper is organized as follows. In Section 3, we will
study the spectra of some classes of bounded Topelitz operators on the Banach
spaces $\mathcal{L}^{p}(T)$ with $1\leqslant p\leqslant\infty$. More
precisely, we will show in Theorems 3.1 and 3.2 that the spectrum of the
Topelitz operator is equal to the closure of the range of its symbol under
certain mild assumptions. This answers an open question posed by Colonna and
Martínez-Avendaño [14]. In Section 4, we investigate the self-adjointness and
positivity of Topelitz operators on the Hilbert space $\mathcal{L}^{2}(T)$.
Surprisingly, we find that there is no nontrivial self-adjoint Topelitz
operator on $\mathcal{L}^{2}(T)$, see Theorem 4.1 for the detailed discussion.
Section 5 is devoting to the characterization for finite rank Toeplitz
operators on $\mathcal{L}^{p}(T)$ with $1\leqslant p\leqslant\infty$. In fact,
we will prove that the Toeplitz operator on $\mathcal{L}^{p}(T)$ is of finite
rank if and only if the support of its symbol is a finite set, see Theorem 5.1
for the proof.

##  2\. Preliminary

In this section, we recall some important definitions and known results which
will be required in the next three sections. We shall make use of the
following derivative of a function on a tree repeatedly.

######  Definition 2.1.

Given a complex-valued function $f$ on a tree $T$. We define the function
$f_{-}$ on $T$ by

$\displaystyle f_{-}(v)=\begin{cases}f(v^{-}),&\mathrm{if}\ \ \ v\neq
o,\vspace{2mm}\\\ 0,&\mathrm{if}\ \ \ v=o.\end{cases}$

In addition, the function $f^{\prime}$ on $T$ is defined by

$f^{\prime}(v)=f(v)-f_{-}(v),\ \ \ v\in T.$

Recall that the operators $\nabla$ and $\Delta$ were introduced in Section 1.
The relationship between these two operators on $\mathcal{L}^{1}(T)$ is given
by the following lemma, see [14, Proposition 5.7] if necessary.

######  Lemma 2.2.

If $f\in\mathcal{L}^{1}(T)$, then $\Delta(\nabla f)=f$ and $\nabla(\Delta
f)=f.$

The following expression for the Toeplitz operator on $\mathcal{L}^{p}(T)$ is
quite useful in the study of the boundedness, compactness, positivity and
spectra of Toeplitz operators, which was proved in [14, Lemma 6.2].

######  Lemma 2.3.

Let $1<p\leqslant\infty$ and let $q$ be the conjugate exponent of $p$. If
$\varphi$ and $\varphi^{\prime}$ are both in $\mathcal{L}^{q}(T)$ and
$f\in\mathcal{L}^{p}(T)$, then we have that $\varphi\nabla
f\in\mathcal{L}^{1}(T)$ and

$T_{\varphi}f=f\varphi_{-}+\Delta(f\varphi^{\prime}).$

The next two lemmas were established in [14, Theorems 6.4 and 6.6], which give
nice sufficient conditions for the boundedness of Toeplitz operators on
$\mathcal{L}^{p}(T)$ with $1\leqslant p\leqslant\infty$.

######  Lemma 2.4.

Assume that $1\leqslant p<\infty$ and $q$ is the conjugate exponent of $p$. If
the functions $\varphi$ and $\widehat{\varphi}$ are both in
$\mathcal{L}^{q}(T)$, then the Toeplitz operator $T_{\varphi}$ is bounded on
$\mathcal{L}^{p}(T)$, where $\widehat{\varphi}$ is defined by

$\widehat{\varphi}(v):=(|v|+1)\varphi^{\prime}(v),\ \ \ \ v\in T.$

######  Lemma 2.5.

If $\varphi$ and $\varphi^{\prime}$ are both in $\mathcal{L}^{1}(T)$, then the
Toeplitz operator $T_{\varphi}$ is bounded on $\mathcal{L}^{\infty}(T)$.

Recall that [14, Theorem 6.10] obtained a characterization on the point
spectrum (the set of eigenvalues) of the Toeplitz operator on
$\mathcal{L}^{p}(T)$ in terms of the range of its symbol. For the sake of
convenience, we quote this theorem in the following.

######  Lemma 2.6.

Let $1\leqslant p\leqslant\infty$ and let $q$ be the conjugate exponent of
$p$. Suppose that $\varphi,\varphi^{\prime}\in\mathcal{L}^{q}(T)$ and the
Toeplitz operator $T_{\varphi}$ is bounded on $\mathcal{L}^{p}(T)$. Then:
$(a)$ $\sigma_{p}(T_{\varphi})=\\{\varphi(w):w\in T\\}$ if there is no nonzero
function $g\in\mathcal{L}^{p}(T)$ with $\nabla g=0$;
$(b)$ $\sigma_{p}(T_{\varphi})=\\{0\\}\cup\\{\varphi(w):w\in T\\}$ if there is
a nonzero function $g\in\mathcal{L}^{p}(T)$ with $\nabla g=0$.

##  3\. The spectra of Toeplitz operators

The study of the spectral properties of operators on infinite trees is
relatively new. In this section, we investigate the spectra of Toeplitz
operators on $\mathcal{L}^{p}(T)$ with $1\leqslant p\leqslant\infty$. Let $q$
be the conjugate exponent of $p$ and $\varphi$ be a function in
$\mathcal{L}^{q}(T)$. Based on the characterization for eigenvalues of
Toeplitz operators (Lemma 2.6), we show that the spectrum of the Toeplitz
operator $T_{\varphi}:\mathcal{L}^{p}(T)\rightarrow\mathcal{L}^{p}(T)$ with
$1\leqslant p<\infty$ equals the closure of $\varphi(T)$ if
$\widehat{\varphi}$ is also in $\mathcal{L}^{q}(T)$. Recall that the function
$\widehat{\varphi}$ was introduced in Lemma 2.4. On the other hand, we also
prove that the spectrum of the Toeplitz operator
$T_{\varphi}:\mathcal{L}^{\infty}(T)\rightarrow\mathcal{L}^{\infty}(T)$ is
coincided with the closure of $\varphi(T)$ when
$\varphi^{\prime}\in\mathcal{L}^{1}(T)$.

Let us consider the case that $1\leqslant p<\infty$ first. As the spectrum of
the Topelitz operator on $\mathcal{L}^{1}(T)$ was obtained in [14] via the
multiplication operator with the same symbol, it is sufficient for us to
discuss the case of $1<p<\infty$.

######  Theorem 3.1.

Let $1\leqslant p<\infty$ and $q$ be the conjugate exponent of $p$. Suppose
that $\varphi$ and $\widehat{\varphi}$ are both in $\mathcal{L}^{q}(T)$, where

$\widehat{\varphi}(v)=(|v|+1)\varphi^{\prime}(v),\ \ \ \ v\in T.$

Then $T_{\varphi}$ is bounded on $\mathcal{L}^{p}(T)$ and in this case we have

$\sigma(T_{\varphi})=\mathrm{clos}\\{\varphi(v):v\in T\\}.$

###### Proof.

As we mentioned above, the conclusion for the case of $p=1$ was obtained in
Section 7 of [14]. So we need only to consider the case for $p>1$. By Lemma
2.4, we see that the Toeplitz operator $T_{\varphi}$ is bounded on
$\mathcal{L}^{p}(T)$. Clearly, $\widehat{\varphi}\in\mathcal{L}^{q}(T)$
implies that $\varphi^{\prime}\in\mathcal{L}^{q}(T)$. Then using Lemma 2.6, we
have

$\mathrm{clos}\\{\varphi(w):w\in
T\\}=\mathrm{clos}[\sigma_{p}(T_{\varphi})]\subset\sigma(T_{\varphi}).$

To prove the reverse inclusion, it is sufficient to show that
$T_{\varphi}-\lambda I$ is invertible on $\mathcal{L}^{p}(T)$ if $\lambda$ is
not in the closure of the range of $\varphi$, where $I$ is the identity
operator on $\mathcal{L}^{p}(T)$. Observe that
$\lambda\notin\mathrm{clos}\\{\varphi(v):v\in T\\}$ implies that $\lambda$ is
not an eigenvalue of $T_{\varphi}$. This means that $T_{\varphi}-\lambda I$ is
an injective from $\mathcal{L}^{p}(T)$ to $\mathcal{L}^{p}(T)$. According to
the Banach inverse operator theorem, we need only to show that
$T_{\varphi}-\lambda I$ is a surjective on $\mathcal{L}^{p}(T)$ if
$\lambda\notin\mathrm{clos}\\{\varphi(v):v\in T\\}$.

Suppose that

$\inf_{v\in T}|\varphi(v)-\lambda|\geqslant\delta$

for some positive constant $\delta$. Let
$\delta_{0}=\min\\{\delta,|\lambda|\\}$. Then we have that $\delta_{0}>0$ and

$|\varphi(v)-\lambda|\geqslant\delta_{0}$

and

$|\varphi_{-}(v)-\lambda|\geqslant\delta_{0}$

for all $v\in T$. Letting $g$ be a function in $\mathcal{L}^{p}(T)$, we define

$\displaystyle
f:=\frac{1}{\lambda}\bigg{[}\Delta\Big{(}\frac{\varphi}{\varphi-\lambda}\nabla
g\Big{)}-g\bigg{]}.$ (3.1)

In the following, we will show that $f\in\mathcal{L}^{p}(T)$ and $f$ is the
preimage of $g$ under $T_{\varphi}-\lambda I$.

To show $f\in\mathcal{L}^{p}(T)$, we need only to prove that

$\Delta\Big{(}\frac{\varphi}{\varphi-\lambda}\nabla
g\Big{)}\in\mathcal{L}^{p}(T).$

Let us show
$\widehat{\big{(}\frac{\varphi}{\varphi-\lambda}\big{)}}\in\mathcal{L}^{q}(T)$
first. Noting that

$\widehat{\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}}(v)=(|v|+1)\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}^{\prime}(v)=(|v|+1)\Big{[}\frac{\varphi(v)}{\varphi(v)-\lambda}-\frac{\varphi_{-}(v)}{\varphi_{-}(v)-\lambda}\Big{]}$

for $v\in T$, we have

$\displaystyle\bigg{\|}\widehat{\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}}\bigg{\|}_{q}$
$\displaystyle=\bigg{[}\sum\limits_{v\in
T}\Big{|}(|v|+1)\Big{(}\frac{\varphi(v)}{\varphi(v)-\lambda}-\frac{\varphi_{-}(v)}{\varphi_{-}(v)-\lambda}\Big{)}\Big{|}^{q}\bigg{]}^{\frac{1}{q}}$
$\displaystyle=\bigg{[}\sum\limits_{v\in
T}\Big{|}(|v|+1)\Big{(}\frac{\varphi(v)}{\varphi(v)-\lambda}-\frac{\varphi_{-}(v)}{\varphi(v)-\lambda}+\frac{\varphi_{-}(v)}{\varphi(v)-\lambda}-\frac{\varphi_{-}(v)}{\varphi_{-}(v)-\lambda}\Big{)}\Big{|}^{q}\bigg{]}^{\frac{1}{q}}.$

Then triangle inequality gives that

$\displaystyle\bigg{\|}\widehat{\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}}\bigg{\|}_{q}$
$\displaystyle\leqslant\bigg{[}\sum\limits_{v\in
T}\Big{|}(|v|+1)\frac{\varphi(v)-\varphi_{-}(v)}{\varphi(v)-\lambda}\Big{|}^{q}\bigg{]}^{\frac{1}{q}}+\bigg{[}\sum\limits_{v\in
T}\Big{|}(|v|+1)\frac{\varphi_{-}(v)[\varphi(v)-\varphi_{-}(v)]}{[\varphi_{-}(v)-\lambda][\varphi(v)-\lambda]}\Big{|}^{q}\bigg{]}^{\frac{1}{q}}$
$\displaystyle=\bigg{[}\sum\limits_{v\in
T}\Big{|}(|v|+1)\frac{\varphi^{\prime}(v)}{\varphi(v)-\lambda}\Big{|}^{q}\bigg{]}^{\frac{1}{q}}+\bigg{[}\sum\limits_{v\in
T}\Big{|}(|v|+1)\frac{\varphi^{\prime}(v)\varphi_{-}(v)}{[\varphi_{-}(v)-\lambda][\varphi(v)-\lambda]}\Big{|}^{q}\bigg{]}^{\frac{1}{q}}$
$\displaystyle\leqslant\frac{1}{\delta_{0}}\Big{(}\sum\limits_{v\in
T}\big{|}(|v|+1)\varphi^{\prime}(v)\big{|}^{q}\Big{)}^{\frac{1}{q}}+\frac{1}{\delta_{0}^{2}}\Big{(}\sum\limits_{v\in
T}\big{|}(|v|+1)\varphi_{-}(v)\varphi^{\prime}(v)\big{|}^{q}\Big{)}^{\frac{1}{q}}$
$\displaystyle\leqslant\frac{1}{\delta_{0}}\Big{(}\sum\limits_{v\in
T}\big{|}(|v|+1)\varphi^{\prime}(v)\big{|}^{q}\Big{)}^{\frac{1}{q}}+\frac{\|\varphi\|_{\infty}}{\delta_{0}^{2}}\Big{(}\sum\limits_{v\in
T}\big{|}(|v|+1)\varphi^{\prime}(v)\big{|}^{q}\Big{)}^{\frac{1}{q}}$
$\displaystyle=\Big{(}\frac{1}{\delta_{0}}+\frac{\|\varphi\|_{\infty}}{\delta_{0}^{2}}\Big{)}\|\widehat{\varphi}\|_{q}.$

This shows that
$\widehat{\big{(}\frac{\varphi}{\varphi-\lambda}\big{)}}\in\mathcal{L}^{q}(T)$,
since $\widehat{\varphi}\in\mathcal{L}^{q}(T)$.

Next, we are going to show that
$\Delta\big{(}\frac{\varphi}{\varphi-\lambda}\nabla
g\big{)}\in\mathcal{L}^{p}(T)$. Noting that $\frac{\varphi}{\varphi-\lambda}$
and $\big{(}\frac{\varphi}{\varphi-\lambda}\big{)}^{\prime}$ are both in
$\mathcal{L}^{q}(T)$, we obtain by Lemma 2.3 that

$\Delta\Big{(}\frac{\varphi}{\varphi-\lambda}\nabla
g\Big{)}=T_{\frac{\varphi}{\varphi-\lambda}}g=\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}_{-}g+\Delta\Big{[}\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}^{\prime}g\Big{]}.$

Furthermore, we have

$\displaystyle\Big{\|}\Delta\Big{[}\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}^{\prime}g\Big{]}\Big{\|}_{p}$
$\displaystyle\leqslant\Big{\|}\Delta\Big{[}\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}^{\prime}g\Big{]}\Big{\|}_{1}=\sum_{v\in
T}\bigg{|}\Delta\Big{[}\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}^{\prime}g\Big{]}(v)\bigg{|}$
$\displaystyle\leqslant\sum_{v\in T}\sum_{u\in
S_{v}}\bigg{|}\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}^{\prime}(u)g(u)\bigg{|}$
$\displaystyle=\sum_{v\in
T}(|v|+1)\bigg{|}\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}^{\prime}(v)g(v)\bigg{|}$
$\displaystyle=\sum_{v\in
T}\bigg{|}\widehat{\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}}(v)\bigg{|}~{}|g(v)|$
$\displaystyle\leqslant\|g\|_{p}\bigg{\|}\widehat{\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}}\bigg{\|}_{q},$

to get that
$\Delta\big{[}\big{(}\frac{\varphi}{\varphi-\lambda}\big{)}^{\prime}g\big{]}\in\mathcal{L}^{p}(T)$.
Since $\frac{\varphi}{\varphi-\lambda}\in\mathcal{L}^{q}(T)$ implies that
$\frac{\varphi}{\varphi-\lambda}\in\mathcal{L}^{\infty}(T)$, we obtain that
$\big{(}\frac{\varphi}{\varphi-\lambda}\big{)}_{-}$ is also bounded. It
follows that $\big{(}\frac{\varphi}{\varphi-\lambda}\big{)}_{-}g$ is in
$\mathcal{L}^{p}(T)$, which gives that
$\Delta\big{(}\frac{\varphi}{\varphi-\lambda}\nabla
g\big{)}\in\mathcal{L}^{p}(T)$. Therefore, we obtain that the function defined
in (3.1) is in $\mathcal{L}^{p}(T)$.

To finish the proof, it remains to verify that

$(T_{\varphi}-\lambda I)f=g.$

Indeed, we have by Lemma 2.3 that

$\frac{\varphi}{\varphi-\lambda}\nabla g\in\mathcal{L}^{1}(T),$

since $g\in\mathcal{L}^{p}(T)$ and
$\frac{\varphi}{\varphi-\lambda},\big{(}\frac{\varphi}{\varphi-\lambda}\big{)}^{\prime}\in\mathcal{L}^{q}(T)$.
It follows from (3.1) and Lemma 2.2 that

$\displaystyle\nabla f$
$\displaystyle=\nabla\Bigg{[}\frac{1}{\lambda}\bigg{(}\Delta\Big{(}\frac{\varphi}{\varphi-\lambda}\nabla
g\Big{)}-g\bigg{)}\Bigg{]}=\frac{1}{\lambda}\frac{\varphi}{\varphi-\lambda}\nabla
g-\frac{\nabla g}{\lambda}=\frac{\nabla g}{\varphi-\lambda}.$

This yields that

$\displaystyle(T_{\varphi}-\lambda I)f$ $\displaystyle=T_{\varphi}f-\lambda
f=\Delta(\varphi\nabla f)-\lambda f$
$\displaystyle=\Delta\Big{(}\frac{\varphi}{\varphi-\lambda}\nabla
g\Big{)}-\Delta\Big{(}\frac{\varphi}{\varphi-\lambda}\nabla g\Big{)}+g$
$\displaystyle=g,$

as desired. This completes the proof of Theorem 3.1. ∎

Now we give a description for the spectrum of
$T_{\varphi}:\mathcal{L}^{\infty}(T)\rightarrow\mathcal{L}^{\infty}(T)$ with
$\varphi,\varphi^{\prime}\in\mathcal{L}^{1}(T)$ in the following theorem.

######  Theorem 3.2.

If $\varphi$ and $\varphi^{\prime}$ are both in $\mathcal{L}^{1}(T)$, then
$T_{\varphi}$ is bounded on $\mathcal{L}^{\infty}(T)$ and in this case we have

$\sigma(T_{\varphi})=\mathrm{clos}\\{\varphi(v):v\in T\\}.$

###### Proof.

Lemma 2.5 tells us that
$T_{\varphi}:\mathcal{L}^{\infty}(T)\rightarrow\mathcal{L}^{\infty}(T)$ is
bounded if $\varphi\in\mathcal{L}^{1}(T)$ and
$\varphi^{\prime}\in\mathcal{L}^{1}(T)$. According to the proof of Theorem
3.1, we need only to show that

$\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}_{-}g\ \ \ \ \mathrm{and}\ \ \ \
\Delta\Big{[}\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}^{\prime}g\Big{]}$

are both in $\mathcal{L}^{\infty}(T)$ if
$\lambda\notin\mathrm{clos}\\{\varphi(v):v\in T\\}$ and
$g\in\mathcal{L}^{1}(T)$. In fact,

$\bigg{\|}\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}_{-}g\bigg{\|}_{\infty}=\sup_{v\in
T}\bigg{|}\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}_{-}(v)g(v)\bigg{|}\leqslant\frac{\|g\|_{\infty}}{\delta_{0}}\|\varphi\|_{\infty},$

where $\delta_{0}$ is the constant defined in the proof of Theorem 3.1.
Moreover, since

$\displaystyle\bigg{\|}\Delta\Big{[}\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}^{\prime}g\Big{]}\bigg{\|}_{\infty}$
$\displaystyle=\sup_{v\in T}\bigg{|}\sum_{u\in
S_{v}}g(u)\Big{(}\frac{\varphi}{\varphi-\lambda}\Big{)}^{\prime}(u)\bigg{|}$
$\displaystyle\leqslant\sup_{v\in T}\sum_{u\in
S_{v}}|g(u)|\bigg{|}\frac{\varphi(u)}{\varphi(u)-\lambda}-\frac{\varphi_{-}(u)}{\varphi_{-}(u)-\lambda}\bigg{|}$
$\displaystyle\leqslant\frac{|\lambda|~{}\|g\|_{\infty}}{\delta_{0}^{2}}\sup_{v\in
T}\Big{[}\sum_{u\in S_{v}}|\varphi(u)-\varphi_{-}(u)|\Big{]}$
$\displaystyle=\frac{|\lambda|~{}\|g\|_{\infty}}{\delta_{0}^{2}}\|\varphi^{\prime}\|_{1},$

we obtain that
$\big{(}\frac{\varphi}{\varphi-\lambda}\big{)}_{-}g\in\mathcal{L}^{\infty}(T)$
and
$\Delta\big{[}\big{(}\frac{\varphi}{\varphi-\lambda}\big{)}^{\prime}g\big{]}\in\mathcal{L}^{\infty}(T)$.

The rest of the proof of this theorem is similar to the previous one, so we
omit the details. This finishes the proof of Theorem 3.2. ∎

######  Remark 3.3.

The conclusions in Theorems 3.1 and 3.2 are comparable to the corresponding
results for analytic Toeplitz operators on the Hardy and Bergman spaces of the
open unit disk, see the books [15] and [24] for more information.

##  4\. The positivity of Toeplitz operators

We say that a bounded linear operator $T$ is self-adjoint (positive) on a
complex Hilbert space $\mathscr{H}$ if $\langle Tx,x\rangle_{\mathscr{H}}$ is
real (nonnegative) for every $x$ in $\mathscr{H}$. In this section, we study
when a bounded Toeplitz operator is self-adjoint (positive) on the Hilbert
space $\mathcal{L}^{2}(T)$.

Observing that the adjoint of the Topelitz operator $T_{\varphi}$ does not
equal $T_{\overline{\varphi}}$ in general, which leads to certain difficulties
in the study of the self-adjointness and positivity of Toeplitz operators on
$\mathcal{L}^{2}(T)$. However, we are able to obtain a complete
characterization for the self-adjoint Toeplitz operators on
$\mathcal{L}^{2}(T)$ by using their point spectra.

######  Theorem 4.1.

Suppose that $\varphi$ is a function in $\mathcal{L}^{2}(T)$ such that
$\widehat{\varphi}\in\mathcal{L}^{2}(T)$. Then the following three conditions
are equivalent:
$(1)$ $T_{\varphi}$ is positive on $\mathcal{L}^{2}(T)$;
$(2)$ $T_{\varphi}$ is self-adjoint on $\mathcal{L}^{2}(T)$;
$(3)$ $\varphi(v)=0$ for all $v\in T$.

###### Proof.

Clearly, $(1)\Rightarrow(2)$ is trivial and $(3)\Rightarrow(1)$ follows
immediately from Lemma 2.3. To complete the proof of this theorem, it remains
to show that $(2)\Rightarrow(3)$.

Now we suppose that $T_{\varphi}$ is self-adjoint on $\mathcal{L}^{2}(T)$.
Then we have by Lemma 2.6 that $\varphi$ must be real-valued, since

$\\{\varphi(v):v\in
T\\}=\sigma_{p}(T_{\varphi})\subset\sigma(T_{\varphi})\subset\mathbb{R}$

or

$\\{0\\}\cup\\{\varphi(v):v\in
T\\}=\sigma_{p}(T_{\varphi})\subset\sigma(T_{\varphi})\subset\mathbb{R}.$

Let $f$ be any function in $\mathcal{L}^{2}(T)$. Since
$\widehat{\varphi}\in\mathcal{L}^{2}(T)$, we see that the series
$\sum\limits_{v\in T}[T_{\varphi}f(v)]\overline{f(v)}$ is absolutely
convergent. Now using Lemma 2.3 again gives

$\displaystyle\begin{split}\langle T_{\varphi}f,f\rangle&=\sum\limits_{v\in
T}[T_{\varphi}f(v)]\overline{f(v)}=\sum\limits_{v\in
T}\big{[}f(v)\varphi_{-}(v)\overline{f(v)}+\Delta(f\varphi^{\prime})(v)\overline{f(v)}\big{]}\\\
&=\sum\limits_{v\in T}|f(v)|^{2}\varphi_{-}(v)+\sum\limits_{v\in
T}\overline{f(v)}\Delta(f\varphi^{\prime})(v).\end{split}$ (4.1)

Furthermore, we have by the definitions of $\varphi^{\prime}$ and the operator
$\Delta$ that

$\displaystyle\begin{split}&\sum\limits_{v\in
T}\overline{f(v)}\Delta(f\varphi^{\prime})(v)\\\ &\ =\sum\limits_{v\in
T}\Big{[}\overline{f(v)}\sum\limits_{u\in
S_{v}}(f\varphi^{\prime})(u)\Big{]}\\\ &\ =\sum\limits_{v\in
T}\Big{\\{}\overline{f(v)}\sum\limits_{u\in
S_{v}}f(u)\big{[}\varphi(u)-\varphi_{-}(u)\big{]}\Big{\\}}\\\ &\
=\sum\limits_{v\in T}|f(v)|^{2}\varphi(v)-\sum\limits_{v\in
T}|f(v)|^{2}\varphi_{-}(v)+\sum\limits_{v\in
T}\Big{\\{}\overline{f(v)}\sum\limits_{u\in
S_{v}\backslash\\{v\\}}\big{[}f(u)\varphi(u)-f(u)\varphi_{-}(u)\big{]}\Big{\\}}.\end{split}$
(4.2)

We first show that $\varphi$ is a constant function. Otherwise, we can choose
two vertices $\xi$ and $\eta$ in $T$ such that
$\varphi(\xi)\neq\varphi(\eta)$. Since the path joining $\xi$ and $\eta$ is a
finite sequence of distinct vertices, we may assume that $\xi\thicksim\eta$.
In other words, we obtain that $\xi\in\mathrm{Chi}(\eta)$ or
$\eta\in\mathrm{Chi}(\xi)$. Without loss of generality, we may assume that
$\eta\in\mathrm{Chi}(\xi)$. Let us consider the function
$g:T\rightarrow\mathbb{C}$ defined by

$\displaystyle g(v)=\begin{cases}1,&\mathrm{if}\ \ v=\xi,\vspace{2mm}\\\
\mathrm{i},&\mathrm{if}\ \ v=\eta,\vspace{2mm}\\\
0,&\text{otherwise}.\vspace{2mm}\\\ \end{cases}$

It follows from (4.1) and (4.2) that

$\displaystyle\langle T_{\varphi}g,g\rangle$
$\displaystyle=|g(\xi)|^{2}\varphi(\xi)+|g(\eta)|^{2}\varphi(\eta)+\overline{g(\xi)}\big{[}g(\eta)\varphi(\eta)-g(\eta)\varphi(\eta^{-})\big{]}$
$\displaystyle=|g(\xi)|^{2}\varphi(\xi)+|g(\eta)|^{2}\varphi(\eta)+\overline{g(\xi)}g(\eta)\big{[}\varphi(\eta)-\varphi(\eta^{-})\big{]}$
$\displaystyle=\big{[}\varphi(\xi)+\varphi(\eta)\big{]}+\big{[}\varphi(\eta)-\varphi(\xi)\big{]}\mathrm{i}.$

This means that $\langle T_{\varphi}g,g\rangle\notin\mathbb{R}$, which
contradicts that $T_{\varphi}$ is self-adjoint. Thus $\varphi$ is a constant
function. However, the condition $\varphi\in\mathcal{L}^{2}(T)$ yields that
$\varphi$ must be zero. This completes the proof. ∎

##  5\. Finite rank Toeplitz operators

The final section is devoted to establishing a equivalent characterization for
finite rank Toeplitz operators on $\mathcal{L}^{p}(T)$, where $1\leqslant
p\leqslant\infty$. More concretely, we will show in the following theorem that
the Toeplitz operator $T_{\varphi}$ having finite rank if and only if $\\{v\in
T:\varphi(v)\neq 0\\}$ is a finite subset of $T$.

######  Theorem 5.1.

Let $1\leqslant p\leqslant\infty$ and $\varphi$ be a function in
$\mathcal{L}^{q}(T)$ such that $\varphi^{\prime}\in\mathcal{L}^{q}(T)$ and the
Toeplitz operator $T_{\varphi}$ is bounded on $\mathcal{L}^{p}(T)$, where
$\frac{1}{p}+\frac{1}{q}=1$. Then $T_{\varphi}$ has finite rank on
$\mathcal{L}^{p}(T)$ if and only if $\varphi(v)=0$ except for finitely many
points $v\in T$.

###### Proof.

We show the sufficiency first. Suppose that $\varphi(v)=0$ for all $v\in T$
except $v_{1},v_{2},\cdots,v_{n}$. For any $f\in\mathcal{L}^{p}(T)$, we have

$\displaystyle T_{\varphi}f(v)=\Delta(\varphi\nabla f)(v)=\sum_{u\in
S_{v}}\varphi(u)\nabla f(u),\ \ \ \ v\in T.$

It follows that $T_{\varphi}f(v)=0$ for all $v\in T$ with
$|v|>\max\\{|v_{k}|:k=1,2,\cdots,n\\}$. Let

$W:=\Big{\\{}w\in T:|w|\leqslant\max_{1\leqslant k\leqslant
n}|v_{k}|\Big{\\}}.$

Then $W$ is a finite set and its elements can be enumerated by

$W=\\{w_{1},w_{2},\cdots,w_{N}\\}.$

Fix a vertex $\xi\in T$ and define

$\displaystyle e_{\xi}(v)=\begin{cases}1,&\mathrm{if}\ \ v=\xi,\vspace{2mm}\\\
0,&\text{otherwise}.\end{cases}$ (5.1)

Since $T_{\varphi}f$ is a vector in $\mathcal{L}^{p}(T)$, it can be expressed
as

$T_{\varphi}f=c_{1}e_{w_{1}}+c_{2}e_{w_{2}}+\cdots+c_{N}e_{w_{N}}$

with constants $c_{1},c_{2},\cdots,c_{N}$. This implies that

$\mathrm{range}(T_{\varphi})\subset\mathrm{span}\big{\\{}e_{w_{1}},e_{w_{2}},\cdots,e_{w_{N}}\big{\\}},$

to obtain that $T_{\varphi}$ is of finite rank on $\mathcal{L}^{p}(T)$.

To show the necessity, we suppose that $T_{\varphi}$ is a finite rank operator
on $\mathcal{L}^{p}(T)$ and

$\mathrm{range}(T_{\varphi})\subset\mathrm{span}\big{\\{}e_{v_{1}},e_{v_{2}},\cdots,e_{v_{n}}\big{\\}},$

where $v_{k}$ is a vertex of $T$ and $e_{v_{k}}$ is defined in (5.1).
According to Lemma 2.3 and the computations in (4.2), we obtain that

$\displaystyle\begin{split}T_{\varphi}e_{w}(v)&=e_{w}(v)\varphi(v^{-})+\sum_{u\in
S_{v}}e_{w}(u)[\varphi(u)-\varphi(u^{-})]\\\
&=e_{w}(v)\varphi(v^{-})+e_{w}(v)[\varphi(v)-\varphi(v^{-})]+\sum_{u\in
S_{v}\backslash\\{v\\}}e_{w}(u)[\varphi(u)-\varphi(u^{-})]\\\
&=e_{w}(v)\varphi(v)+\sum_{u\in
S_{v}\backslash\\{v\\}}e_{w}(u)[\varphi(u)-\varphi(u^{-})]\end{split}$ (5.2)

for every $v\neq o$ and $w\in T$.

If $\varphi$ does not vanish at infinitely many vertices of $T$, then we can
choose a vertex $w\in T$ such that
$|w|>\max\big{\\{}|v_{1}|,|v_{2}|,\cdots,|v_{n}|\big{\\}}$ and $\varphi(w)\neq
0$. It follows from (5.2) that

$T_{\varphi}e_{w}(w)=\varphi(w).$

We claim that
$T_{\varphi}e_{w}\notin\mathrm{span}\big{\\{}e_{v_{1}},e_{v_{2}},\cdots,e_{v_{n}}\big{\\}}$.
Indeed, if

$T_{\varphi}e_{w}=c_{1}e_{v_{1}}+c_{2}e_{v_{2}}+\cdots+c_{n}e_{v_{n}}$

for some scalars $c_{1},c_{2},\cdots,c_{n}$, then we would have

$0=c_{1}e_{v_{1}}(w)+c_{2}e_{v_{2}}(w)+\cdots+c_{n}e_{v_{n}}(w)=T_{\varphi}e_{w}(w)=\varphi(w)\neq
0,$

which is a contradiction. This yields that

$\mathrm{dim}\big{[}\mathrm{range}(T_{\varphi})\big{]}>n.$

But this contradicts that the dimension of $\mathrm{range}(T_{\varphi})$ is
less than or equal to $n$. Therefore, the proof of Theorem 5.1 is finished. ∎

######  Remark 5.2.

It is worth noting that Theorem 5.1 is analogous to the corresponding results
for finite rank Toeplitz operators on the Bergman and Fock spaces, see [13]
and [25, Theorem 6.42] respectively.

### Acknowledgment

This work was partially supported by NSFC (grant number: 11701052) and
Chongqing Natural Science Foundation (cstc2019jcyj-msxmX0337). The third
author was partially supported by the Fundamental Research Funds for the
Central Universities (grant numbers: 2020CDJQY-A039, 2020CDJ-LHSS-003).

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