# The Douglas question on the Bergman and Fock spaces

Jian-hua Chen  School of Mathematical and Computational Science, Hunan
University of Science and Technology, Xiangtan, Hunan, 411201, P. R. China
<EMAIL_ADDRESS>,  Qianrui Leng  College of Mathematics and
Statistics, Chongqing University, Chongqing, 401331, P. R. China
<EMAIL_ADDRESS>and  Xianfeng Zhao  College of Mathematics and Statistics,
Chongqing University, Chongqing, 401331, P. R. China<EMAIL_ADDRESS>
###### Abstract.

Let $\mu$ be a positive Borel measure and $T_{\mu}$ be the bounded Toeplitz
operator induced by $\mu$ on the Bergman or Fock space. In this paper, we
mainly investigate the invertibility of the Toeplitz operator $T_{\mu}$ and
the Douglas question on the Bergman and Fock spaces. In the Bergman-space
setting, we obtain several necessary and sufficient conditions for the
invertibility of $T_{\mu}$ in terms of the Berezin transform of $\mu$ and the
reverse Carleson condition in two classical cases: (1) $\mu$ is absolutely
continuous with respect to the normalized area measure on the open unit disk
$\mathbb{D}$; (2) $\mu$ is the pull-back measure of the normalized area
measure under an analytic self-mapping of $\mathbb{D}$. Nonetheless, we show
that there exists a Carleson measure for the Bergman space such that its
Berezin transform is bounded below but the corresponding Toeplitz operator is
not invertible. On the Fock space, we show that $T_{\mu}$ is invertible if and
only if $\mu$ is a reverse Carleson measure, but the invertibility of
$T_{\mu}$ is not completely determined by the invertibility of the Berezin
transform of $\mu$. These suggest that the answers to the Douglas question for
Toeplitz operators induced by positive measures on the Bergman and Fock spaces
are both negative in general cases.

###### Key words and phrases:

Toeplitz operator; invertibility; Douglas question; positive measure; reverse
Carleson measure

###### 2010 Mathematics Subject Classification:

47B35.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Necessary knowledge for the Bergman space
    2. 2.2 Necessary knowledge for the Fock space
  3. 3 The Douglas question on the Bergman space
  4. 4 The Douglas question on the Fock space

##  1\. Introduction

Let us begin with certain notation and basic knowledge about the Bergman
space, the Fock space and Toeplitz operators on them. Let
$dA(z)=\frac{1}{\pi}dxdy$ denote the normalized Lebesgue area measure on the
complex plane $\mathbb{C}$. $L^{2}(\mathbb{D},dA)$ denotes the space of the
Lebesgue measurable functions on the open unit disk
$\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$ with the following norm:

$\|f\|_{L^{2}(\mathbb{D},dA)}=\bigg{[}\int_{\mathbb{D}}|f(z)|^{2}dA(z)\bigg{]}^{\frac{1}{2}}<+\infty.$

The Bergman space $L_{a}^{2}$ is the closed subspace of $L^{2}(\mathbb{D},dA)$
consisting of analytic functions on $\mathbb{D}$, which is a Hilbert space
with the inner product

$\langle f,g\rangle_{L_{a}^{2}}=\int_{\mathbb{D}}f(z)\overline{g(z)}dA(z).$

Recall that $L_{a}^{2}$ is a reproducing kernel Hilbert space and the
reproducing kernel at $z\in\mathbb{D}$ is given by

$K_{z}(w)=\frac{1}{(1-\overline{z}w)^{2}},\ \ \ \ w\in\mathbb{D}.$

The unit vector $k_{z}=\frac{K_{z}}{\|K_{z}\|_{L_{a}^{2}}}$ is called the
normalized reproducing kernel for $L_{a}^{2}$.

Letting $P$ be the orthogonal projection from $L^{2}(\mathbb{D},dA)$ onto
$L_{a}^{2}$, then the integral representation of $P$ is given by

$Pf(z)=\int_{\mathbb{D}}f(w)\overline{K_{z}(w)}dA(w)=\int_{\mathbb{D}}\frac{f(w)}{(1-z\overline{w})^{2}}dA(w),\
\ \ \ f\in L^{2}(\mathbb{D},dA).$

Given a function $\varphi\in L^{\infty}(\mathbb{D},dA)$, the Toeplitz operator
$T_{\varphi}$ with symbol $\varphi$ on the Bergman space $L_{a}^{2}$ is
defined by

$T_{\varphi}f=P(\varphi f),\ \ \ \ f\in L_{a}^{2}.$

Using the integral representation of $P$, we can write $T_{\varphi}$ as an
integral operator on the Bergman space:

$T_{\varphi}f(z)=\int_{\mathbb{D}}\frac{\varphi(w)f(w)}{(1-z\overline{w})^{2}}dA(w),$

where $f\in L_{a}^{2}$ and $z\in\mathbb{D}$.

Toeplitz operators can also be defined for finite Borel measures on
$\mathbb{D}$. In fact, if $\mu$ is a finite (complex) Borel measure on the
open unit disk $\mathbb{D}$, then the Toeplitz operator induced by the measure
$\mu$ is defined by

$T_{\mu}f(z)=\int_{\mathbb{D}}\frac{f(w)}{(1-z\overline{w})^{2}}d\mu(w),\ \ \
f\in L_{a}^{2}.$

In this case, the Berezin transform of $\mu$ (or $T_{\mu}$) is the function
defined by

$\widetilde{\mu}(z)=\int_{\mathbb{D}}|k_{z}(w)|^{2}d\mu(w),\ \ \ \
z\in\mathbb{D}.$

If $d\mu=\varphi dA$ for a function $\varphi\in L^{1}(\mathbb{D},dA)$, then we
simply write $T_{\mu}=T_{\varphi}$ and

$\widetilde{\mu}(z)=\int_{\mathbb{D}}\varphi(w)|k_{z}(w)|^{2}dA(w)$

is usually denoted by $\widetilde{\varphi}$, which is also called the Berezin
transform of $\varphi$. In particular, when $\varphi\in
L^{\infty}(\mathbb{D},dA)$, we return to the classical Toeplitz operator
defined in the preceding paragraph.

We now turn to the introduction of the Fock space and Toeplitz operators on
such space. Let

$d\lambda(z)=\frac{1}{2}\mathrm{e}^{-\frac{|z|^{2}}{2}}dA(z)$

be the Gaussian measure on the complex plane $\mathbb{C}$. The Fock space
$\mathcal{F}^{2}$ consists of all entire functions $f$ in
$L^{2}(\mathbb{C},d\lambda)$. One can show easily that $\mathcal{F}^{2}$ is a
closed subspace of $L^{2}(\mathbb{C},d\lambda)$. Consequently, the Fock space
$\mathcal{F}^{2}$ is a Hilbert space with the inner product:

$\langle
f,g\rangle_{\mathcal{F}^{2}}=\int_{\mathbb{C}}f(z)\overline{g(z)}d\lambda(z).$

Moreover, $\mathcal{F}^{2}$ is also a reproducing kernel Hilbert space and the
reproducing kernel at $z\in\mathbb{C}$ is given by

$K_{z}^{(\mathcal{F}^{2})}(w)=\mathrm{e}^{\frac{\overline{z}w}{2}},\ \ \ \
w\in\mathbb{C}.$

Similar to the Bergman space, the orthogonal projection
$P^{(\mathcal{F}^{2})}$ from $L^{2}(\mathbb{C},d\lambda)$ onto
$\mathcal{F}^{2}$ is the following integral operator:

$\displaystyle
P^{(\mathcal{F}^{2})}f(z)=\int_{\mathbb{C}}f(w)\overline{K_{z}^{(\mathcal{F}^{2})}(w)}d\lambda(w)=\int_{\mathbb{C}}f(w)\mathrm{e}^{\frac{z\overline{w}}{2}}dA(w),\
\ \ \ f\in L^{2}(\mathbb{C},d\lambda).$

Given a function $\varphi\in L^{\infty}(\mathbb{C},dA)$, the Toeplitz operator
$T_{\varphi}$ with symbol $\varphi$ on the Fock space is defined by

$\displaystyle T_{\varphi}f=P^{(\mathcal{F}^{2})}(\varphi f),\ \ \ \
f\in\mathcal{F}^{2},$

which can be written as

$\displaystyle
T_{\varphi}f(z)=\int_{\mathbb{C}}f(w)\varphi(w)\mathrm{e}^{\frac{z\overline{w}}{2}}d\lambda(w),$

where $f\in\mathcal{F}^{2}$ and $z\in\mathbb{C}$.

Let $\mu$ be a complex Borel measure on $\mathbb{C}$ satisfying the _condition
(M)_ , that is,

$\displaystyle\int_{\mathbb{C}}\big{|}K_{z}^{(\mathcal{F}^{2})}(w)\big{|}^{2}e^{-\frac{|w|^{2}}{2}}d|\mu|(w)<+\infty$
(1.1)

for all $z\in\mathbb{C}$, where $|\mu|$ denotes the total variation of $\mu$.
Under this assumption, the Toeplitz operator induced by $\mu$ can be defined
on a dense subspace of $\mathcal{F}^{2}$:

$T_{\mu}f(z)=\frac{1}{2}\int_{\mathbb{C}}f(w)\overline{K_{z}^{(\mathcal{F}^{2})}(w)}\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w),\
\ \ \ z\in\mathbb{C}.$

Condition (M) also guarantees that the Berezin transform of $\mu$ is well-
defined:

$\widetilde{\mu}(z)=\frac{1}{2}\int_{\mathbb{C}}|k_{z}^{(\mathcal{F}^{2})}(w)|^{2}\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w)=\frac{1}{2}\int_{\mathbb{C}}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w),$

where

$k_{z}^{(\mathcal{F}^{2})}(w)=\frac{K_{z}^{(\mathcal{F}^{2})}(w)}{\|K_{z}^{(\mathcal{F}^{2})}\|_{\mathcal{F}^{2}}}=\mathrm{e}^{\frac{w\overline{z}}{2}-\frac{|z|^{2}}{4}}\
\ \ \ \ (z,w\in\mathbb{C})$

is the normalized reproducing kernel for $\mathcal{F}^{2}$. When $d\mu=\varphi
dA$ with $\varphi\in L^{\infty}(\mathbb{C},d\lambda)$, we get back to
$T_{\varphi}$, and in this case $\widetilde{\mu}$ is again denoted by
$\widetilde{\varphi}$, which is also called the Berezin transform of
$\varphi$. Notice that the dual use of the notation “$T_{\varphi}$” and
“$T_{\mu}$” for Toeplitz operators on the Bergman and Fock spaces should cause
no confusion.

For more knowledge about the Bergman and Fock spaces, the Berezin transform
and Toeplitz operators on such spaces, one can consult for instance [1], [27],
[2] and [20]. Usually, Toeplitz operators on the Bergman space are called
“Bergman-Toeplitz operators” and Toeplitz operators on the Fock space are
called “Fock-Toeplitz operators” for short.

Toeplitz operators induced by measures were investigated by many authors in
the past few decades. However, Luecking [12] is the first to consider Toeplitz
operators on the Bergman space with measures as symbols. Later, using the
Berezin transform and _Carleson measures_ (which will be introduced in the
next section), Zhu characterized the boundedness and compactness of Toeplitz
operators induced by positive measures on weighted Bergman spaces of bounded
symmetric domains, see [25] and [27]. The characterization of Schatten class
Toeplitz operators with measures as symbols on the Bergman space of the unit
ball was obtained by Zhu [26, 27]. In 2008, Luecking [13] studied the finite
rank Toeplitz operator induced by a complex measure on the Bergman space.

An especially important but quite difficult problem in the Toeplitz operator
theory is to determine the invertibility of Toeplitz operators in terms of the
properties of their symbols. The original Douglas question is an influential
open question concerning the invertibility problem of Hardy-Toeplitz
operators, which says that: _if the harmonic extension (poisson integral) of
$\varphi$ is bounded below on the unit disk $\mathbb{D}$, then is the Toeplitz
operator with symbol $\varphi$ invertible on the Hardy space?_ The original
Douglas question was studied by many authors, for instance, Tolokonnikov [17],
Nikolskii [15], Wolff [19] and Karaev [10]. Observe that the harmonic
extension of a bounded function $\varphi$ defined on $\partial\mathbb{D}$ is
equal to the Berezin transform of the Toeplitz operator with symbol $\varphi$
on the Hardy space. This leads to the following natural question for Bergman-
Toeplitz operators analogous to the Douglas question on the Hardy space:

######  Question 1.1.

Let $\varphi$ be in $L^{\infty}(\mathbb{D},dA)$. Then is $T_{\varphi}$
invertible on the Bergman space $L_{a}^{2}$ if the Berezin transform
$|\widetilde{\varphi}(z)|\geqslant\delta$ for some positive constant $\delta$
and for all $z\in\mathbb{D}$?

The above question is answered positively for Bergman-Toeplitz operators with
bounded (conjugate) analytic symbols, and real harmonic symbols in [14]. In
[11], by constructing a reverse Carleson measure for the Bergman space
$L_{a}^{2}$, Luecking established a number of necessary and sufficient
conditions for Toeplitz operators with bounded positive symbols to be
invertible on the Bergman space. Based on Luecking’s work, Zhao and Zheng
showed that the answer to Question 1.1 is also affirmative for Bergman-
Toeplitz operators with bounded positive symbols [21]. Čučković and Vasaturo
[3] obtained a necessary and sufficient condition for the invertibility of
Toeplitz operators whose symbols are averaging functions of Carleson measures
for the (standard) weighted Bergman space. Recently, Guo, Zhao and Zheng [22,
7] provided some partial answers to this question for Bergman-Toeplitz
operators with some harmonic polynomials. However, Question 1.1 remains open
for general harmonic symbols.

In the present paper, we are interested in the invertibility problem of
Bergman-Toeplitz and Fock-Toeplitz operators with positive measures as
symbols. In particular, we will focus on the Douglas question for Toeplitz
operators induced by positive measures on the Bergman and Fock spaces:

######  Question 1.2.

Suppose that $\mu$ is a finite positive Borel measure on the unit disk
$\mathbb{D}$ and $T_{\mu}$ is bounded on the Bergman space $L_{a}^{2}$. Then
is $T_{\mu}$ invertible on $L_{a}^{2}$ if the Berezin transform of $\mu$ is
bounded below on $\mathbb{D}$?

######  Question 1.3.

Suppose that $\mu$ is a positive Borel measure on the complex plane
$\mathbb{C}$ and $T_{\mu}$ is bounded on the Fock space $\mathcal{F}^{2}$.
Then is $T_{\mu}$ invertible on $\mathcal{F}^{2}$ if the Berezin transform of
$\mu$ is bounded below on $\mathbb{C}$?

As mentioned above, the answer to Question 1.2 is positive in the case that
$d\mu=\varphi dA$, where $\varphi$ is a nonnegative function in
$L^{\infty}(\mathbb{D},dA)$. Apart from this, very little is known about the
invertibility of Toeplitz operators induced by measures, even if the measure
is positive. Using the Berezin transform and the reverse Carleson condition,
in Section 3 we establish a number of necessary and sufficient conditions for
Toeplitz operators with positive measures as symbols to be invertible on the
Bergman space in two classical cases. More precisely, in the case that $\mu$
is absolutely continuous with respect to the Lebesgue measure $dA$ and the
Radon-Nikodym derivative of $d\mu$ with respect to $dA$ is bounded, we show in
Theorem 3.4 that $T_{\mu}$ is invertible if and only if the Berezin transform
of $\mu$ is invertible in $L^{\infty}(\mathbb{D},dA)$, if and only if $\mu$
satisfies the _reverse Carleson condition for $L_{a}^{2}$_ (see (2.2) for the
definition). For the case that $\mu$ is the pull-back measure of $dA$ under an
analytic self-mapping of $\mathbb{D}$, we show in Theorem 3.6 that the
invertibility of $T_{\mu}$ and the Berezin transform of $\mu$ are equivalent.
Note that the Berezin transform of $\mu$ is bounded below on $\mathbb{D}$
provided that $T_{\mu}$ is bounded below on the Bergman space. Combining this
with the conclusions in Theorems 3.4 and 3.6, one may conjecture that the
answer to Question 1.2 is affirmative. Surprisingly, we will show that there
exists a finite positive Borel measure on $\mathbb{D}$ such that the Berezin
transform of $\mu$ is invertible in $L^{\infty}(\mathbb{D},dA)$, but the
corresponding Toeplitz operator $T_{\mu}$ is not invertible on the Bergman
space (see Theorem 3.11). This implies that the answer to Question 1.2 is
negative in general.

In the setting of the Fock space, Question 1.3 was answered positively by Wang
and Zhao [18] when $d\mu=\varphi dA$, where $\varphi$ is a bounded positive
function on the complex plane. However, the answer to this question is unknown
for general positive measures. In the final section, we investigate the
invertibility of Fock-Toeplitz operators induced by positive measures and
discuss the analogues of Theorems 3.4 and 3.11 in the Fock-space setting.
Specifically, we establish in Theorem 4.4 that the Fock-Toeplitz operator
$T_{\mu}$ is invertible if and only if $\mu$ is a reverse Carleson measure for
the Fock space, where $\mu$ is a positive Borel measure on the complex plane
$\mathbb{C}$. Based on this result, in Theorem 4.7 we establish several
equivalent characterizations for $T_{\mu}$ to be invertible on the Fock space
via the Berezin transform and the _reverse Carleson condition for
$\mathcal{F}^{2}$_ (see (2.3) for the definition). In addition, we show in
Theorem 4.9 that there exists a positive Borel measure $\nu$ on $\mathbb{C}$
such that $T_{\nu}$ is not invertible on the Fock space but the Berezin
transform of $\nu$ is bounded below on $\mathbb{C}$. Therefore, the answer to
the Douglas question for Toeplitz operators induced by positive measures on
the Fock space is also negative.

##  2\. Preliminaries

Before studying the invertibility of Toeplitz operators induced by positive
measures on the Bergman and Fock spaces, we need to fix some notation and
review certain necessary preliminary knowledge first.

###  2.1. Necessary knowledge for the Bergman space

For $w\in\mathbb{D}$, let $\varphi_{w}$ be the Möbius transform defined by

$\varphi_{w}(z)=\frac{w-z}{1-\overline{w}z},\ \ \ \ z\in\mathbb{D}.$

Note that the function $\varphi_{w}$ is a one-to-one analytic mapping of
$\mathbb{D}$ onto $\mathbb{D}$. Let $w$ and $z$ be two points in $\mathbb{D}$.
The pseudo-hyperbolic distance $\rho(w,z)$ and the Bergman distance
$\beta(w,z)$ between $w$ and $z$ are defined by

$\rho(z,w)=|\varphi_{w}(z)|\ \ \ \ \ \textrm{and}\ \ \ \ \
\beta(z,w)=\tanh^{-1}{\rho(z,w)},$

respectively. An application of Schwarz’s lemma shows that $\rho$ is a metric
on the unit disk $\mathbb{D}$, see [5, Lemma 1.4] if necessary. Thus, $\beta$
is also a metric on $\mathbb{D}$. In addition, the metrics $\rho$ and $\beta$
are both invariant under the action of Möbius transforms.

Let $w\in\mathbb{C}$ and $r>0$. In the following, we use $B(w,r)$ to denote
the Euclidean disk with center $w$ and radius $r$:

$B(w,r)=\\{z\in\mathbb{C}:|w-z|<r\\}.$

The pseudo-hyperbolic disk $E(w,r)$ with center $w\in\mathbb{D}$ and radius
$r\in(0,1)$ is defined by

$E(w,r)=\\{z\in\mathbb{D}:\rho(w,z)<r\\}.$

Moreover, the Bergman disk $D(w,r)$ with center $w\in\mathbb{D}$ and radius
$r$ is defined by

$D(w,r)=\\{z\in\mathbb{D}:\beta(w,z)<r\\}.$

From [27, Proposition 4.4], the Lebesgue measure of $E(z,r)$ is comparable to
the Lebesgue measure of $E(w,r)$ when $\rho(z,w)$ stays bounded:

$\displaystyle A(E(z,r))\asymp A(E(w,r))$ (2.1)

when $\rho(z,w)<c$, where $c\in(0,1)$ is a constant. Here and in the
following, the notation $A\lesssim B$ for two nonnegative quantities $A$ and
$B$ means that there is an inessential positive constant $C_{1}$ such that
$A\leqslant C_{1}B$. Similarly, the notation $A\gtrsim B$ stands for that
there is an inessential positive constant $C_{2}$ such that $A\geqslant
C_{2}B$. By $A\asymp B$ for two nonnegative quantities $A$ and $B$, we mean
that $A\lesssim B$ and $A\gtrsim B$.

Let $\mu$ be a finite positive Borel measure on $\mathbb{D}$. We say that
$\mu$ is a _Carleson measure for_ $L_{a}^{2}$ if there exists an absolute
constant $C>0$ such that

$\int_{\mathbb{D}}|f(z)|^{2}d\mu(z)\leqslant
C\int_{\mathbb{D}}|f(z)|^{2}dA(z)$

for all $f\in L_{a}^{2}$; We say that $\mu$ is a _reverse Carleson measure
for_ $L_{a}^{2}$ if there exists an absolute constant $C>0$ such that

$\int_{\mathbb{D}}|f(z)|^{2}d\mu(z)\geqslant
C\int_{\mathbb{D}}|f(z)|^{2}dA(z)$

for all $f\in L_{a}^{2}$. Moreover, we call that $\mu$ satisfies the _reverse
Carleson condition_ if there is an $r\in(0,1)$ and a positive constant $C$ so
that

$\displaystyle C^{-1}\leqslant\frac{\mu(E(z,r))}{A(E(z,r))}\leqslant C$ (2.2)

for all $z\in\mathbb{D}$.

It is well-known that $\mu$ is a Carleson measure for $L_{a}^{2}$ if and only
if for every radius $r\in(0,1)$, we can find a positive constant $C$ depending
only on $r$ such that

$\frac{\mu(E(z,r))}{A(E(z,r))}\leqslant C$

for all $z\in\mathbb{D}$, see [27] if needed. Using the reverse Carleson
condition for $L_{a}^{2}$, Ghatage and Tjani [6] provided some necessary and
sufficient conditions for a composition operator to have closed range on the
Bergman space.

Let $G$ be a measurable subset of $\mathbb{D}$ and $\chi_{G}$ be the
characteristic function of $G$. It was established by Luecking [11] that
$\chi_{G}dA$ is a reverse Carleson measure for $L_{a}^{2}$ if and only if
there is a $\delta>0$ and an $\eta\in(0,1)$ such that

$A(G\cap E(a,\eta))>\delta A(E(a,\eta))$

for all $a\in\mathbb{D}$, which is equivalent to that the measure $\chi_{G}dA$
satisfies the reverse Carleson condition for $L_{a}^{2}$. Moreover, Ghatage
and Tjani [6] showed that the Carleson measure for the Bergman space has the
following important property:

######  Lemma 2.1.

_([6, Proposition 4.1]) _If $\mu$ is a Carleson measure for the Bergman space,
then

$\lim_{r\rightarrow 1^{-}}\sup_{z\in\mathbb{D}}\int_{\mathbb{D}\backslash
E(z,r)}\frac{(1-|z|^{2})^{2}}{|1-\overline{z}w|^{4}}d\mu(w)=0.$

We end this subsection by introducing the definitions of separated, sampling
and interpolation sequences for the Bergman space, which will be needed later
on. For more information concerning these concepts, one can consult [4], [9]
and [16].

######  Definition 2.2.

Let $\\{u_{k}\\}_{k=1}^{\infty}$ be a sequence in the open unit disk
$\mathbb{D}$.

  * •

$\\{u_{k}\\}_{k=1}^{\infty}$ is called uniformly discrete or separated if
there is a constant $\varepsilon\in(0,1)$ such that

$\rho(u_{k},u_{j})>\varepsilon\ \ \ \text{for}\ \ \ k\neq j.$

In this case, the separation constant of $\\{u_{k}\\}_{k=1}^{\infty}$ is
defined by

$\delta(\\{u_{k}\\}_{k=1}^{\infty})=\inf_{k\neq j}{\rho(a_{k},a_{j})}.$

  * •

We call $\\{u_{k}\\}_{k=1}^{\infty}$ an $r$-net if

$\mathbb{D}=\bigcup_{k=1}^{\infty}{E(u_{k},r)}$

for some $r\in(0,1)$.

  * •

If $u_{k}\neq u_{j}$ for $k\neq j$ and

$\displaystyle\int_{\mathbb{D}}|f(z)|^{2}dA(z)\asymp\sum_{k=1}^{\infty}{(1-|u_{k}|^{2})^{2}|f(u_{k})|^{2}},\
\ \ \ f\in L_{a}^{2},$

then $\\{u_{k}\\}_{k=1}^{\infty}$ is called a sampling sequence for
$L_{a}^{2}$.

  * •

If $u_{k}\neq u_{j}$ for $k\neq j$ and the interpolation problem
$f(u_{k})=w_{k}$ _(_ $k=1,2,\cdots$_)_ has a solution $f\in L_{a}^{2}$
whenever

$\displaystyle\sum_{k=1}^{\infty}{(1-|u_{k}|^{2})^{2}|w_{k}|^{2}}<+\infty,$

then we say that $\\{u_{k}\\}_{k=1}^{\infty}$ is an interpolation sequence for
$L_{a}^{2}$.

There is an important class of uniformly discrete nets which is called the
lattice. A lattice for $L_{a}^{2}$ is a sequence $\\{u_{k}\\}_{k=1}^{\infty}$
of distinct points in $\mathbb{D}$ such that

$\displaystyle\mathbb{D}=\bigcup_{k=1}^{\infty}{D(u_{k},R)}=\bigcup_{k=1}^{\infty}{E(u_{k},r)}$

and $\beta(u_{k},u_{j})\geqslant\frac{R}{2}$ for $k\neq j$, where $R>0$ and
$r=\tanh{(R)}$. Clearly, $\\{u_{k}\\}_{k=1}^{\infty}$ is an $r$-net with the
separation constant greater than or equal to $\tanh{(\frac{R}{2})}$.

###  2.2. Necessary knowledge for the Fock space

Let $\mu$ be a positive Borel measure $\mu$ on the complex plane $\mathbb{C}$.
Then $\mu$ is called a _Carleson measure for the Fock space $\mathcal{F}^{2}$_
if there exists a constant $C>0$ such that

$\displaystyle\int_{\mathbb{C}}|f(w)|^{2}\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w)\leqslant
C\int_{\mathbb{C}}|f(w)|^{2}\mathrm{e}^{-\frac{|w|^{2}}{2}}dA(w),\ \ \ \ \
f\in\mathcal{F}^{2}.$

On the other hand, $\mu$ is called a _reverse Carleson measure for
$\mathcal{F}^{2}$_ if there exists a constant $C>0$ such that

$\displaystyle\int_{\mathbb{C}}|f(w)|^{2}\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w)\geqslant
C\int_{\mathbb{C}}|f(w)|^{2}\mathrm{e}^{-\frac{|w|^{2}}{2}}dA(w),\ \ \ \ \
f\in\mathcal{F}^{2}.$

Like in the case of the Bergman space, we say that a positive Borel measure
$\mu$ on $\mathbb{C}$ satisfies the _reverse Carleson condition for
$\mathcal{F}^{2}$_ if there exists an $r>0$ and a positive constant $C$ such
that

$\displaystyle C^{-1}\leqslant\mu(B(z,r))\leqslant C,\ \ \ \ \
z\in\mathbb{C}.$ (2.3)

There are several concepts for the Fock space parallel to those in Definition
2.2, which can be found in Chapter 4 of the book [28].

######  Definition 2.3.

Let $\\{u_{k}\\}_{k=1}^{\infty}$ be a sequence in the complex plane
$\mathbb{C}$.

  * •

$\\{u_{k}\\}_{k=1}^{\infty}$ is called uniformly discrete or separated if
there is a constant $\varepsilon>0$ such that

$|u_{k}-u_{j}|>\varepsilon\ \ \ \text{for}\ \ \ k\neq j.$

  * •

We call $\\{u_{k}\\}_{k=1}^{\infty}$ an $r$-net if

$\mathbb{C}=\bigcup_{k=1}^{\infty}{B(u_{k},r)}$

for some $r>0$.

  * •

If $u_{k}\neq u_{j}$ for $k\neq j$ and

$\displaystyle\int_{\mathbb{D}}|f(z)|^{2}d\lambda(z)\asymp\sum_{k=1}^{\infty}{|f(u_{k})|^{2}\mathrm{e}^{-\frac{|u_{k}|^{2}}{2}}}$

for $f\in\mathcal{F}^{2}$, then $\\{u_{k}\\}_{k=1}^{\infty}$ is called a
sampling sequence for the Fock space $\mathcal{F}^{2}$.

In the setting of the Fock space, we can also define a lattice. Let $\omega$
be any complex number and $\omega_{1}$ and $\omega_{2}$ be any two nonzero
complex numbers such that their ratio is not real. The set

$\Lambda(\omega,\omega_{1},\omega_{2})=\\{\omega+m\omega_{1}+n\omega_{2}:m,n\in\mathbb{Z}\\}$

is called the lattice for $\mathcal{F}^{2}$ generated by $\omega$,
$\omega_{1}$ and $\omega_{2}$, where $\mathbb{Z}$ denotes the set of integers.
In the following, we only consider the case when $\omega=0$, $\omega_{1}=r$
and $\omega_{2}=\mathrm{i}r$ with $r>0$. In this case, we denote

$\displaystyle\Lambda(0,r,\mathrm{i}r)=\\{a_{k}\\}_{k=1}^{\infty}.$ (2.4)

Clearly, $\Lambda(0,r,\mathrm{i}r)$ is a separated $2r$-net for
$\mathcal{F}^{2}$.

##  3\. The Douglas question on the Bergman space

In Section 3, we study the invertibility of Bergman-Toeplitz operators with
positive measures as symbols and the Douglas question for such class of
Toeplitz operators. This section is organized as follows. Firstly, we discuss
two classic kinds of Bergman-Toeplitz operators:

  * Case I. $d\mu=\varphi dA$, where $\varphi$ is a nonnegative function in $L^{\infty}(\mathbb{D},dA)$;

  * Case II. $\mu$ is the pull-back measure of $dA$ under an analytic self-mapping of $\mathbb{D}$.

Both of the above two cases provide a positive answer to the Douglas question,
i.e., the invertibility of $T_{\mu}$ and $\widetilde{\mu}$ are equivalent.
Secondly, using Lemmas 3.2 and 3.3 (which give equivalent characterizations
for the invertibility of $T_{\mu}$ and $\widetilde{\mu}$) we transform the
study of the Douglas question into the discussion of properties of $\mu$. Then
by constructing a special kind of measures, we can answer Question 1.2
negatively.

Let us begin with the following three lemmas, which provide characterizations
for the boundedness and invertibility of a Toeplitz operator and its Berezin
transform. The latter two are particularly important for the investigation of
the Douglas question on the Bergman space.

######  Lemma 3.1.

_([27, Theorem 7.5])_ Let $\mu$ be a finite positive Borel measure on the unit
disk $\mathbb{D}$. Then the following are equivalent:
$(a)$ $T_{\mu}$ is bounded on the Bergman space $L_{a}^{2}$;
$(b)$ $\widetilde{\mu}$ is bounded on $\mathbb{D}$;
$(c)$ $\mu$ is a Carleson measure for $L_{a}^{2}$.

######  Lemma 3.2.

_([3, Proposition 2])_ Let $\mu$ be a finite positive Borel measure on
$\mathbb{D}$. If the Toeplitz operator $T_{\mu}$ is bounded on the Bergman
space $L_{a}^{2}$, then the following are equivalent:
$(a)$ $T_{\mu}$ is invertible on $L_{a}^{2}$;
$(b)$ $\mu$ is a reverse Carleson measure for $L_{a}^{2}$.

######  Lemma 3.3.

_([6, Theorem 4.1])_ A Carleson measure $\mu$ for the Bergman space satisfies
the reverse Carleson condition if and only if the Berezin transform
$\widetilde{\mu}$ is bounded above and below on $\mathbb{D}$.

Applying Lemma 3.3 and combining the conclusions in [11, 21] and Lemma 3.2, we
obtain the following five equivalent characterizations for the invertibility
of Bergman-Toeplitz operators in Case I.

######  Theorem 3.4.

Let $\varphi$ be a bounded positive function on $\mathbb{D}$. If $d\mu=\varphi
dA$, then the following are equivalent:
$(a)$ $T_{\mu}=T_{\varphi}$ is invertible on $L_{a}^{2}$;
$(b)$ $\widetilde{\mu}=\widetilde{\varphi}$ is invertible in
$L^{\infty}(\mathbb{D},dA)$, i.e., $\widetilde{\mu}$ is bounded below on
$\mathbb{D}$;
$(c)$ There exists an $r>0$ such that
$\chi_{\\{z\in\mathbb{D}:\varphi(z)>r\\}}dA$ is a reverse Carleson measure for
$L_{a}^{2}$;
$(d)$ $\mu$ is a reverse Carleson measure for $L_{a}^{2}$;
$(e)$ $\mu$ satisfies the reverse Carleson condition _( 2.2)_.

###### Proof.

$(a)\Leftrightarrow(d)$ and $(b)\Leftrightarrow(e)$ are special cases of
Lemmas 3.2 and 3.3. $(a)\Leftrightarrow(c)$ was established in [11, Corollary
3]. Lastly, $(a)\Leftrightarrow(b)$ was proven in [21, Theorem 3.2]. ∎

Next, we turn to the study of the invertibility of Bergman-Toeplitz operators
in Case II. Suppose that $\varphi$ is an analytic self-mapping of
$\mathbb{D}$, i.e., $\varphi$ is an analytic mapping from $\mathbb{D}$ to
$\mathbb{D}$. Let $\mu_{\varphi}$ be the pull-back measure of the normalized
area measure on $\mathbb{D}$ under the mapping $\varphi$ defined on Borel
subsets of $\mathbb{D}$ by

$\mu_{\varphi}(E)=A(\varphi^{-1}(E)).$

In the following, we will characterize the invertibility of
$T_{\mu_{\varphi}}$ in terms of the Berezin transform
$\widetilde{\mu_{\varphi}}$ and the composition operator $C_{\varphi}$ (which
is defined by $C_{\varphi}f=f\circ\varphi$) on the Bergman space $L_{a}^{2}$.

Recalling that Littlewood’s subordination theorem tells us that $C_{\varphi}$
is bounded on the Bergman space, we have

$\widetilde{\mu_{\varphi}}(z)=\int_{\mathbb{D}}|k_{z}(w)|^{2}d\mu_{\varphi}(w)=\int_{\mathbb{D}}|k_{z}(\varphi(w))|^{2}dA(w)=\|C_{\varphi}k_{z}\|^{2}_{L_{a}^{2}}\leqslant\|C_{\varphi}\|^{2}.$

It follows that $\widetilde{\mu_{\varphi}}$ is bounded on $\mathbb{D}$.
According to Lemma 3.1, we have that $\mu_{\varphi}$ is a Carleson measure for
$L_{a}^{2}$ and the Toeplitz operator $T_{\mu_{\varphi}}$ is bounded.
Furthermore, the following relationship between $C_{\varphi}$ and
$T_{\mu_{\varphi}}$ was obtained in [27] (see the identity in (11.5) of
Section 11.3):

$T_{\mu_{\varphi}}f=C_{\varphi}^{*}C_{\varphi}f,\ \ \ \ f\in L_{a}^{2}.$

The following lemma characterizes when the composition operator $C_{\varphi}$
has closed range on the Bergman space, which was established in [6, Theorems
4.2 and 4.4].

######  Lemma 3.5.

Let $\varphi$ be a non-constant analytic self-mapping of the unit disk
$\mathbb{D}$. Then the following statements are equivalent:
$(a)$ $\mu_{\varphi}$ satisfies the reverse Carleson condition;
$(b)$ $\widetilde{\mu_{\varphi}}$ is bounded below on $\mathbb{D}$;
$(c)$ $C_{\varphi}$ has closed range on the Bergman space $L_{a}^{2}$.

Using Lemma 3.5, we are able to establish several necessary and sufficient
conditions concerning the invertibility of the Bergman-Toeplitz operator
$T_{\mu_{\varphi}}$ via the reverse Carleson measure for $L_{a}^{2}$ and the
Berezin transform of $\mu_{\varphi}$.

######  Theorem 3.6.

Let $\varphi$ be a non-constant analytic self-mapping of $\mathbb{D}$ and
$d\mu_{\varphi}=dA\circ\varphi^{-1}$. Then the following five conditions are
equivalent:
$(a)$ $T_{\mu_{\varphi}}$ is invertible on $L_{a}^{2}$;
$(b)$ $\widetilde{\mu_{\varphi}}$ is bounded below on $\mathbb{D}$;
$(c)$ $\mu_{\varphi}$ satisfies the reverse Carleson condition _( 2.2)_;
$(d)$ $C_{\varphi}$ has closed range on $L_{a}^{2}$;
$(e)$ $\mu_{\varphi}$ is a reverse Carleson measure for $L_{a}^{2}$.

###### Proof.

Note that $(a)\Rightarrow(b)$ is obvious and
$(b)\Leftrightarrow(c)\Leftrightarrow(d)$ were established in Lemma 3.5. In
addition, $(a)\Leftrightarrow(e)$ was obtained in Lemma 3.2. Therefore, it is
sufficient to show that $(d)\Rightarrow(a)$.

Suppose that $(d)$ holds, i.e., the composition operator $C_{\varphi}$ has
closed range on the Bergman space. Since $C_{\varphi}$ is injective on
$L_{a}^{2}$, we conclude by the inverse mapping theorem that $C_{\varphi}$ is
bounded below on $L_{a}^{2}$. Thus there exists a constant $\delta>0$ such
that

$\|C_{\varphi}f\|_{L_{a}^{2}}\geqslant\delta\|f\|_{L_{a}^{2}},\ \ \ \ f\in
L_{a}^{2}.$

It follows that

$\langle
C_{\varphi}^{*}C_{\varphi}f,f\rangle_{L_{a}^{2}}=\|C_{\varphi}f\|_{L_{a}^{2}}^{2}\geqslant\delta^{2}\|f\|_{L_{a}^{2}}^{2}=\delta^{2}\langle
f,f\rangle_{L_{a}^{2}}$

for all $f\in L_{a}^{2}$. Thus we have

$C_{\varphi}^{*}C_{\varphi}\geqslant\delta^{2}I,$

which yields that $C_{\varphi}^{*}C_{\varphi}$ is invertible on $L_{a}^{2}$.
Hence $T_{\mu_{\varphi}}$ is also invertible on the Bergman space, completing
the proof of Theorem 3.6. ∎

For Toeplitz operators in Cases I and II, the key point of the proofs of
Theorems 3.4 and 3.6 is to show that $T_{\mu}$ is invertible if
$\widetilde{\mu}$ is, which relies on the specific expression of $\mu$, i.e.,
$d\mu=\varphi dA$ or $d\mu_{\varphi}=dA\circ\varphi^{-1}$.

In the rest of this section, we are going to give a negative answer to the
Douglas question for Bergman-Toeplitz operators induced by positive measures
(Question 1.2). Indeed, we will show in the end of this section that the
invertibility of a Bergman-Toeplitz operator with a positive measure $\mu$ as
the symbol is not completely determined by the invertibility of the Berezin
transform of $\mu$. According to Lemmas 3.2 and 3.3, it is sufficient to
construct a finite positive measure $\mu$ on the open unit disk $\mathbb{D}$
such that _$\mu$ is not a reverse Carleson measure for the Bergman space but
satisfies the reverse Carleson condition (2.2)_.

Because the invertibility of a Toeplitz operator and its Berezin transform is
equivalent in Case I (see Theorem 3.4), to solve the Douglas question for
Bergman-Toeplitz operators induced by general positive measures we try to
consider the measures which are singular with respect to $dA$. This inspires
the following observation.

######  Proposition 3.7.

Suppose that $r\in(0,1)$ and $\Gamma=\\{a_{k}\\}_{k=1}^{\infty}$ is a
uniformly discrete $r$-net. Define

$\displaystyle\mu=\sum_{k=1}^{\infty}{(1-|a_{k}|^{2})^{2}\delta_{a_{k}}},$
(3.1)

where $\delta_{a_{k}}$ is the point mass concentrated at $a_{k}$. Then $\mu$
is a finite positive Borel measure satisfying the reverse Carleson condition
for $L_{a}^{2}$.

###### Proof.

Clearly, $\mu$ is a finite Borel measure on $\mathbb{D}$ and

$\mu(E(z,r))\lesssim A(E(z,r))$

for all $z\in\mathbb{D}$. Let $z$ be any point in $\mathbb{D}$. Since
$\\{E(a_{k},r)\\}_{k=1}^{\infty}$ is a covering of $\mathbb{D}$, $z$ is in
some $E(a_{k_{1}},r)$. It follows that

$\displaystyle\mu(E(z,r))\geqslant(1-|a_{k_{1}}|^{2})^{2}\asymp
A(E(a_{k_{1}},r)).$

Because $\rho(z,a_{k_{1}})$ stays bounded, by (2.1) we further have

$\displaystyle\mu(E(z,r))\gtrsim A(E(z,r)),$

finishing the proof. ∎

######  Proposition 3.8.

The measure $\mu$ defined by _( 3.1)_ is a reverse Carleson measure for
$L_{a}^{2}$ if and only if $\Gamma$ is a sampling sequence.

###### Proof.

Noting that

$\displaystyle\int_{\mathbb{D}}|f(z)|^{2}d\mu(z)=\sum_{k=1}^{\infty}{(1-|a_{k}|^{2})^{2}|f(a_{k})|^{2}},\qquad
f\in L_{a}^{2},$

the assertion follows immediately. ∎

Given an $R>0$. It follows from [27, Lemma 4.8] that there is an $R$-lattice
$\Gamma_{0}=\\{b_{k}\\}_{k=1}^{\infty}$. According to Proposition 3.7, the
measure

$\displaystyle\mu_{0}=\sum_{k=1}^{\infty}{(1-|b_{k}|^{2})^{2}\delta_{b_{k}}}$
(3.2)

also satisfies the reverse Carleson condition. Consequently, we conclude by
Proposition 3.8 that _$\mu_{0}$ is a reverse Carleson measure for the Bergman
space $L_{a}^{2}$ if and only if $\Gamma_{0}$ is a sampling sequence_.

Recently, using a characterization of sampling sequence for $L_{a}^{2}$ (see
[8, Theorem 5.23]), Zhuo and Ye [24] showed that the measure $\mu_{0}$ defined
in (3.2) is not a reverse Carleson measure for $L_{a}^{2}$ if the lower Seip
density (see (5.22) in [8] for the definition) of $\Gamma_{0}$ is less than or
equal to $\frac{1}{2}$, see Theorem 1.1 and Example 1.2 in [24] for the
details.

Next, we will further discuss how the values of $R$ influence $\mu_{0}$ to be
a reverse Carleson measure for $L_{a}^{2}$ by using the following criteria on
sampling and interpolation sequences for $L_{a}^{2}$.

######  Lemma 3.9.

_([4, Theorems 10 and 11 in Chapter 6]) _ Suppose that
$\\{u_{k}\\}_{k=1}^{\infty}$ is a uniformly discrete sequence with separation
constant $\delta$.
$(a)$ If $\\{u_{k}\\}_{k=1}^{\infty}$ is an $r$-net with $0<r<\frac{1}{2}$,
then $\\{u_{k}\\}_{k=1}^{\infty}$ is a sampling sequence;
$(b)$ If $\delta$ satisfies that

$\displaystyle(2\pi+1)\frac{\sqrt{1-\delta}}{(1-\sqrt{1-\delta})^{2}}<\frac{1}{2},$

then $\\{u_{k}\\}_{k=1}^{\infty}$ is an interpolation sequence.

It follows immediately from Lemma 3.9 that $\mu_{0}$ is a reverse Carleson
measure for $L_{a}^{2}$ when $R\rightarrow 0^{+}$. On the other hand, note
that the separation constant $\delta(\Gamma_{0})$ of $\Gamma_{0}$ is greater
than or equal to $\tanh(\frac{R}{2})$. This means that $\Gamma_{0}$ is an
interpolation sequence when $R$ is large enough, so $\Gamma_{0}$ is not a
sampling sequence in this case. This yields that $\mu_{0}$ is not a reverse
Carleson measure for $L_{a}^{2}$ if $R$ is sufficiently large. To summarize,
we have the following proposition.

######  Proposition 3.10.

Suppose that $R>0$ and $\Gamma_{0}=\\{b_{k}\\}_{k=1}^{\infty}$ is a sequence
satisfying

$\displaystyle\mathbb{D}=\bigcup_{k=1}^{\infty}{D(b_{k},R)}$

and

$\displaystyle\beta(b_{k},b_{j})\geqslant\frac{R}{2},\ \ \ \ k\neq j.$

Let $\mu_{0}$ be the measure defined by

$\mu_{0}=\sum_{k=1}^{\infty}{(1-|b_{k}|^{2})^{2}\delta_{b_{k}}}.$

Then we have:
$(a)$ $\mu_{0}$ is a reverse Carleson measure for the Bergman space if $R$ is
sufficiently close to $0$;
$(b)$ $\mu_{0}$ is not a reverse Carleson measure for the Bergman space if $R$
is large enough.

Now we are in the position to give a negative answer to Question 1.2.

######  Theorem 3.11.

There exists a finite positive Borel measure $\nu$ on the open unit disk
$\mathbb{D}$ such that the Toeplitz operator $T_{\nu}$ is not invertible on
the Bergman space, but the Berezin transform of $\nu$ is bounded below on
$\mathbb{D}$.

###### Proof.

Let $\nu$ be the positive measure constructed in (b) of Proposition 3.10. Then
we have by Proposition 3.7 that $\nu$ satisfies the reverse Carleson
condition. It follows from Lemma 3.3 that $\widetilde{\nu}$ is bounded below
on the unit disk $\mathbb{D}$.

However, Lemma 3.2 tells us that the Toeplitz operator $T_{\nu}$ is not
invertible on the Bergman space $L_{a}^{2}$, since $\nu$ is not a reverse
Carleson measure for $L_{a}^{2}$. ∎

##  4\. The Douglas question on the Fock space

In the final section, we show that the analogues of Lemma 3.2, Theorems 3.4
and 3.11 also hold in the Fock-space setting. Let $\mu$ be a positive Borel
measure on the complex plane $\mathbb{C}$ satisfying condition (M) and
$T_{\mu}$ be the Toeplitz operator induced by $\mu$ on the Fock space
$\mathcal{F}^{2}$. Let us first review some characterizations of the Carleson
measure for the Fock space.

######  Lemma 4.1.

_([25, Theorme 3.29])_ Suppose that $\mu$ is positive Borel measure on
$\mathbb{C}$. Then the following statements are equivalent:
_( $a$)_ $\mu$ is a Carleson measure for $\mathcal{F}^{2}$;
_( $b$)_ There exists a constant $C>0$ such that

$\displaystyle\int_{\mathbb{C}}|f(w)|\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w)\leqslant
C\int_{\mathbb{C}}|f(w)|\mathrm{e}^{-\frac{|w|^{2}}{2}}dA(w)$

for every entire function $f$;
_( $c$)_ There exists a constant $C>0$ such that

$\int_{\mathbb{C}}|k_{z}^{(\mathcal{F}^{2})}(w)|^{2}\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w)=\int_{\mathbb{C}}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)\leqslant
C$

for all $z\in\mathbb{C}$;
_( $d$)_ For every $r>0$, there exists a constant $C>0$ (depending only on
$r$) such that

$\mu(B(z,r))\leqslant C$

for all $z\in\mathbb{C}$.

The next proposition is parallel to Lemma 3.1.

######  Proposition 4.2.

Suppose $\mu$ is a positive Borel measure on $\mathbb{C}$ satisfying the
condition (M). Then the following are equivalent:
_$(a)$_ $T_{\mu}$ is bounded on the Fock space $\mathcal{F}^{2}$;
_$(b)$_ $\widetilde{\mu}$ is bounded on $\mathbb{C}$;
_$(c)$_ $\mu$ is a Carleson measure for $\mathcal{F}^{2}$.

###### Proof.

This can be proved by using the same technique as the one used in the proof of
Lemma 3.1. However, we shall include a detailed proof here for the sake of
completeness.

We first show $(a)\Rightarrow(b)$. Suppose that $T_{\mu}$ is bounded on the
Fock space, then the Cauchy-Schwarz inequality gives

$\widetilde{\mu}(z)=\frac{1}{2}\int_{\mathbb{C}}|k_{z}^{(\mathcal{F}^{2})}(w)|^{2}\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w)=\big{|}\langle
T_{\mu}k_{z}^{(\mathcal{F}^{2})},k_{z}^{(\mathcal{F}^{2})}\rangle_{\mathcal{F}^{2}}\big{|}\leqslant\|T_{\mu}\|,\
\ \ \ z\in\mathbb{C},$

where the second equality comes from the property of reproducing kernels for
$\mathcal{F}^{2}$.

Note that $(b)\Rightarrow(c)$ is a direct consequence of “$(c)\Rightarrow(a)$”
in Lemma 4.1. It remains to show that $(c)\Rightarrow(a)$. If $\mu$ is a
Carleson measure for $\mathcal{F}^{2}$, then for any analytic polynomial $f$
we have by Lemma 4.1 that

$\displaystyle|T_{\mu}f(z)|$
$\displaystyle\leqslant\frac{1}{2}\int_{\mathbb{C}}|f(w)|~{}|K^{(\mathcal{F}^{2})}_{w}(z)|\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w)$
$\displaystyle\lesssim\frac{1}{2}\int_{\mathbb{C}}|f(w)K^{(\mathcal{F}^{2})}_{w}(z)|\mathrm{e}^{-\frac{|w|^{2}}{2}}dA(w)$
$\displaystyle=\int_{\mathbb{C}}|f(w)K^{(\mathcal{F}^{2})}_{w}(z)|d\lambda(w)$
$\displaystyle=S(|f|)(z),\ \ \ \ z\in\mathbb{C},$

where $S$ is the linear operator defined by

$\displaystyle
Sh(z)=\int_{\mathbb{C}}h(w)|K_{z}^{(\mathcal{F}^{2})}(w)|d\lambda(w),\ \ \ \
h\in L^{2}(\mathbb{C},d\lambda).$ (4.1)

According to [25, Theorem 2.20], the operator $S$ is bounded on
$L^{2}(\mathbb{C},d\lambda)$. Then the boundedness of $T_{\mu}$ follows from
the boundedness of $S$, to complete the proof. ∎

The following identity is quite useful in the study of the bounded Fock-
Toeplitz operators with measures as symbols.

######  Lemma 4.3.

Suppose that $\mu$ is a positive Borel measure satisfying condition (M) and
$T_{\mu}$ is bounded on the Fock space $\mathcal{F}^{2}$. Then

$\displaystyle\langle
T_{\mu}f,g\rangle_{\mathcal{F}^{2}}=\frac{1}{2}\int_{\mathbb{C}}f(z)\overline{g(z)}\mathrm{e}^{-\frac{|z|^{2}}{2}}d\mu(z)$

for $f$ and $g$ in $\mathcal{F}^{2}$.

###### Proof.

Using the integral expression of $T_{\mu}f$, we have

$\displaystyle\langle
T_{\mu}f,g\rangle_{\mathcal{F}^{2}}=\frac{1}{2}\int_{\mathbb{C}}\int_{\mathbb{C}}f(w)\overline{g(z)}K_{w}^{(\mathcal{F}^{2})}(z)\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w)d\lambda(z).$

In order to use Fubini’s theorem to finish the proof, we need to show that

$\displaystyle\int_{\mathbb{C}}\int_{\mathbb{C}}|f(w)|~{}|g(z)|~{}|K_{w}^{(\mathcal{F}^{2})}(z)|\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w)d\lambda(z)<+\infty.$

Indeed, since $T_{\mu}$ is bounded, $\mu$ is a Carleson measure by Proposition
4.2. According to Lemma 4.1, there exists a positive constant $C$ such that

$\displaystyle\int_{\mathbb{C}}|f(w)|~{}|K_{w}^{(\mathcal{F}^{2})}(z)|\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w)$
$\displaystyle\leqslant
C\int_{\mathbb{C}}|f(w)K_{w}^{(\mathcal{F}^{2})}(z)|\mathrm{e}^{-\frac{|w|^{2}}{2}}dA(w)$
$\displaystyle=2C\int_{\mathbb{C}}|f(w)K_{w}^{(\mathcal{F}^{2})}(z)|d\lambda(w)$
$\displaystyle=2CS(|f|)(z),$

where $S$ is the bounded operator defined in (4.1). Noting that both $g$ and
$|f|$ are in $L^{2}(\mathbb{C},d\lambda)$, we arrive at

$\displaystyle\int_{\mathbb{C}}\int_{\mathbb{C}}|f(w)|~{}|g(z)|~{}|K_{w}^{(\mathcal{F}^{2})}(z)|\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w)d\lambda(z)$
$\displaystyle\leqslant 2C\int_{\mathbb{C}}|g(z)|~{}S(|f|)(z)d\lambda(z)$
$\displaystyle<+\infty.$

Applying Fubini’s theorem, we obtain that

$\displaystyle\langle T_{\mu}f,g\rangle_{\mathcal{F}^{2}}$
$\displaystyle=\int_{\mathbb{C}}T_{\mu}f(z)\overline{g(z)}d\lambda(z)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{C}}\int_{\mathbb{C}}f(w)\overline{g(z)}K_{w}^{(\mathcal{F}^{2})}(z)\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w)d\lambda(z)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{C}}f(w)\mathrm{e}^{-\frac{|w|^{2}}{2}}\int_{\mathbb{C}}\overline{g(z)}K_{w}^{(\mathcal{F}^{2})}(z)d\lambda(z)d\mu(w)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{C}}f(w)\overline{g(w)}\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w),$

as desired. ∎

Now we can establish the analogue of Lemma 3.2 on the Fock space by using
Lemma 4.3 and Proposition 4.2.

######  Theorem 4.4.

Assume that $\mu$ is a positive Borel measure on $\mathbb{C}$ satisfying the
condition (M). If $T_{\mu}$ is bounded on the Fock space $\mathcal{F}^{2}$,
then $T_{\mu}$ is invertible on $\mathcal{F}^{2}$ if and only if $\mu$ is a
reverse Carleson measure for $\mathcal{F}^{2}$.

###### Proof.

For simplicity, we denote

$\displaystyle d\mu_{0}(w)=\frac{1}{2}\mathrm{e}^{-\frac{|w|^{2}}{2}}d\mu(w),\
\ \ \ w\in\mathbb{C}.$

If $T_{\mu}$ is invertible on $\mathcal{F}^{2}$, then we can find a positive
constant $\delta$ such that $T_{\mu}\geqslant\delta I$. It follows from Lemma
4.3 that

$\displaystyle\delta\|f\|_{\mathcal{F}^{2}}^{2}\leqslant\langle
T_{\mu}f,f\rangle_{\mathcal{F}^{2}}=\langle
f,f\rangle_{L^{2}(\mathbb{C},d\mu_{0})}=\|f\|_{L^{2}(\mathbb{C},d\mu_{0})}^{2},\
\ \ \ \ f\in\mathcal{F}^{2}.$

This proves the necessity.

Conversely, assume that $\mu$ is a reverse Carleson measure for
$\mathcal{F}^{2}$. Since $T_{\mu}$ is bounded on $\mathcal{F}^{2}$, we have by
Proposition 4.2 that $\mu$ is also a Carleson measure for $\mathcal{F}^{2}$.
This yields that

$\displaystyle\|f\|_{\mathcal{F}^{2}}\asymp\|f\|_{L^{2}(\mathbb{C},d\mu_{0})},\qquad
f\in\mathcal{F}^{2}.$ (4.2)

Furthermore, we have that $\mathcal{F}^{2}$ is a closed subspace of
$L^{2}(\mathbb{C},d\mu_{0})$. Let $Q$ be the orthogonal projection from
$L^{2}(\mathbb{C},d\mu_{0})$ onto $\mathcal{F}^{2}$. Then (4.2) gives

$\displaystyle\|f\|_{\mathcal{F}^{2}}\asymp\sup\left\\{|\langle
f,g\rangle_{L^{2}(\mathbb{C},d\mu_{0})}|:g\in L^{2}(\mathbb{C},d\mu_{0})\ \ \
\mathrm{and}\ \ \ \|g\|_{L^{2}(\mathbb{C},d\mu_{0})}\leqslant 1\right\\}$

for any $f\in\mathcal{F}^{2}$.

Let $g$ be in $L^{2}(\mathbb{C},d\mu_{0})$ with
$\|g\|_{L^{2}(\mathbb{C},d\mu_{0})}\leqslant 1$ and $h=Qg$. It follows from
Lemma 4.3 and (4.2) that

$\displaystyle|\langle f,g\rangle_{L^{2}(\mathbb{C},d\mu_{0})}|$
$\displaystyle=|\langle f,h\rangle_{L^{2}(\mathbb{C},d\mu_{0})}|=|\langle
T_{\mu}f,h\rangle_{\mathcal{F}^{2}}|$
$\displaystyle\leqslant\|T_{\mu}f\|_{\mathcal{F}^{2}}\|h\|_{\mathcal{F}^{2}}\asymp\|T_{\mu}f\|_{\mathcal{F}^{2}}\|h\|_{L^{2}(\mathbb{C},d\mu_{0})}$
$\displaystyle\leqslant\|T_{\mu}f\|_{\mathcal{F}^{2}}.$

Therefore

$\displaystyle\|f\|_{\mathcal{F}^{2}}\lesssim\|T_{\mu}f\|_{\mathcal{F}^{2}},\
\ \ \ f\in\mathcal{F}^{2},$

which implies that the positive Toeplitz operator $T_{\mu}$ is invertible on
$\mathcal{F}^{2}$. ∎

The next proposition shows that a Carleson measure for the Fock space has the
nice property similar to the conclusion in Lemma 2.1 on the Bergman space.

######  Proposition 4.5.

If $\mu$ is a Carleson measure for the Fock space $\mathcal{F}^{2}$, then

$\displaystyle\lim_{R\rightarrow+\infty}{\sup_{z\in\mathbb{C}}{\int_{\mathbb{C}\setminus
B(z,R)}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)}}=0.$

###### Proof.

Fix $R>2\sqrt{2}$. For $m,n\in\mathbb{Z}$, we define

$\displaystyle
Q_{m,n}=\left\\{z\in\mathbb{C}:m\leqslant\mathrm{Re}(z)\leqslant m+1,\
n\leqslant\mathrm{Im}(z)\leqslant n+1\right\\}.$

Denote $\\{Q_{m,n}:m,n\in\mathbb{Z}\\}$ by $\\{Q_{k}\\}_{k=1}^{\infty}$ and
define

$\displaystyle I=I(z,R)=\left\\{k\geqslant 1:Q_{k}\nsubseteq B(z,R)\right\\}.$

It follows that

$\displaystyle\int_{\mathbb{C}\setminus
B(z,R)}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)=\sum_{k\in
I}{\int_{Q_{k}\setminus
B(z,R)}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)}\leqslant\sum_{k\in
I}{\int_{Q_{k}}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)}.$

Let $H(z,\alpha,\beta)$ with $0<\alpha<\beta$ be the annulus

$\displaystyle\left\\{w\in\mathbb{C}:\alpha\leqslant|w-z|\leqslant\beta\right\\}.$

If we set

$\displaystyle I_{j}=I_{j}(z,R)=\left\\{k\geqslant 1:Q_{k}\cap
H(z,R+j,R+j+1)\neq\varnothing\right\\},\ \ \ j=0,1,2,\cdots,$

then clearly $I=\bigcup\limits_{j=0}^{\infty}{I_{j}}$ and

$\displaystyle\int_{\mathbb{C}\setminus
B(z,R)}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)\leqslant\sum_{j=0}^{\infty}{\sum_{k\in
I_{j}}{\int_{Q_{k}}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)}}.$

For each $k\in I_{j}$, since $Q_{k}\cap H(z,R+j,R+j+1)\neq\varnothing$, by the
triangle inequality we have

$\displaystyle R+j-\sqrt{2}\leqslant|w-z|\leqslant R+j+1+\sqrt{2},\qquad w\in
Q_{k}.$ (4.3)

This implies that

$\displaystyle Q_{k}\subseteq H(z,R+j-\sqrt{2},R+j+1+\sqrt{2})$

when $k\in I_{j}$, which yields that

$\displaystyle\mathrm{card}(I_{j})$ $\displaystyle\leqslant
A\big{(}H(z,R+j-\sqrt{2},R+j+1+\sqrt{2})\big{)}$
$\displaystyle\leqslant\pi(1+2\sqrt{2})(2R+2j+1)<60(R+j).$ (4.4)

Moreover, we obtain by (4.3) that

$\displaystyle\int_{Q_{k}}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)\leqslant\mathrm{e}^{-\frac{(R+j-\sqrt{2})^{2}}{2}}\mu(Q_{k}),\
\ \ \ \ k\in I_{j}.$

Since each cube $Q_{k}$ is contained in some disk $B(u,1)$ and $\mu$ is a
Carleson measure for $\mathcal{F}^{2}$, Lemma 4.1 tells us that there is a
constant $C>0$ such that

$\displaystyle\mu(Q_{k})\leqslant\mu(B(u,1))\leqslant C.$

Thus we have

$\displaystyle\int_{Q_{k}}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)\leqslant
C\mathrm{e}^{-\frac{(R+j-\sqrt{2})^{2}}{2}},\ \ \ \ \ k\in I_{j}.$

It follows from (4.4) that

$\displaystyle\sum_{k\in
I_{j}}{\int_{Q_{k}}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)}$
$\displaystyle\leqslant C\sum_{k\in
I_{j}}{\mathrm{e}^{-\frac{(R+j-\sqrt{2})^{2}}{2}}}$ $\displaystyle\leqslant
60C(R+j)\mathrm{e}^{-\frac{(R+j-\sqrt{2})^{2}}{2}}.$

Now we arrive at

$\displaystyle\int_{\mathbb{C}\setminus
B(z,R)}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)$ $\displaystyle\leqslant
60C\sum_{j=0}^{\infty}{(R+j)\mathrm{e}^{-\frac{(R+j-\sqrt{2})^{2}}{2}}}$
$\displaystyle\leqslant
60C\sum_{j=0}^{\infty}{(R+j)\mathrm{e}^{-\frac{(R+j)^{2}}{8}}}$
$\displaystyle\leqslant
60C\sum_{j=0}^{\infty}{(R+j)\mathrm{e}^{-\frac{R^{2}+j^{2}}{8}}}$
$\displaystyle\lesssim\mathrm{e}^{-\frac{R^{2}}{8}}\left(R\sum_{j=0}^{\infty}{\mathrm{e}^{-\frac{j^{2}}{8}}}+\sum_{j=0}^{\infty}{j\mathrm{e}^{-\frac{j^{2}}{8}}}\right),$

where the second inequality is due to $R>2\sqrt{2}$. This completes the proof
of Proposition 4.5. ∎

According to Proposition 4.5, we obtain the following characterization of a
Carleson measure for the Fock space that satisfies the reverse Carleson
condition, which is parallel to Lemma 3.3 in the setting of the Bergman space.

######  Theorem 4.6.

A Carleson measure $\mu$ for the Fock space $\mathcal{F}^{2}$ satisfies the
reverse Carleson condition if and only if the Berezin transform
$\widetilde{\mu}$ is bounded above and below on $\mathbb{C}$.

###### Proof.

Assume that $\mu$ satisfies the reverse Carleson condition for the Fock space.
Then there is an $r>0$ and a constant $c>0$ such that $\mu(B(z,r))\geqslant c$
for $z\in\mathbb{C}$. It follows that

$\displaystyle\widetilde{\mu}(z)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{C}}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)\geqslant\frac{1}{2}\int_{B(z,r)}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)$
$\displaystyle\geqslant\frac{\mathrm{e}^{-\frac{r^{2}}{2}}}{2}\mu(B(z,r))\geqslant\frac{c\mathrm{e}^{-\frac{r^{2}}{2}}}{2}$

for all $z\in\mathbb{C}$, to get that $\widetilde{\mu}$ is bounded below on
$\mathbb{C}$.

Conversely, suppose that $\widetilde{\mu}$ is bounded below, i.e., there is a
constant $\varepsilon_{0}>0$ such that

$\displaystyle\varepsilon_{0}\leqslant\widetilde{\mu}(z),\qquad
z\in\mathbb{C}.$

Then it is sufficient to show that there is a $\rho_{0}>0$ and a constant
$C>0$ such that $\mu({B(z,\rho_{0})})\geqslant C$ for all $z\in\mathbb{C}$. To
this end, we write

$\displaystyle\widetilde{\mu}(z)=\frac{1}{2}\int_{B(z,\rho)}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)+\frac{1}{2}\int_{\mathbb{C}\setminus
B(z,\rho)}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)$

for $\rho>0$. By Proposition 4.5, there exists some $\rho_{0}$ large enough
such that

$\displaystyle\frac{1}{2}\int_{\mathbb{C}\setminus
B(z,\rho_{0})}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)<\frac{\varepsilon_{0}}{2},\qquad
z\in\mathbb{C}.$

From this we deduce that

$\displaystyle\frac{\varepsilon_{0}}{2}\leqslant\frac{1}{2}\int_{B(z,\rho_{0})}\mathrm{e}^{-\frac{|z-w|^{2}}{2}}d\mu(w)\leqslant\frac{\mu(B(z,\rho_{0}))}{2}$

for all $z\in\mathbb{C}$, which implies that $\mu$ satisfies the reverse
Carleson condition for $\mathcal{F}^{2}$. ∎

Combining Theorem 4.4, Theorem 4.6 and the conclusions in [18, Theorem 1.1],
we obtain the following necessary and sufficient conditions for the
invertibility of the Toeplitz operator $T_{\varphi}$ on the Fock space, where
$\varphi$ is a bounded positive function on the complex plane $\mathbb{C}$.

######  Theorem 4.7.

Let $\varphi$ be a bounded positive function on $\mathbb{C}$. If $d\mu=\varphi
dA$, then the following statements are equivalent:
$(a)$ $T_{\mu}=T_{\varphi}$ is invertible on $\mathcal{F}^{2}$;
$(b)$ $\widetilde{\mu}=\widetilde{\varphi}$ is bounded below on $\mathbb{C}$;
$(c)$ There exists an $r>0$ such that
$\chi_{\\{z\in\mathbb{D}:\varphi(z)>r\\}}d\lambda$ is a reverse Carleson
measure for $\mathcal{F}^{2}$;
$(d)$ $\mu$ is a reverse Carleson measure for $\mathcal{F}^{2}$;
$(e)$ $\mu$ satisfies the reverse Carleson condition for $\mathcal{F}^{2}$.

In view of the discussions for Bergman-Toeplitz operators in the previous
section, it is not difficult to guess that Theorem 4.7 does not hold for
Toeplitz operators induced by general positive measures on the Fock space. In
fact, using the conclusions in Theorems 4.4 and 4.6 and following the idea of
the construction in Proposition 3.7, it is natural for us to consider the
measure of the form:

$\displaystyle\mu=\sum_{k=1}^{\infty}{\delta_{a_{k}}},$

where $\delta_{a_{k}}$ is the point mass concentrated at $a_{k}$. Furthermore,
one can extend Propositions 3.7 and 3.8 to the Fock-space setting without many
difficulties.

In order to answer Question 1.3, it suffices to consider the measure defined
above with $\\{a_{k}\\}_{k=1}^{\infty}$ an $r$-lattice for $\mathcal{F}^{2}$,
which was investigated by Zhuo and Lou in Section 1 of [23]. Let us quote
their result as the following lemma.

######  Lemma 4.8.

Let $\\{a_{k}\\}_{k=1}^{\infty}$ be the sequence given in _( 2.4)_, that is,

$\Lambda(0,r,\mathrm{i}r)=\\{a_{k}\\}_{k=1}^{\infty}.$

Define the positive Borel measure

$\displaystyle\mu=\sum_{k=1}^{\infty}{\delta_{a_{k}}},$

where $\delta_{a_{k}}$ is the point mass concentrated at $a_{k}$. Then $\mu$
satisfies the reverse Carleson condition for the Fock space $\mathcal{F}^{2}$,
but $\mu$ is not a reverse Carleson measure for $\mathcal{F}^{2}$ provided
$r\geqslant\sqrt{2\pi}$.

Now we are ready to give a negative answer to Question 1.3. In fact, combining
Theorem 4.4, Theorem 4.6 and Lemma 4.8, we obtain the following analogue of
Theorem 3.11 on the Fock space.

######  Theorem 4.9.

Let $r\geqslant\sqrt{2\pi}$ and
$\Lambda(0,r,\mathrm{i}r)=\\{a_{k}\\}_{k=1}^{\infty}$. Then
$\nu=\sum\limits_{k=1}^{\infty}\delta_{a_{k}}$ is not a reverse Carleson
measure for the Fock space $\mathcal{F}^{2}$ but satisfies the condition (M)
and the reverse Carleson condition for $\mathcal{F}^{2}$. Consequently, the
Berezin transform of $\nu$ is bounded below on $\mathbb{C}$ but the
corresponding Toeplitz operator $T_{\nu}$ is not invertible on the Fock space
$\mathcal{F}^{2}$.

###### Proof.

Using Lemma 4.8, we have that $\nu$ satisfies the reverse Carleson condition
for the Fock space $\mathcal{F}^{2}$, so $\nu$ is a Carleson measure for
$\mathcal{F}^{2}$. It follows from Lemma 4.1 that

$\displaystyle\int_{\mathbb{C}}|K_{z}^{(\mathcal{F}^{2})}(w)|^{2}\mathrm{e}^{-\frac{|w|^{2}}{2}}d\nu(w)=\|K_{z}^{(\mathcal{F}^{2})}\|_{\mathcal{F}^{2}}^{2}\int_{\mathbb{C}}|k_{z}^{(\mathcal{F}^{2})}(w)|^{2}\mathrm{e}^{-\frac{|w|^{2}}{2}}d\nu(w)<+\infty$

for each $z\in\mathbb{C}$, implying that $\nu$ satisfies condition (M).

By Theorems 4.4 and 4.6, we easily deduce that $\widetilde{\nu}$ is bounded
below on the complex plane but $T_{\nu}$ is not invertible on the Fock space
$\mathcal{F}^{2}$. This completes the proof of Theorem 4.9. ∎

### Acknowledgment

The first author was supported by NSFC (grant number: 12161141013) and Hunan
Provincial Natural Science Foundation of China (grant number: 2022JJ30233).
The third author was supported by NSFC (grant number: 12371125) and the
Fundamental Research Funds for the Central Universities (grant number:
2022CDJJCLK002).

## References

  * [1] S. Axler, Bergman spaces and their operators, Surveys of some recent results in operator theory, vol. I, Pitman Res. Notes Math. Ser., vol. 171, Longman Sci. Tech., Harlow, 1988, 1-50.
  * [2] S. Axler, D. Zheng, Compact operators via the Berezin transform, _Indiana Univ. Math. J._ , 1998, 47: 387-400.
  * [3] Ž. Čučković, A. Vasaturo, Carleson measures and Douglas’ question on the Bergman space, _Rend. Circ. Mat. Palermo, Ser. II_ , 2018, 67: 323-336.
  * [4] P. Duren, S. Alexander, _Bergman Spaces_ , Mathematical surveys, 2004.
  * [5] J. B. Garnett, _Bounded Analytic Functions_ , Graduate Texts in Mathematics, vol. 236, Springer, New York, 1981.
  * [6] P. Ghatage, M. Tjani, Closed range composition operators on Hilbert function spaces, _J. Math. Anal. Appl._ , 2015, 431: 841-866.
  * [7] K. Guo, X. Zhao, D. Zheng, The spectral picture of Bergman-Toeplitz operators with harmonic polynomial symbols, _Ark. Mat._ , 2023, 61: 343-374.
  * [8] H. Hedenmalm, B. Korenblum, K. Zhu, _Theory of Bergman Spaces_ , Graduate Texts in Mathematics, vol. 199, Springer, New York, 2000.
  * [9] C. Horowitz, Zeros of functions in the Bergman spaces, _Duke Math. J._ , 1974, 41: 693-710.
  * [10] M. T. Karaev, Berezin symbol and invertibility of operators on the functional Hilbert spaces, _J. Funct. Anal._ , 2006, 238: 181-192.
  * [11] D. Luecking, Inequalities on Bergman spaces, _Illinois J. Math._ , 1981, 25: 1-11.
  * [12] D. Luecking, Trace idea criteria for Toeplitz operators, _J. Funct. Anal._ , 1987, 73: 345-368.
  * [13] D. Luecking, Finite rank Toeplitz operators on the Bergman space, _Proc. Amer. Math. Soc._ , 2008, 136: 1717-1723.
  * [14] G. McDonald, C. Sundberg, Toeplitz operators on the disc, _Indiana Univ. Math. J.,_ 1979, 28: 595-611.
  * [15] N. K. Nikolskii, _Treatise on the shift operator: spectral function theory_ , Springer, 1986.
  * [16] K. Seip, Beurling type density theorems in the unit disk, _Invent. Math._ , 1993, 113: 21-39.
  * [17] V. A. Tolokonnikov, Estimates in the Carleson corona theorem, ideals of the algebra $H^{\infty}$, a problem of S.-Nagy, _Zapiski Nauchnykh Seminarov POMI_ , 1981, 113: 178-198.
  * [18] Z. Wang, X. Zhao, Invertibility of Fock Toeplitz operators with positive symbols, _J. Math. Anal. Appl._ , 2016, 435: 1335-1351.
  * [19] T. H. Wolff, Counterexamples to two variants of the Helson-Szegö theorem, _J. Anal. Math._ , 2002, 88: 41-62.
  * [20] X. Zhao, D. Zheng, Positivity of Toeplitz operators via Berezin transform, _J. Math. Anal. Appl._ , 2014, 416: 881-900.
  * [21] X. Zhao, D. Zheng, Invertibility of Toeplitz operators via Berezin transforms, _J. Operator Theory_ , 2016, 75: 475-495.
  * [22] X. Zhao, D. Zheng, The spectrum of Bergman Toeplitz operators with some harmonic symbols, _Sci. China. Math._ , 2016, 4: 731-740.
  * [23] Z. Lou, Z. Lou, Sampling measure on doubling Fock spaces, _J. Geom. Anal._ , 2023, 33: Paper No. 313.
  * [24] Z. Zhuo, S. Ye, Sampling measure on Bergman spaces, _Proc. Amer. Math. Soc._ , 2023, 151: 4817-4825.
  * [25] K. Zhu, Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains, _J. Operator Theory_ , 1988, 20: 329-357.
  * [26] K. Zhu, Schatten class Toeplitz operators on weighted Bergman spaces of the unit ball, _New York J. Math._ , 2007, 13: 299-316.
  * [27] K. Zhu, _Operator Theory in Function Spaces_ , second edition, Mathematical Surveys and Monographs, 138, American Mathematical Society, 2007.
  * [28] K. Zhu, _Analysis on Fock Spaces_ , Graduate Texts in Mathematics, vol. 263, Springer, New York, 2012.