# Reducing Labelled Data Requirement for Pneumonia Segmentation using Image
Augmentations

Jitesh Seth1,2 Rohit Lokwani2 Viraj Kulkarni2 Aniruddha Pant2 Amit Kharat2
1Indian Institute of Science Education and Research, Pune 2DeepTek Inc

###### Abstract

Deep learning semantic segmentation algorithms can localise abnormalities or
opacities from chest radiographs. However, the task of collecting and
annotating training data is expensive and requires expertise which remains a
bottleneck for algorithm performance. We investigate the effect of image
augmentations on reducing the requirement of labelled data in the semantic
segmentation of chest X-rays for pneumonia detection. We train fully
convolutional network models on subsets of different sizes from the total
training data. We apply a different image augmentation while training each
model and compare it to the baseline trained on the entire dataset without
augmentations. We find that rotate and mixup are the best augmentations
amongst rotate, mixup, translate, gamma and horizontal flip, wherein they
reduce the labelled data requirement by 70% while performing comparably to the
baseline in terms of AUC and mean IoU in our experiments.

##  1 Introduction

The progress in computer vision has had a substantial impact on radiology
McBee2018 ; Ginneken2017 . Deep learning approaches such as Convolutional
Neural Networks (CNNs) have shown great success in the classification and
segmentation of radiographs like chest X-rays and CT scans Lakhani2017 ;
Dunnmon2018 . Research on pathology identification in chest X-rays (CXRs) has
mainly focussed on classification, where models predict a class label from a
broad set of pathologies. Such an approach does not directly inform us of the
regions in the CXRs responsible for the class label. It requires an additional
step of plotting saliency maps or gradient class activation maps to interpret
how the CNN is making decisions Pasa2019 .

Segmentation and object detection algorithms provide an advantage over
standard CNNs because they can predict the regions of interest. Architectures
such as U-Net Ronneberger2015  and RetinaNet Lin2018  localise the regions
responsible for a specific pathology. Besides, semantic segmentation models
are more useful than classification models in two ways: they require less
training data since they have pixel-level labels for every image, and they can
assist radiologists in their work by localising the abnormalities Hurt2020 .

However, the cost and effort to label the data is often a significant
constraint on training models that perform well in practice Prevedello2019 .
There have been efforts on both fronts - to increase the amount of labelled
data available and to create techniques that can learn better from small
amounts of labelled data. The former include large databases such as CheXpert
Irvin2019 , and ChestX-ray8 Wang2017 , whereas the latter consists of image
augmentations, semi-supervised learning, special architectures such as U-Nets
Ronneberger2015 , GANs Goodfellow2014  and others.

Image augmentations is a vital data processing technique to improve the
performance of machine learning models. They can make the CNN indifferent to
naturally present variations in the data, such as position, scale, or
different radiography equipment. Augmentations can be of varied types. Rotate,
scale and flip and similar augmentations are called geometrical augmentations.
Photometric augmentations transform the colour space of the images. More
complex transformations include elastic deformation or mixing multiple images.
However, inflating the dataset with numerous augmentations would add to the
training time and compute requirements without necessarily increasing
performance. A drawback of augmentations is that they may cause overfitting by
making the CNN invariant to some features but highly tailored to the training
data in others Shorten2019 .

The problem of augmentations on the performance of segmentation models in the
medical domain is not sufficiently addressed in research. Knowledge of
specific augmentations which reduce the labelled data requirements will help
researchers and data scientists fine-tune their models faster and better.
Moreover, data augmentation studies investigate the increase in model
performance rather than the decrease in the labelled training data
requirements. Our paper specifically addresses the reduction in the training
data requirement using image augmentations.

In this paper, we implement five different augmentations on the training of
CNN models on Chest X-rays. We propose three criteria for identifying
augmentations that reduce labelled data requirement. First, the model should
perform comparably to the baseline on a subset of the data. Second, the models
with augmentation trained on partial data should perform better than models
without any augmentations trained on the same data. Third, the model should
satisfy the criteria above for multiple test sets. We validated our results
with an in-sample and out-of-sample test set.

Figure 1: Original CXR with opacities labelled(left) and result of
augmentations with updated bounding boxes.

##  2 Related Work

This section covers the recent advancements in Computer-aided diagnosis (CADx)
using deep learning, image augmentations and semantic segmentation in the
field of CXRs. CADx for chest radiography started around since 1960s
Ginneken2001 , but deep learning has transformed and dominated the field in a
few years. Van Ginneken et al. Ginneken2017  has comprehensively summarised
the evolution of CADx from rule-based and machine learning approaches to deep
learning ones. Kermany et al. Kermany2018  showed the generalizability of deep
neural networks by using the same neural network to classify retinal Optical
Coherence Tomography images and pediatric pneumonia classification in CXRs.
The latter model achieved an accuracy of 92.8%, and the area under the ROC
curve was 96.8%.

Recent studies have proven that image augmentations improve the machine
learning model performance. Sirazitdinov et al. Sirazitdinov2019  showed the
effect of augmentations on the classification of chest radiographs. They
concluded that a combination of increasing brightness, random rotation and
horizontal flips led to the best performance on the ChestX-Ray14 dataset, with
an AUC-ROC of 0.808 (compared to 0.785 without any augmentations). However,
they do not quantify the extent of each augmentation, thus decreasing its
reproducibility. After the invention of mixup Zhang2018 , Eaton-Rosen et al.
EatonRosen2018  applied it to a dataset of MRI images. They provide a
graphical overview of mixup compared to other augmentations and a baseline for
a large (199 images) and a small (10 images) dataset.

Souza et al. Souza_2019  created an automatic method for segmentation and
reconstruction of lungs, which can take into account lung opacities from
pneumonia or tuberculosis, reconstruct the lung boundaries, and finally,
segment the lungs. They used the segmented lungs for a classification model,
which achieved an accuracy of 96.97%, an average Dice coefficient of 0.94 on
the Montgomery County’s Tuberculosis Control dataset Jaeger2014 . Selvan et
al. Selvan2020  tackled the same problem by treating high opacity regions as
missing data and using a variational auto-encoder for data imputation. They
achieved an accuracy of 88.15% and a Dice coefficient of 0.8503 on a curated
CXR dataset. Thus segmentation was mainly used to demarcate the lungs for use
in classification models. However, we can also use semantic segmentation
algorithms to demarcate lung opacities. This opportunity became plausible
after the publication of the Radiology Society of North America’s (RSNA)
pneumonia dataset Shih2019 , also used in this study.

Wu et al. took the concept of lung segmentation and opacity detection one step
forward. They segmented both the lungs, divided them into three zones each and
predicted the presence of pneumonia in each zone using the patient’s radiology
report. Thus they created an object detection dataset from radiology reports.
Using this dataset, they trained a RetinaNet model and tested it on the RSNA
data set. The model had a mean IoU of 0.29 per pneumonia positive image. Hurt
et al. have shown that semantic segmentation of CXRs can be used as a
probability map to interpret the radiographs. Their segmentation model on the
RSNA dataset showed a dice coefficient of 0.603, and the classification had an
AUC of 0.854.

An essential aspect of deep learning in CADx is the usability of the models -
we do not desire clever models that might end up being clinically irrelevant
Lundervold2019 . For example, both in Pan and Cadrin-Chênevert’s, and Cheng’s
model of the RSNA dataset Pan2019 , they systematically decreased the
predicted bounding boxes by 12-17%, which increased the performance for the
particular test set, but there is no medically relevant reason to do the same
in practical settings. Another practical constraint with Pan’s models was
extensive ensembling, which requires the availability of high-end GPUs.

Figure 2: Two examples of CXR with the ground truth mask (left) and predicted
mask by the baseline model (right)

##  3 Data and Methodology

###  3.1 Data

For this study, we use the publicly available pneumonia dataset jointly
annotated by RSNA and the Society of Thoracic Radiology(STR) Shih2019 . The
dataset consists of DICOM images of Chest X-Rays (CXRs) having dimensions
$1024\times 1024$. Pneumonia positive images contain bounding box information
of lung opacities. The dataset consists of about 30,000 frontal CXRs with
bounding boxes around the lung opacities.

From this dataset, we select all the pneumonia-positive CXRs (n=6012) and a
subset of the negative CXRs (n=8488) and divide them into training(n=10000),
validation (n=1500) and test (n=3000) sets. Each CXR belongs to a unique
individual. All three sets have the same prevalence of the positive class
(41.4%).

We also test all trained models on an out-of-sample test set curated from
Padchest Bustos2020  and four private hospitals and population screening
programmes from India and Indonesia. This set contains 1125 pneumonia positive
CXRs with the corresponding polygonal or rectangular annotations and 1875
pneumonia negative CXRs.

###  3.2 Model Architecture and Evaluation

We use a U-net-like CNN with depthwise separable convolutions style
connections implemented using the Keras library for this study Team . We
resize the input into an image of shape (512,512,3). The network has two
parts, an encoder part and a decoder part. The encoder part uses residual
connections, depthwise separable convolution and 2D convolution and max
pooling. The decoder portion uses transpose convolution, 2D upsampling and 2D
convolution.

We use binary cross-entropy as the loss function, and the evaluation metric is
the mean intersection over union (IoU). We use Adam to optimise the loss
function and the learning rate is scheduled to decrease when plateauing for 5
epochs. We train each model until it has not shown improvement in validation
loss for at least 5 epochs. We save the model weights with the lowest
validation loss for evaluation.

After the model training, we evaluate it on the in-sample and out-of-sample
test sets. These are compared to the ground truth masks to calculate the mean
IoU and the loss. We use the segmentation results to classify the CXR as
positive or negative. We calculate the area under the curve (AUC) of the
receiver operating characteristics (ROC) based on this classification.

###  3.3 Augmentations

We have chosen five image augmentations for this study. These are random
rotation between $-10^{\circ}$ to $10^{\circ}$, changing the gamma between
0.75 to 1.25, translating the image randomly between 0-5% of its length in x
and y direction, horizontal flips, and mixup Zhang2018 . In mixup, two images
and their masks (represented by $x_{1}$ and $x_{2}$) are combined using the
formula:

$x=x_{1}\lambda+x_{2}(1-\lambda)$

Where $\lambda\sim\beta(0.2,0.2)$. These augmentations have negligible
computational cost. Mixup results in better-performing segmentation models
according to recent studies EatonRosen2018 .

We train the baseline (i.e. no augmentations) on 100% of the training set. For
each of the six conditions - five augmentations and one with no augmentation
(hereafter referred to as “NoAug”), models were trained using 30%, 50%, 70%
and 90% of the training set.

###  3.4 Statistical Analysis

We compare each model with the baseline and the NoAug model on the same amount
of data. We use the non-parametric DeLong test delong1988  to compare the AUC
for the models’ classification performance. We set the significance level at
0.05. We use the pROC library in R Robin2011  to perform the same.

Figure 3: (Top) original CXRs for mixup; (bottom left) result of mixup
augmentation and (bottom right) corresponding mask.

Figure 4: Mean IoU on the 4 internal test set and 4 external test set for the
various models trained. 4 The legend

Figure 5: AUC ROC of the 5 internal and 5 external test set for the various models. Test Set | Internal Test set | External Test Set
---|---|---
Augmentation | 30% | 50% | 30% | 50%
None | 7.66$\times 10^{-12}$ / 1 | 0.4826 / 0.7587 | 2.20 $\times 10^{-16}$ / 1 | 6.04 $\times 10^{-8}$ / 1
Translate |  0.2288 / 0.1144 |  0.176 / 0.912 | 2.34 $\times 10^{-5}$ / 1 |  0.084 / 0.958
Rotate |  0.1576 / 0.0788 |  0.6309 / 0.3154 |  0.0599 / 0.0300 |  0.2606 / 0.8697
Gamma | 0.0147 / 0.9927 | 1.86 / 9.29$\mathbf{\times 10^{-4}}$ | 1.11 $\times 10^{-13}$ / 1 | 5.15 $\times 10^{-4}$ / 2.58 $\mathbf{\times 10^{-4}}$
Mixup |  0.908 / 0.454 | 0.0546 / 0.0273 | 2.70 $\times 10^{-8}$ / 1 |  0.4118 / 0.2059
Flip |  0.7151 / 0.3575 | 0.02505 / 0.9875 |  0.2316 / 0.8842 | 6.984 $\times 10^{-4}$ / 0.9997
Table 1: p-values of DeLong Test compared to the baseline for the external
test set. The first value corresponds to two-tailed DeLong test and the second
corresponds to the one-tailed DeLong test. Bold indicates that the value
satisfied our criteria, as explained in Results

##  4 Results

For comparison of the models’ segmentation performance, we plotted each
model’s mean IoU in Fig. 4. We also compared the different models’
classification performances by using AUC, in Fig. 5.

It is essential to note the behaviour of the NoAug models. In both Fig. 4 and
Fig. 5, we see that the performance of NoAug models is lower than baseline for
30% and 50% data. However, the NoAug models with 70% and 90% data perform as
good as, or even better, than the baseline. Thus it is irrelevant to study
augmentations on 70% or more fraction of the data for this study. The p-values
for the one-tailed DeLong Test between AUC of NoAug and baseline are $\sim
10^{-3}$ and $\sim 10^{-4}$ respectively for the internal test set.

To check the first criterion, we did two DeLong hypothesis tests to see if the
augmentation models’ performance is comparable to the baseline. The first has
the null hypothesis that the AUC of augmentation models is equal to that of
the baseline trained on 100% data. In this case, a p-value larger than 0.05
would mean that we fail to reject the null hypothesis, and thus we can say the
two AUC are comparable. In the second test, the null hypothesis is that the
AUC of the augmentation models is lesser than the baseline. Here a p-value of
less than 0.05 would mean that we can reject the null hypothesis, and the AUC
of augmentation models is significantly larger than that of the baseline. We
find that except for gamma with 30% data and flip with 50% data, all models
trained on 30% and 50% data pass either of the two tests on the internal test
set. However, for the external test set, only rotate and flip with 30% data,
and translate, rotate, gamma and mixup with 50% data pass the hypothesis tests
(see table 1 for the p-values).

We checked the second criterion by performing a one-tailed DeLong Test on the
AUC of the augmentation against NoAug models trained on the same data on both
test sets. We find that for the external test set, all augmentations with 30%
and 50% data perform significantly better than NoAug (p $<10^{-2}$). For the
internal test set, all augmentations trained on 30% data performed better than
NoAug with 30% data. However, on 50% data, only the gamma and mixup performed
better than NoAug.

We propose that good augmentations should satisfy both the criteria above for
both the test sets. We found that rotate and flip with 30% data and gamma and
mixup with 50% data accomplished this.

On the other hand, the mean IoU plot informs us of the best data augmentations
at pixel-level. As we can see in the internal test set (Fig. 4), rotate and
mixup perform quite good at 30%, whereas mixup and flip perform better at 50%.
For the external test set(Fig. 4), we see that rotate and mixup at 30% and
flip and gamma at 50% performing better than NoAug and almost as good as the
baseline.

Therefore, we find that rotate and mixup are the best augmentations for
semantic segmentation on our dataset. These augmentations are capable of
reducing the labelled data requirements by even 70%.

Test Set | Internal Test set | External Test Set
---|---|---
Augmentation | 30% | 50% | 30% | 50%
Baseline on 100% | 0.8569 | 0.8569 | 0.9298 | 0.9298
None | 0.8251 | 0.854 | 0.859 | 0.9058
Translate | 0.8615 | 0.8515 | 0.9106 | 0.9232
Rotate | 0.8626 | 0.8587 | 0.9373 | 0.9251
Gamma | 0.8464 | 0.868 | 0.8954 | 0.9428
Mixup | 0.8574 | 0.8638 | 0.9042 | 0.9329
Flip | 0.8584 | 0.8483 | 0.9249 | 0.9135
Table 2: The AUC ROC of the different augmentations calculated on 30% and 50%
data for both internal and external test sets.

##  5 Conclusion

Users of deep learning algorithms often overlook image augmentations as an
extra step for a minor boost in performance. Our study has shown that
augmentations are capable of reducing the amount of labelled training data
required. To the best of our knowledge, this is the first study addressing
augmentations in this manner.

There is an assumption that augmentations are most useful when the training
and test data are from the same distribution Shorten2019 . Using two test sets
from vastly different populations (internal test set being from the RSNA data
set, sourced from the USA and the out of sample test set sourced from India)
and achieving good results on both, we have shown that this assumption need
not hold.

While this study looked at individual augmentations, there is still scope for
improving the model performance and reducing the labelled data requirement
further by combining multiple augmentations.

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# Hankel matrices acting on the Dirichlet space

Guanlong Bao, Kunyu Guo, Fangmei Sun and Zipeng Wang  Guanlong Bao
Department of Mathematics, Shantou University
Shantou, Guangdong, 515821, China<EMAIL_ADDRESS>Kunyu Guo
School of Mathematical Sciences, Fudan University
Shanghai, 200433, China<EMAIL_ADDRESS>Fangmei Sun
School of Mathematical Sciences, Fudan University
Shanghai, 200433, China<EMAIL_ADDRESS>Zipeng Wang
College of Mathematics and Statistics, Chongqing University
Chongqing, 401331, China<EMAIL_ADDRESS>
###### Abstract.

The characterization of the boundedness of operators induced by Hankel
matrices on analytic function spaces can be traced back to the work of Z.
Nehari and H. Widom on the Hardy space, and has been extensively studied on
many other analytic function spaces recently. However, this question remains
open in the context of the Dirichlet space [20]. By Carleson measures, the
Widom type condition and the reproducing kernel thesis, this paper provides a
comprehensive solution to this question. As a beneficial product,
characterizations of the boundedness and compactness of operators induced by
Cesàro type matrices on the Dirichlet space are given. In addition, we also
show that a random Dirichlet function almost surely induces a compact Hankel
type operator on the Dirichlet space.

###### Key words and phrases:

Hankel matrix, Cesàro matrix, Carleson measure, Dirichlet space, Bergman space

###### 2010 Mathematics Subject Classification:

47B35, 30H20, 46E15

##  1\. Introduction

Let $\mathbb{N}$ be the set of nonnegative integers. Suppose
$\boldsymbol{\lambda}=\\{\lambda_{n}\\}_{n\in\mathbb{N}}$ is a sequence of
complex numbers. An infinite Hankel matrix
$\boldsymbol{H}_{\boldsymbol{\lambda}}$ induced by the sequence
$\boldsymbol{\lambda}$ is given by
$\boldsymbol{H}_{\boldsymbol{\lambda}}=(\lambda_{j+k})_{j,k\in\mathbb{N}}.$
The Hankel matrix $\boldsymbol{H}_{\boldsymbol{\lambda}}$ is initially defined
for all finitely supported sequences in $\ell^{2}$. The celebrated Nehari
theorem [35] illustrates that $\boldsymbol{H}_{\boldsymbol{\lambda}}$
represents a bounded operator on $\ell^{2}$ if and only if there exists a
function $\psi$ in $L^{\infty}$ on the unit circle $\mathbb{T}$ such that
$\lambda_{n}$, $n\geq 0$, is the $n$-th Fourier coefficient of $\psi$.

For $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ in $H(\mathbb{D})$, the class of
functions analytic in the open unit disk $\mathbb{D}$, the Hankel matrix
$\boldsymbol{H}_{\boldsymbol{\lambda}}$ acts on the function $f$ via

$\mathcal{H}_{\boldsymbol{\lambda}}(f)(z):=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}\lambda_{n+k}a_{k}\right)z^{n},\
\ z\in\mathbb{D},$

whenever the right-hand side makes sense and defines an analytic function in
$\mathbb{D}$. For a space $X\subseteq H(\mathbb{D})$ equipped with a norm
$\|\cdot\|_{X}$, the Hankel type operator $\mathcal{H}_{\boldsymbol{\lambda}}$
is bounded on $X$ if $\mathcal{H}_{\boldsymbol{\lambda}}(f)$ is well defined
in $H(\mathbb{D})$ for any $f\in X$ and there exits a positive constant $C$
independent of $g$ such that
$\|\mathcal{H}_{\boldsymbol{\lambda}}(g)\|_{X}\leq C\|g\|_{X}$ for all $g$ in
$X$.

By the Fourier transform, the Hankel matrix
$\boldsymbol{H}_{\boldsymbol{\lambda}}$ represents a bounded operator on
$\ell^{2}$ if and only if $\mathcal{H}_{\boldsymbol{\lambda}}$ is bounded on
the Hardy space $H^{2}$. See [36] for details.

When ${\boldsymbol{\lambda}}$ is the moment sequence
$\\{\mu_{n}\\}_{n\in\mathbb{N}}$ of a finite positive Borel measure $\mu$ on
$[0,1)$, where $\mu_{n}=\int_{[0,1)}t^{n}d\mu(t)$, the related Hankel matrix
is denoted by $\boldsymbol{H}_{\mu}$ in the literature. Write
$\mathcal{H}_{\mu}$ for the corresponding Hankel type operator. H. Widom [42]
proved that $\mathcal{H}_{\mu}$ is bounded on $H^{2}$ if and only if
$\mu_{j}=O(\frac{1}{j+1})$. It is worth noting that
$(\frac{1}{j+1})_{j\in\mathbb{N}}$ is the moment sequence of the Lebesgure
measure on $[0,1)$, which corresponds to the classical Hilbert matrix
$\boldsymbol{H}=((j+k+1)^{-1})_{j,k\in\mathbb{N}}$. See [11, 16, 17, 18, 28,
29, 31, 32, 33, 34, 45] for developments of the Hilbert matrix acting on
analytic function spaces.

Since then, operator questions regarding the Hankel matrix have been
investigated in various other function spaces of analytic functions on the
unit disk, including Hardy spaces, weighted Bergman spaces, Dirichlet type
spaces, and Möbius invariant function spaces. In [15], E. Diamantopoulos’ work
illustrates that H. Widom’s condition remains true for the Dirichlet-type
space $\mathcal{D}_{\alpha}$ with $0<\alpha<2$. He also noted that the
boundedness of $\mathcal{H}_{\mu}$ on the classical Dirichlet space
$\mathcal{D}$ and the Bergman space $A^{2}$ coincides. Under a milder
condition, using Carleson measures for $\mathcal{D}$, P. Galanopoulos and J.
Peláez [23] completely characterized the boundedness of $\mathcal{H}_{\mu}$ on
the Dirichlet space. C. Chatzifountas, D. Girela and J. Peláez studied the
boundedness and compactness of $\mathcal{H}_{\mu}$ between distinct Hardy
spaces in [13]. D. Girela and N. Merchán [24] developed a method to give
complete descriptions of the boundedness and compactness of
$\mathcal{H}_{\mu}$ on some Hardy spaces and Möbius invariant function spaces.
G. Bao, F. Ye and K. Zhu [8] considered bounded $\mathcal{H}_{\mu}$ on
analytic functions spaces in terms of so-called Hankel measures.

In their recent work [20], P. Galanopoulos, D. Girela, A. Mas, and N. Merchán
demonstrated that a 1-logarithmic 1-Carleson measure ensures the boundedness
of $\mathcal{H}_{\mu}$ on the Dirichlet space. However, they also pointed out
that the characterization of the finite positive Borel measure $\mu$ on
$[0,1)$ for which the operator $\mathcal{H}_{\mu}$ is bounded on the Dirichlet
space remains unresolved. Via Carleson measures, Widom type conditions and the
reproducing kernel thesis, the current work answers this question.

Recall that the Dirichlet space $\mathcal{D}$ is a Hilbert space of analytic
functions on $\mathbb{D}$ equipped with the Dirichlet inner product

$\langle
f,g\rangle_{\mathcal{D}}=f(0)\overline{g(0)}+\int_{\mathbb{D}}f^{\prime}(z)\overline{g^{\prime}(z)}dA(z).$
(1.1)

Here $dA(z)=\pi^{-1}dxdy$ for $z=x+iy$ is the normalized Lebesgue measure on
$\mathbb{D}$. A finite positive Borel measure $\nu$ on $\mathbb{D}$ is said to
be a Carleson measure for the Dirichlet space if there is a positive constant
$C$ such that

$\left(\int_{\mathbb{D}}|f(z)|^{2}d\nu(z)\right)^{\frac{1}{2}}\leq
C\|f\|_{\mathcal{D}}$ (1.2)

for all $f\in\mathcal{D}$. Denote by $CM(\mathcal{D})$ the set of Carleson
measures for $\mathcal{D}$. The smallest such $C$ in (1.2) is denoted by
$\|\nu\|_{CM(\mathcal{D})}$, the Carleson measure norm of $\nu$. If the
identity map $I_{d}:\mathcal{D}\to L^{2}(\mathbb{D},d\nu)$ is compact, then we
say that $\nu$ is a vanishing Carleson measure for $\mathcal{D}$. See [2, 5,
19, 23, 41, 43] for this definition of Carleson measures for $\mathcal{D}$ and
related investigations.

In this paper, we first obtain the following result.

######  Theorem 1.1.

Suppose $\boldsymbol{\lambda}=\\{\lambda_{n}\\}_{n\in\mathbb{N}}$ is a
sequence of complex numbers. Then the Hankel type operator
$\mathcal{H}_{\boldsymbol{\lambda}}$ is bounded on the Dirichlet space
$\mathcal{D}$ if and only if $h_{\overline{\boldsymbol{\lambda}}}$ is analytic
on $\mathbb{D}$ and the measure
$|h_{\overline{{\boldsymbol{\lambda}}}}^{\prime}(z)|^{2}dA(z)$ is a Carleson
measure for the Dirichlet space $\mathcal{D}$, where
$h_{\overline{{\boldsymbol{\lambda}}}}(z)=\sum_{n=0}^{\infty}\overline{\lambda_{n}}z^{n}$.

In the case of $\mathcal{H}_{\mu}$, by Theorem 1.1, the operator
$\mathcal{H}_{\mu}$ is bounded on $\mathcal{D}$ if and only if
$|h^{\prime}_{\mu}(z)|^{2}dA(z)$ is a Carleson measure for $\mathcal{D}$,
where

$h_{\mu}(z)=\int_{[0,1)}\frac{1}{1-tz}d\mu(t).$

Note that in [41] D. Stegenga’s beautiful characterization of
$CM(\mathcal{D})$ is related to the logarithmic capacity of a finite union of
intervals of the unit circle. Based on certain integrals involve the Carleson
box and the heightened box, N. Arcozzi, R. Rochberg and E. Sawyer [2, Theorem
1] gave a complete description of $CM(\mathcal{D})$. There still lacks a
direct relation between moment sequences and the boundedness of
$\mathcal{H}_{\mu}$ on the Dirichlet space. We will establish these relations
through the use of Widom type conditions and the reproducing kernel thesis.

Corresponding to the Dirichlet inner product (1.1), the reproducing kernel of
the Dirichlet space $\mathcal{D}$ at a point $w\in\mathbb{D}$ is

$K_{w}(z)=1+\log\frac{1}{1-z\overline{w}}$

and its normalized reproducing kernel

$k_{w}(w)=\frac{K_{w}(z)}{\sqrt{K_{w}(w)}}.$

Now, we state our characterization of the bounded Hankel type operator
$\mathcal{H}_{\mu}$ on the Dirichlet spaces as follows.

######  Theorem 1.2.

Suppose $\mu$ is a finite positive Borel measure on $[0,1)$. Then the
following conditions are equivalent.

  1. (i)

The Hankel type operator $\mathcal{H}_{\mu}$ is bounded on $\mathcal{D}$.

  2. (ii)

The reproducing kernel thesis holds; that is,

$\sup_{t\in[0,1)}\|\mathcal{H}_{\mu}(k_{t})\|_{\mathcal{D}}<\infty,$

where $k_{t}$ is the normalized reproducing kernel of $\mathcal{D}$ at $t$ in
$[0,1)$.

  3. (iii)

The Widom type condition is true; that is,

$\sum^{\infty}_{n=m}n\mu_{n}^{2}=O\left(\frac{1}{\log(m+2)}\right).$

To prove Theorem 1.2, we need Cesàro type matrices. As benefit products, the
necessary and sufficient condition for the boundedness of Cesàro type matrix
acting on the Dirichlet space is obtained.

Let ${\boldsymbol{\eta}}=\\{\eta_{n}\\}_{n\in\mathbb{N}}$ be a sequence of
complex numbers. Recall that the Cesàro type matrix
$\boldsymbol{C}_{\boldsymbol{\eta}}$ is the following infinite lower
triangular matrix:

$\boldsymbol{C}_{\boldsymbol{\eta}}=\begin{pmatrix}\eta_{0}&0&0&0&\cdots\\\
\eta_{1}&\eta_{1}&0&0&\cdots\\\ \eta_{2}&\eta_{2}&\eta_{2}&0&\cdots\\\
\vdots&\vdots&\vdots&\ddots&\ddots\end{pmatrix}.$

In the same way as operators induced by Hankel matrices, the matrix
$\boldsymbol{C}_{\boldsymbol{\eta}}$ induces a Cesàro type operator
$\mathcal{C}_{\boldsymbol{\eta}}$ as follows:

$\displaystyle\mathcal{C}_{\boldsymbol{\eta}}(f)(z)=\sum_{n=0}^{\infty}\Big{(}\eta_{n}\sum_{k=0}^{n}a_{k}\Big{)}z^{n},\quad
z\in\mathbb{D},$ (1.3)

for $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ in $H(\mathbb{D})$, whenever the
right-hand side of (1.3) makes sense and defines an analytic function in
$\mathbb{D}$.

If $\eta_{n}=1/(n+1)$ for every $n$, then the Cesàro type matrix
$\boldsymbol{C}_{\boldsymbol{\eta}}$ is the classical Cesàro matrix
$\boldsymbol{C}$. In [14], N. Danikas and A. Siskakis showed that the operator
induced by $\boldsymbol{C}$ is bounded from $H^{\infty}$ to $BMOA$. A.
Siskakis [38, 39] studied the action of $\boldsymbol{C}$ on Hardy spaces.

Given a finite positive Borel measure $\mu$ on $[0,1)$, if

$\eta_{n}=\mu_{n}=\int_{[0,1)}\frac{1}{1-tz}d\mu(t)$

for each $n$, then we write $\mathcal{C}_{\boldsymbol{\eta}}$ as
$\mathcal{C}_{\mu}$. The operator $\mathcal{C}_{\mu}$ is said to be the
Cesàro-like operator which was introduced in [22, 26] recently. We refer to
[7, 9, 20, 21] for more results about $\mathcal{C}_{\mu}$ on some spaces of
analytic functions. From Theorem 5 in [20], if $\mu$ is a 1-logarithmic
1-Carleson measure, then $\mathcal{C}_{\mu}$ is a bounded on $\mathcal{D}$.
Conversely, if $\mathcal{C}_{\mu}$ is bounded on $\mathcal{D}$, then $\mu$ is
a $\frac{1}{2}$-logarithmic 1-Carleson measure. We obtain the following
result.

######  Theorem 1.3.

Suppose ${\boldsymbol{\eta}}=\\{\eta_{n}\\}_{n=0}^{\infty}$ is a sequence of
complex numbers. Then the following conditions are equivalent.

  1. (i)

The Cesàro type operator $\mathcal{C}_{\boldsymbol{\eta}}$ is bounded on
$\mathcal{D}$.

  2. (ii)

The reproducing kernel thesis holds; that is,

$\sup_{t\in[0,1)}\|\mathcal{C}_{\boldsymbol{\eta}}k_{t}\|_{\mathcal{D}}<\infty,$

where $k_{t}$ is the normalized reproducing kernel of $\mathcal{D}$ at $t$ in
$[0,1)$.

  3. (iii)

The Widom type condition is true; that is,

$\sum^{\infty}_{n=m}n|\eta_{n}|^{2}=O\left(\frac{1}{\log(m+2)}\right).$

Following our previous work on the boundedness of Hankel type operators and
Cesàro type operators on the Dirichlet space, we also establish their
compactness counterparts. These results are included in Section 4.

Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of independent identically
distributed (i.i.d.) real random variables on a probability space
$(\Omega,\mathcal{A},\mathbb{P})$. For a sequence $\\{\alpha_{n}\\}$ of
complex numbers, one formally defines a random power series

$\varphi_{\omega}(z)=\sum_{n=0}^{\infty}X_{n}(\omega)\alpha_{n}z^{n}.$

By Z. Nehari’s theorem, almost surely the random analytic function
$\varphi_{\omega}$ induces a bounded Hankel type operator on the Hardy space
$H^{2}$ if and only if the symbol function $\varphi_{\omega}$ is a BMOA
function almost surely. The famous unsolved J. Anderson’s question [1] is to
seek the necessary and sufficient condition to characterize random BMOA
functions. One can consult [27] for recent progress and more details in this
subject.

According to Theorem 1.1, the symbol function
$h_{\overline{{\boldsymbol{\lambda}}}}$ of a bounded Hankel type operator
$\mathcal{H}_{\boldsymbol{\lambda}}$ on $\mathcal{D}$ must be in the Dirichlet
space. However, our next corollary illustrates that, under milder conditions,
a random Dirichlet function almost surely induced a compact Hankel type
operator, and hence almost surely induced a bounded Hankel type operator.

######  Theorem 1.4.

Suppose $\boldsymbol{\lambda}=\\{\lambda_{n}\\}_{n\in\mathbb{N}}$ is a
sequence of complex numbers. Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of
i.i.d. real random variables satisfying that $\mathbb{E}[X_{n}]=0$ and
$\mathbb{E}[X_{n}^{4}]$ is finite. If
$h_{\boldsymbol{\lambda}}\in\mathcal{D}$, then almost surely the Hankel type
operator $\mathcal{H}_{\boldsymbol{\omega}}$ is compact on $\mathcal{D}$,
where
$\boldsymbol{\omega}=\left\\{X_{n}\overline{\lambda_{n}}\right\\}_{n\in\mathbb{N}}$.

Let $E\subseteq\mathbb{Z}$ be a set. As defined by W. Rudin in [37], the
integrable function $f$ on the unite circle $\mathbb{T}$ is called an
$E$-function if $\hat{f}(n)$, the $n$-th Fourier coefficient of $f$, is equal
to zero for all $n\in\mathbb{Z}\setminus E$. Suppose $p>2$, the set
$E\subseteq\mathbb{Z}$ is called a Rudin’s $\Lambda(p)$ set if there is a
constant $C$ such that $\|f\|_{L^{p}(\mathbb{T})}\leq
C\|f\|_{L^{2}(\mathbb{T})}$ for all trigonometric polynomials whose
coefficients are equal to $0$ on $\mathbb{Z}\setminus E$. The Rudin’s
$\Lambda(p)$ set is a natural generalization of the classical lacunary
sequences (the Hadamard set) which plays important roles in many aspects of
analysis (see [37], [25] and [10] for more details). Recall that an increasing
sequence $\\{n_{k}\\}_{k\in\mathbb{N}}$ of positive integers is said to be
lacunary if there exists $q$ in $(1,\infty)$ such that $n_{k+1}/n_{k}>q$ for
all $k$. It is also known that there exist Rudin’s $\Lambda(p)$ sets rather
than Hadamard sets.

Our next result presents a characterization of bounded and compact Hankel type
operators on $\mathcal{D}$ when the non zero Fourier coefficients of the
symbol function are located in a Rudin’s $\Lambda(p)$ set.

######  Corollary 1.5.

Suppose $p>2$, the set $E$ is a Rudin’s $\Lambda(p)$ set and
$\boldsymbol{\lambda}=\\{\lambda_{n}:n\in E\\}$ is a sequence of complex
numbers. Then the following statements are equivalent.

  1. (i)

The Hankel type operator $\mathcal{H}_{\boldsymbol{\lambda}}$ is compact on
$\mathcal{D}$.

  2. (ii)

The Hankel type operator $\mathcal{H}_{\boldsymbol{\lambda}}$ is bounded on
$\mathcal{D}$.

  3. (iii)

The symbol function $h_{\overline{{\boldsymbol{\lambda}}}}(z)=\sum_{n\in
E}\overline{\lambda_{n}}z^{n}$ belongs to $\mathcal{D}$; that is,

$\sum_{n\in E}(n+1)|\lambda_{n}|^{2}<\infty.$

Throughout this paper, the symbol $A\thickapprox B$ means that $A\lesssim
B\lesssim A$. We say that $A\lesssim B$ if there exists a positive constant
$C$ such that $A\leq CB$.

##  2\. Bounded Hankel type operators $\mathcal{H}_{\boldsymbol{\lambda}}$ on
$\mathcal{D}$

In this section we will prove Theorem 1.1. We also consider the Möbius
invariance of the norm of bounded $\mathcal{H}_{\boldsymbol{\lambda}}$ on the
Dirichlet space. For $f\in H(\mathbb{D})$, we say that $f\in\mathcal{X}$ if

$\|f\|^{2}_{\mathcal{X}}=|f(0)|^{2}+\||f^{\prime}|^{2}dA\|_{CM(\mathcal{D})}<\infty.$

Denote by $\mathcal{X}_{0}$ the norm closure in $\mathcal{X}$ of the space of
polynomials. Equivalently, $f\in\mathcal{X}_{0}$ if and only if
$|f^{\prime}|^{2}dA$ is a vanishing Carleson measure for $\mathcal{D}$. We
refer to [3, 44] for the theory of $\mathcal{X}$ and $\mathcal{X}_{0}$.

Given a function $b$ in $H(\mathbb{D})$, define a Hankel type bilinear form on
the Dirichlet space, initially for $f$, $g$ in $\mathcal{P}$, as

$T_{b}(f,g):=\langle fg,b\rangle_{\mathcal{D}}.$

Here $\mathcal{P}$ is the space of polynomials. The norm of the bilinear form
$T_{b}$ is

$\|T_{b}\|_{\mathcal{D}\times\mathcal{D}}=\sup\left\\{|T_{b}(f,g)|:\ \
\|f\|_{\mathcal{D}}=\|g\|_{\mathcal{D}}=1,\ f,g\in\mathcal{P}\right\\}.$

For a bounded bilinear form $T_{b}$ on $\mathcal{D}$, $T_{b}$ is said to be
compact on $\mathcal{D}$ if $T_{b}(B_{\mathcal{D}}\times B_{\mathcal{D}})$ is
precompact in $\mathbb{C}$, where
$B_{\mathcal{D}}=\\{f\in\mathcal{D}:\|f\|_{\mathcal{D}}\leq 1\\}$. In other
words, $T_{b}$ is compact on $\mathcal{D}$ if and only if for all bounded
sequences $\\{(f_{n},g_{n})\\}\subseteq\mathcal{D}\times\mathcal{D}$, the
sequence $\\{T_{b}(f_{n},g_{n})\\}$ has a convergent subsequence. See [6] for
more descriptions on compact bilinear operators.

With respect to the normalized basis $\\{(n+1)^{-1/2}z^{n}\\}$ of the
Dirichlet space $\mathcal{D}$, the matrix representations of $T_{b}$ is

$\left(\frac{j+k}{\sqrt{j+1}\sqrt{k+1}}\overline{b_{j+k}}\right),\ \ \ \
j,k=0,1,\cdots,$

where $b_{j+k}$ is the $(j+k)$-th Taylor coefficient of $b$. N. Arcozzi, R.
Rochberg, E. Sawyer and B. Wick characterized the boundedness and compactness
of the bilinear form on the Dirichlet space as follows.

######  Theorem A.

[4, Theorem 1.1] Let $b\in H(\mathbb{D})$. Then the following assertions hold.

  1. (i)

$T_{b}$ extends to a bounded bilinear form on $\mathcal{D}$ if and only if
$b\in\mathcal{X}$.

  2. (ii)

$T_{b}$ extends to a compact bilinear form on $\mathcal{D}$ if and only if
$b\in\mathcal{X}_{0}$.

###### Proof of Theorem 1.1.

Given a sequence of complex number
${\boldsymbol{\lambda}}=\\{\lambda_{n}\\}^{\infty}_{n=0}$, suppose
$h_{\overline{{\boldsymbol{\lambda}}}}$ is analytic on $\mathbb{D}$ and the
measure $|h_{\overline{{\boldsymbol{\lambda}}}}^{\prime}(z)|^{2}dA(z)$ is a
Carleson measure for $\mathcal{D}$. Then
$h_{\overline{\boldsymbol{\lambda}}}\in\mathcal{X}$. It follows that
$h_{\boldsymbol{\lambda}}\in\mathcal{D}$, and

$\|h_{\boldsymbol{\lambda}}\|^{2}_{\mathcal{D}}=|\lambda_{0}|^{2}+\sum_{n=0}^{\infty}(n+1)|\lambda_{n+1}|^{2}<\infty.$

Let $f(z)=\sum^{\infty}_{k=0}c_{k}z^{k}$ belong to $\mathcal{D}$. Then for any
$j\in\mathbb{N}$,

$\displaystyle\left|\sum_{k=0}^{\infty}\lambda_{j+k}c_{k}\right|$
$\displaystyle\lesssim\|f\|_{\mathcal{D}}\bigg{(}\sum_{k=0}^{\infty}\frac{|\lambda_{j+k}|^{2}}{k+1}\bigg{)}^{1/2}$
$\displaystyle\lesssim\|f\|_{\mathcal{D}}\bigg{(}\sum_{k=0}^{\infty}(j+k+1)|\lambda_{j+k}|^{2}\bigg{)}^{1/2}$
$\displaystyle\lesssim\|f\|_{\mathcal{D}}\|h_{\boldsymbol{\lambda}}\|_{\mathcal{D}}.$

Write $a_{j}=\sum_{k=0}^{\infty}\lambda_{j+k}c_{k}$. Then
$\\{a_{j}\\}_{j\in\mathbb{N}}$ is a bounded sequence of complex numbers and
hence $\mathcal{H}_{\boldsymbol{\lambda}}(f)(z)=\sum_{j=0}^{\infty}a_{j}z^{j}$
is analytic on $\mathbb{D}$. Note that

$\|\mathcal{H}_{\boldsymbol{\lambda}}f\|_{\mathcal{D}}^{2}=|a_{0}|^{2}+\sum_{j=0}^{\infty}(j+1)|a_{j+1}|^{2}.$

Let $m$ be a positive integer. For any polynomial
$g(z)=\sum^{m}_{j=0}b_{j}z^{j}$ with $\|g\|_{\mathcal{D}}\leq 1$, we have

$\displaystyle\bigg{|}a_{0}\overline{b_{0}}+\sum_{j=0}^{m-1}(j+1)a_{j+1}\overline{b_{j+1}}\bigg{|}$
$\displaystyle\leq$
$\displaystyle|\lambda_{0}||c_{0}||b_{0}|+|b_{0}|\Big{|}\sum^{\infty}_{k=1}\lambda_{k}c_{k}\Big{|}+\Big{|}\sum^{m-1}_{j=0}\overline{b_{j+1}}(j+1)\sum^{\infty}_{k=0}\lambda_{j+1+k}c_{k}\Big{|}$
$\displaystyle=$
$\displaystyle|\lambda_{0}||c_{0}||b_{0}|+|b_{0}|\left|\int_{\mathbb{D}}\frac{f(z)-f(0)}{z}\overline{h^{\prime}_{\overline{\boldsymbol{\lambda}}}(z)}dA(z)\right|$
$\displaystyle+\left|\int_{\mathbb{D}}f(z)g^{\prime}_{1}(z)\
\overline{h^{\prime}_{\overline{\boldsymbol{\lambda}}}(z)}dA(z)\right|$
$\displaystyle\leq$
$\displaystyle|\lambda_{0}|\|f\|_{\mathcal{D}}\|g\|_{\mathcal{D}}+\|g\|_{\mathcal{D}}\Big{(}\int_{\mathbb{D}}\Big{|}\frac{f(z)-f(0)}{z}\Big{|}^{2}|h^{\prime}_{\overline{\boldsymbol{\lambda}}}(z)|^{2}dA(z)\Big{)}^{\frac{1}{2}}$
$\displaystyle+\Big{(}\int_{\mathbb{D}}|f(z)|^{2}|h^{\prime}_{\overline{\boldsymbol{\lambda}}}(z)|^{2}dA(z)\Big{)}^{\frac{1}{2}}\Big{(}\int_{\mathbb{D}}\left|g_{1}^{\prime}(z)\right|^{2}dA(z)\Big{)}^{\frac{1}{2}},$

where $g_{1}(z)=\sum^{m}_{j=0}\overline{b_{j}}z^{j}$. Since
$|h^{\prime}_{\overline{\boldsymbol{\lambda}}}(z)|^{2}dA(z)$ is a Carleson
measure for $\mathcal{D}$, there exists a positive constant $C$ such that

$\bigg{|}a_{0}\overline{b_{0}}+\sum_{j=0}^{m-1}(j+1)a_{j+1}\overline{b_{j+1}}\bigg{|}\leq
C\|f\|_{\mathcal{D}}.$

Therefore

$\bigg{(}|a_{0}|^{2}+\sum_{j=0}^{m-1}(j+1)|a_{j+1}|^{2}\bigg{)}^{1/2}\leq
C\|f\|_{\mathcal{D}}$

and $\|\mathcal{H}_{\boldsymbol{\lambda}}f\|_{\mathcal{D}}\leq
C\|f\|_{\mathcal{D}}.$

On the other hand, suppose the operator $\mathcal{H}_{\boldsymbol{\lambda}}$
is bounded on $\mathcal{D}$. Then for any $f$ and $g$ in $\mathcal{D}$,

$|\langle\mathcal{H}_{\boldsymbol{\lambda}}f,g\rangle_{\mathcal{D}}|\lesssim\|f\|_{\mathcal{D}}\|g\|_{\mathcal{D}}.$

This implies that $h_{\overline{{\boldsymbol{\lambda}}}}$ is analytic in
$\mathbb{D}$ and

$\int_{\mathbb{D}}|h_{\overline{{\boldsymbol{\lambda}}}}^{\prime}(z)|^{2}dA(z)<\infty.$

Moreover, for any polynomial $f$ and $g$,

$\displaystyle\int_{\mathbb{D}}f(z)g^{\prime}(z)\
\overline{h^{\prime}_{\overline{\boldsymbol{\lambda}}}(z)}dA(z)=\int_{\mathbb{D}}(\mathcal{H}_{\boldsymbol{\lambda}}f)^{\prime}(z)g^{\prime}(\overline{z})dA(z),$
(2.1)

and

$\displaystyle\int_{\mathbb{D}}f^{\prime}(z)g(z)\overline{h^{\prime}_{\overline{\boldsymbol{\lambda}}}(z)}dA(z)=\int_{\mathbb{D}}(\mathcal{H}_{\boldsymbol{\lambda}}g)^{\prime}(z)f^{\prime}(\overline{z})dA(z).$
(2.2)

Consequently,

$\left|f(0)g(0)\overline{h_{\overline{\boldsymbol{\lambda}}}(0)}\right|+\left|\int_{\mathbb{D}}(f(z)g(z))^{\prime}\overline{h^{\prime}_{\overline{\boldsymbol{\lambda}}}(z)}dA(z)\right|\lesssim\|f\|_{\mathcal{D}}\|g\|_{\mathcal{D}}.$

Therefore, $T_{h_{\overline{\boldsymbol{\lambda}}}}$ is a bounded bilinear
form on $\mathcal{D}$. It follows from Theorem A that
$h_{\overline{\boldsymbol{\lambda}}}\in\mathcal{X}$ and
$|h_{\overline{{\boldsymbol{\lambda}}}}^{\prime}(z)|^{2}dA(z)$ is a Carleson
measure for $\mathcal{D}$. This completes the whole proof of Theorem 1.1. ∎

Denote by $\text{Aut}(\mathbb{D})$ the Möbius group which consists of all one-
to-one analytic functions that map $\mathbb{D}$ onto itself. Let
$\widetilde{\mathcal{D}}$ be the space of functions $f$ in $\mathcal{D}$ with
$f(0)=0$. For the bounded $\mathcal{H}_{\boldsymbol{\lambda}}$ on
$\widetilde{\mathcal{D}}$, the norm of $\mathcal{H}_{\boldsymbol{\lambda}}$ is
Möbius invariant in the following sense.

######  Proposition 2.1.

Let $\boldsymbol{\lambda}=\\{\lambda_{n}\\}_{n\in\mathbb{N}}$ be a sequence of
complex numbers. Suppose the Hankel type operator
$\mathcal{H}_{\boldsymbol{\lambda}}$ is bounded on $\widetilde{\mathcal{D}}$.
Then there exist positive constants $C_{1}$ and $C_{2}$ depending only on
$\boldsymbol{\lambda}$ such that

$C_{1}\|\mathcal{H}_{\boldsymbol{\lambda}}\|\leq\|\mathcal{H}_{\boldsymbol{\lambda_{\phi}}}\|\leq
C_{2}\|\mathcal{H}_{\boldsymbol{\lambda}}\|$

for all $\phi\in\text{Aut}(\mathbb{D})$, where
$\boldsymbol{\lambda_{\phi}}=\left\\{\overline{\lambda_{\phi,n}}\right\\}_{n\in\mathbb{N}}$
with

$h_{\overline{\boldsymbol{\lambda}}}\circ\phi(z)=\sum_{n=0}^{\infty}\lambda_{\phi,n}z^{n}.$

###### Proof.

By Theorem 1.1, the boundedness of $\mathcal{H}_{\boldsymbol{\lambda}}$ on
$\widetilde{\mathcal{D}}$ yields

$\||h^{\prime}_{\overline{\boldsymbol{\lambda}}}|^{2}dA\|_{CM(\widetilde{\mathcal{D}})}<\infty.$

From the proof of Theorem 1.1, there are positive constants $C_{1}$ and
$C_{2}$ depending only on $\boldsymbol{\lambda}$ such that

$C_{1}\||h^{\prime}_{\overline{\boldsymbol{\lambda}}}|^{2}dA\|_{CM(\widetilde{\mathcal{D}})}\leq\|\mathcal{H}_{\boldsymbol{\lambda}}\|\leq
C_{2}\||h^{\prime}_{\overline{\boldsymbol{\lambda}}}|^{2}dA\|_{CM(\widetilde{\mathcal{D}})}.$
(2.3)

Let $\phi\in\text{Aut}(\mathbb{D})$. For $g\in\widetilde{\mathcal{D}}$, by the
change of variables, we see

$\displaystyle\int_{\mathbb{D}}|g(z)|^{2}|(h_{\overline{\boldsymbol{\lambda}}}\circ\phi)^{\prime}(z)|^{2}dA(z)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{D}}|g(\phi^{-1}(w))|^{2}|h^{\prime}_{\overline{\boldsymbol{\lambda}}}(w)|^{2}dA(w).$

Also,

$\int_{\mathbb{D}}|g^{\prime}(z)|^{2}dA(z)=\int_{\mathbb{D}}|(g\circ\phi^{-1})^{\prime}(w)|^{2}dA(w).$

Thus there is $C>0$ such that

$\int_{\mathbb{D}}|g(z)|^{2}|(h_{\overline{\boldsymbol{\lambda}}}\circ\phi)^{\prime}(z)|^{2}dA(z)\leq
C\int_{\mathbb{D}}|g^{\prime}(z)|^{2}dA(z)$

for all $g\in\widetilde{\mathcal{D}}$ if and only if

$\int_{\mathbb{D}}|k(w)|^{2}|h^{\prime}_{\overline{\boldsymbol{\lambda}}}(w)|^{2}dA(w)\leq
C\int_{\mathbb{D}}|k^{\prime}(z)|^{2}dA(z)$

for all $k\in\widetilde{\mathcal{D}}$. This means

$\||h^{\prime}_{\overline{\boldsymbol{\lambda}}}|^{2}dA\|_{CM(\widetilde{\mathcal{D}})}=\||(h_{\overline{\boldsymbol{\lambda}}}\circ\phi)^{\prime}|^{2}dA\|_{CM(\widetilde{\mathcal{D}})}.$

Combining this with (2.3), we get the desired result. ∎

##  3\. Bounded Hankel type operators $\mathcal{H}_{\mu}$ and Cesàro type
operators on the Dirichlet space

This section is devoted to an intrinsic description of the boundedness of
$\mathcal{H}_{\mu}$ on $\mathcal{D}$. Indeed, we can characterize bounded
Hankel type operators $\mathcal{H}_{\mu}$ on $\mathcal{D}$ induced by a
decreasing sequence of positive numbers. The proof is based on our
characterization of bounded Cesàro type operators on $\mathcal{D}$.

The following Hilbert’s double theorem for the Dirichlet space plays a crucial
role in our proofs. By using Schur’s test, one can show that inequality (see
the proof of Theorem 2 in [40] for more details).

######  Theorem B.

[40, p. 814] Let $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ in $\mathcal{D}$. Then
there exists a positive constant $C$ independent of $f$ such that

$\sum^{\infty}_{m=1}\sum_{n=1}^{\infty}\frac{|a_{n}||a_{m}|}{\log(n+m+1)}\leq
C\sum^{\infty}_{n=1}n|a_{n}|^{2}.$

We first prove Theorem 1.3.

###### Proof of Theorem 1.3.

Since $\|k_{w}\|_{\mathcal{D}}=1$ for any $w\in\mathbb{D}$, we obtain the
implication $(i)\Rightarrow(ii)$.

Next, for each $t\in[0,1)$, recall that the normalized reproducing kernel
$k_{t}$ for the Dirichlet space

$k_{t}(z)=\left(1+\log\frac{1}{1-t^{2}}\right)^{-\frac{1}{2}}\left(1+\sum_{n=1}^{\infty}\frac{1}{n}t^{n}z^{n}\right).$

By direct computations,

$\displaystyle\|\mathcal{C}_{\boldsymbol{\eta}}k_{t}\|^{2}_{\mathcal{D}}$
$\displaystyle=$
$\displaystyle\left(1+\log\frac{1}{1-t^{2}}\right)^{-1}\left(|\eta_{0}|^{2}+\sum^{\infty}_{n=0}(n+1)|\eta_{n+1}|^{2}\Big{(}1+\sum^{n+1}_{k=1}\frac{t^{k}}{k}\Big{)}^{2}\right).$

Thus for any positive integer $m$ and $t\in[0,1)$,

$\displaystyle\|\mathcal{C}_{\boldsymbol{\eta}}k_{t}\|^{2}_{\mathcal{D}}\gtrsim$
$\displaystyle\left(\log\frac{e}{1-t}\right)^{-1}\sum^{\infty}_{n=m}(n+1)|\eta_{n+1}|^{2}\Big{(}\sum^{m}_{k=1}\frac{t^{k}}{k}\Big{)}^{2}$
$\displaystyle\gtrsim$
$\displaystyle\left(\log\frac{e}{1-t}\right)^{-1}t^{2m}(\log(m+1))^{2}\sum^{\infty}_{n=m}(n+1)|\eta_{n+1}|^{2}.$

In particular, let $t=\frac{m}{m+1}\in(0,1)$,

$\frac{1}{\log(m+1)}\gtrsim\sum^{\infty}_{n=m}(n+1)|\eta_{n+1}|^{2}.$

By (ii), we have
$\|\mathcal{C}_{\boldsymbol{\eta}}k_{t}\|^{2}_{\mathcal{D}}\lesssim 1$ and

$\sum^{\infty}_{n=m}n|\eta_{n}|^{2}=O\left(\frac{1}{\log(m+2)}\right),$

which gives $(ii)\Rightarrow(iii)$.

$(iii)\Rightarrow(i)$. The Widom type condition gives

$|\eta_{n}|^{2}\lesssim\frac{1}{(n+1)\log(n+2)},$

for all $n\in\mathbb{N}$. For $f(z)=\sum^{\infty}_{k=0}a_{k}z^{k}$ in
$\mathcal{D}$, we get

$\displaystyle\left|\eta_{n}\sum^{n}_{k=0}a_{k}\right|\leq$
$\displaystyle|\eta_{n}|\left(\sum^{n}_{k=0}\frac{1}{k+1}\right)^{\frac{1}{2}}\left(\sum^{n}_{k=0}(k+1)|a_{k}|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle|\eta_{n}|\left(\log(n+2)\right)^{\frac{1}{2}}\|f\|_{\mathcal{D}}$
$\displaystyle\lesssim$ $\displaystyle\|f\|_{\mathcal{D}}$

for all nonnegative integers $n$. Thus $\mathcal{C}_{\boldsymbol{\eta}}(f)$ is
well defined in $H(\mathbb{D})$.

Because of $(iii)$,

$\displaystyle\sum^{\infty}_{n=0}(n+1)|\eta_{n+1}|^{2}<\infty.$ (3.1)

Let $g(z)=\sum^{\infty}_{k=0}a_{k}z^{k}$ be in the Dirichlet space
$\mathcal{D}$. Then

$\displaystyle\|\mathcal{C}_{\boldsymbol{\eta}}g\|^{2}_{\mathcal{D}}$
$\displaystyle\leq$
$\displaystyle|\eta_{0}a_{0}|^{2}+2\sum^{\infty}_{n=0}(n+1)|\eta_{n+1}|^{2}|a_{0}|^{2}+2\sum^{\infty}_{n=0}(n+1)|\eta_{n+1}|^{2}\Big{(}\sum^{n+1}_{k=1}|a_{k}|\Big{)}^{2}.$

Using (3.1), (iii), and Theorem B, we deduce

$\displaystyle\|\mathcal{C}_{\boldsymbol{\eta}}g\|^{2}_{\mathcal{D}}$
$\displaystyle\lesssim$
$\displaystyle\|g\|_{\mathcal{D}}^{2}+\small{\sum^{\infty}_{n=0}(n+1)|\eta_{n+1}|^{2}\bigg{(}\sum^{\infty}_{k=1}|a_{k}|\chi_{\\{k\leq
n+1\\}}(k)\bigg{)}\bigg{(}\sum^{\infty}_{j=1}|a_{j}|\chi_{\\{j\leq
n+1\\}}(j)\bigg{)}}$ $\displaystyle\thickapprox$
$\displaystyle\|g\|_{\mathcal{D}}^{2}+\sum^{\infty}_{k=1}\sum^{\infty}_{j=1}|a_{k}||a_{j}|\bigg{(}\sum^{\infty}_{n=\max\\{k-1,j-1\\}}(n+1)|\eta_{n+1}|^{2}\bigg{)}$
$\displaystyle\lesssim$
$\displaystyle\|g\|_{\mathcal{D}}^{2}+\sum^{\infty}_{k=1}\sum^{\infty}_{j=1}\frac{|a_{k}||a_{j}|}{\log(k+j+1)}$
$\displaystyle\lesssim$ $\displaystyle\|g\|^{2}_{\mathcal{D}},$

where $\chi_{\\{k\leq n+1\\}}(k)$ is the characteristic function on ${\\{k\leq
n+1\\}}\cap\mathbb{N}^{+}$. Here $\mathbb{N}^{+}$ is the set of positive
integers. Hence $\mathcal{C}_{\boldsymbol{\eta}}$ is bounded on $\mathcal{D}$.
The proof of Theorem 1.3 is complete. ∎

The proof of Theorem 1.2 follows from the next more general result.

######  Theorem 3.1.

Suppose ${\boldsymbol{\lambda}}=\\{\lambda_{n}\\}^{\infty}_{n=0}$ is a
decreasing sequence of positive real numbers. Then the following conditions
are equivalent.

  1. (i)

The Hankel type operator $\mathcal{H}_{\boldsymbol{\lambda}}$ is bounded on
$\mathcal{D}$.

  2. (ii)

The reproducing kernel thesis holds; that is,

$\sup_{t\in[0,1)}\|\mathcal{H}_{\boldsymbol{\lambda}}k_{t}\|_{\mathcal{D}}<\infty,$

where $k_{t}$ is the normalized reproducing kernel of $\mathcal{D}$ at $t$ in
$[0,1)$.

  3. (iii)

The Widom type condition is true; that is,

$\sum^{\infty}_{n=m}n\lambda_{n}^{2}=O\left(\frac{1}{\log(m+2)}\right).$

###### Proof.

If $\mathcal{H}_{\boldsymbol{\lambda}}$ is bounded on $\mathcal{D}$, it is
clear that (ii) holds.

$(ii)\Rightarrow(iii)$. Since $\\{\lambda_{n}\\}$ is a decreasing sequence of
positive numbers,

$\displaystyle\|\mathcal{H}_{\boldsymbol{\lambda}}k_{t}\|_{\mathcal{D}}^{2}\geq$
$\displaystyle\left(1+\log\frac{1}{1-t^{2}}\right)^{-1}\sum^{\infty}_{n=1}(n+1)\Big{(}\sum^{\infty}_{k=1}\lambda_{n+k+1}\frac{t^{k}}{k}\Big{)}^{2}$
$\displaystyle\geq$
$\displaystyle\left(1+\log\frac{1}{1-t^{2}}\right)^{-1}\sum^{\infty}_{n=m}(n+1)\Big{(}\sum^{n}_{k=1}\lambda_{n+k+1}\frac{t^{k}}{k}\Big{)}^{2}$
$\displaystyle\geq$
$\displaystyle\left(1+\log\frac{1}{1-t^{2}}\right)^{-1}\sum^{\infty}_{n=m}(n+1)\lambda_{2n+1}^{2}\Big{(}\sum^{m}_{k=1}\frac{t^{k}}{k}\Big{)}^{2}$
$\displaystyle\gtrsim$
$\displaystyle\left(\log\frac{e}{1-t}\right)^{-1}t^{2m}(\log(m+1))^{2}\sum^{\infty}_{n=m}(n+1)\lambda_{2n+1}^{2},$

for all $t\in[0,1)$ and all positive integers $m$.

For any fixed integer $m$, taking $t=\frac{m}{m+1}$ in the above, we get

$\sum^{\infty}_{n=m}(2n+1)\lambda_{2n+1}^{2}\lesssim\frac{1}{\log(m+1)}$

and

$\sum^{\infty}_{n=m}(2n+2)\lambda_{2n+2}^{2}\lesssim\frac{1}{\log(m+1)}.$

Then the desired result holds.

$(iii)\Rightarrow(i)$. It follows from (iii) that

$\displaystyle\lambda_{k}^{2}\lesssim\frac{1}{k\log(k+1)}$ (3.2)

for all positive integers $k$. Let $f(z)=\sum^{\infty}_{k=0}a_{k}z^{k}$ belong
to $\mathcal{D}$. By (3.2) and the monotonicity of $\\{\lambda_{k}\\}$,

$\displaystyle\Big{|}\sum^{\infty}_{k=0}\lambda_{n+k}a_{k}\Big{|}\leq$
$\displaystyle\lambda_{n}|a_{0}|+\sum^{\infty}_{k=1}\lambda_{n+k}|a_{k}|$
$\displaystyle\lesssim$
$\displaystyle\|f\|_{\mathcal{D}}+\|f\|_{\mathcal{D}}\Big{(}\sum^{\infty}_{k=1}\frac{\lambda^{2}_{n+k}}{k+1}\Big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim$
$\displaystyle\|f\|_{\mathcal{D}}+\|f\|_{\mathcal{D}}\Big{(}\sum^{\infty}_{k=1}\frac{1}{(k+1)^{2}\log(k+1)}\Big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim$ $\displaystyle\|f\|_{\mathcal{D}}$

for all nonnegative integers $n$. Then $\mathcal{H}_{\boldsymbol{\lambda}}(f)$
is analytic on the unit disk $\mathbb{D}$ for any $f$ in $\mathcal{D}$.

Observe that by condition (iii), we have

$\displaystyle\sum^{\infty}_{n=0}(n+1)\lambda_{n+1}^{2}<\infty.$ (3.3)

Then the Hölder inequality gives
$\sum^{\infty}_{k=0}\lambda_{k}|a_{k}|\lesssim\|f\|_{\mathcal{D}}.$
Consequently,

$\displaystyle\|\mathcal{H}_{\boldsymbol{\lambda}}f\|^{2}_{\mathcal{D}}\leq$
$\displaystyle\left(\sum^{\infty}_{k=0}\lambda_{k}|a_{k}|\right)^{2}+\sum^{\infty}_{n=0}(n+1)\Big{(}\sum^{\infty}_{k=0}\lambda_{n+k+1}|a_{k}|\Big{)}^{2}$
$\displaystyle\lesssim$
$\displaystyle\|f\|_{\mathcal{D}}^{2}+\sum^{\infty}_{n=0}(n+1)\Big{(}\sum^{n+1}_{k=0}\lambda_{n+k+1}|a_{k}|\Big{)}^{2}$
$\displaystyle+\sum^{\infty}_{n=0}(n+1)\Big{(}\sum^{\infty}_{k=n+1}\lambda_{n+k+1}|a_{k}|\Big{)}^{2}.$
(3.4)

By Theorem 1.3 and the monotonicity of the sequence $\boldsymbol{\lambda}$, it
is true that

$\displaystyle\sum^{\infty}_{n=0}(n+1)\Big{(}\sum^{n+1}_{k=0}\lambda_{n+k+1}|a_{k}|\Big{)}^{2}$
$\displaystyle\leq$
$\displaystyle\sum^{\infty}_{n=0}(n+1)\lambda_{n+1}^{2}\Big{(}\sum^{n+1}_{k=0}|a_{k}|\Big{)}^{2}$
$\displaystyle\leq$
$\displaystyle\|\mathcal{C}_{\boldsymbol{\lambda}}f_{2}\|^{2}\lesssim\|f\|^{2}_{\mathcal{D}},$
(3.5)

where $f_{2}(z)=\sum^{\infty}_{k=0}|a_{k}|z^{k}$ with the same Dirichlet norm
of $f$.

Note that the monotonicity of the sequence $\boldsymbol{\lambda}$ again, the
Hölder inequality and (3.3). Then

$\displaystyle\sum^{\infty}_{n=0}(n+1)\Big{(}\sum^{\infty}_{k=n+1}\lambda_{n+k+1}|a_{k}|\Big{)}^{2}$
$\displaystyle\leq$
$\displaystyle\sum^{\infty}_{n=0}(n+1)\Big{(}\sum^{\infty}_{k=n+1}\lambda_{k}|a_{k}|\Big{)}^{2}$
$\displaystyle\lesssim$
$\displaystyle\|f\|^{2}_{\mathcal{D}}\sum^{\infty}_{n=0}(n+1)\bigg{(}\sum^{\infty}_{k=n+1}\frac{\lambda^{2}_{k}}{k}\bigg{)}$
$\displaystyle\thickapprox$
$\displaystyle\|f\|^{2}_{\mathcal{D}}\sum^{\infty}_{k=1}\frac{\lambda^{2}_{k}}{k}\sum^{k-1}_{n=0}(n+1)$
$\displaystyle\thickapprox$
$\displaystyle\|f\|^{2}_{\mathcal{D}}\sum^{\infty}_{k=1}k\lambda^{2}_{k}\lesssim\|f\|^{2}_{\mathcal{D}}.$
(3.6)

Joining (3), (3), and (3), we get the boundedness of
$\mathcal{H}_{\boldsymbol{\lambda}}$ on $\mathcal{D}$. The proof is complete.
∎

##  4\. Compact Hankel and Cesàro type operators on $\mathcal{D}$

In this section, we give corresponding results about the compactness of Hankel
and Cesàro type operators on $\mathcal{D}$.

The result below is the compact version of Theorem 1.1.

######  Theorem 4.1.

Suppose $\boldsymbol{\lambda}=\\{\lambda_{n}\\}_{n\in\mathbb{N}}$ is a
sequence of complex numbers. Then the Hankel type operator
$\mathcal{H}_{\boldsymbol{\lambda}}$ is compact on the Dirichlet space
$\mathcal{D}$ if and only if $h_{\overline{{\boldsymbol{\lambda}}}}$ is
analytic on $\mathbb{D}$ and the measure
$|h_{\overline{{\boldsymbol{\lambda}}}}^{\prime}(z)|^{2}dA(z)$ is a vanishing
Carleson measure for the Dirichlet space $\mathcal{D}$.

###### Proof.

Suppose $h_{\overline{\boldsymbol{\lambda}}}\in\mathcal{X}_{0}$. Then the
identity map

$I_{d}:\mathcal{D}\to
L^{2}(\mathbb{D},|h^{\prime}_{\overline{\boldsymbol{\lambda}}}|^{2}dA)$

is compact. Let $\\{f_{k}\\}_{k=1}^{\infty}$ be a bounded sequence in
$\mathcal{D}$ such that $\\{f_{k}\\}_{k=1}^{\infty}$ tends to 0 uniformly in
compact subsets of $\mathbb{D}$ as $k\to\infty$. From the proof of Theorem
1.1, we obtain

$\displaystyle|\langle\mathcal{H}_{\boldsymbol{\lambda}}f_{k},g\rangle_{\mathcal{D}}|$
$\displaystyle\lesssim$
$\displaystyle|\lambda_{0}||f_{k}(0)||g(0)|+\|g\|_{\mathcal{D}}\Big{(}\int_{\mathbb{D}}\Big{|}\frac{f_{k}(z)-f_{k}(0)}{z}\Big{|}^{2}|h^{\prime}_{\overline{\boldsymbol{\lambda}}}(z)|^{2}dA(z)\Big{)}^{\frac{1}{2}}$
$\displaystyle+\|g\|_{\mathcal{D}}\Big{(}\int_{\mathbb{D}}|f_{k}(z)|^{2}|h^{\prime}_{\overline{\boldsymbol{\lambda}}}(z)|^{2}dA(z)\Big{)}^{\frac{1}{2}}$

for all $g\in\mathcal{D}$. Hence, for any $\varepsilon>0$, there is an integer
$k_{0}$ such that

$|\langle\mathcal{H}_{\boldsymbol{\lambda}}f_{k},g\rangle_{\mathcal{D}}|\lesssim\|g\|_{\mathcal{D}}\
\ \varepsilon$

for all $k>k_{0}$ and all $g\in\mathcal{D}$. Then
$\|\mathcal{H}_{\boldsymbol{\lambda}}f_{k}\|_{\mathcal{D}}\to 0$ as
$k\to\infty$. Thus $\mathcal{H}_{\boldsymbol{\lambda}}$ is compact on
$\mathcal{D}$.

Conversely, suppose $\mathcal{H}_{\boldsymbol{\lambda}}$ is compact on
$\mathcal{D}$. For a bounded sequence
$\\{(f_{n},g_{n})\\}_{n=1}^{\infty}\subseteq\mathcal{D}\times\mathcal{D}$,
both $\\{f_{n}\\}$ and $\\{g_{n}\\}$ are bounded sequences in $\mathcal{D}$.
Then both $\\{\mathcal{H}_{\boldsymbol{\lambda}}(f_{n})\\}$ and
$\\{\mathcal{H}_{\boldsymbol{\lambda}}(g_{n})\\}$ have convergent
subsequences. Because of (2.1) and (2.2),
$T_{h_{\overline{\boldsymbol{\lambda}}}}$ extends to a compact bilinear form
on $\mathcal{D}$. Then Theorem A yields
$h_{\overline{\boldsymbol{\lambda}}}\in\mathcal{X}_{0}$. The proof is
complete. ∎

For the compactness of $\mathcal{C}_{\boldsymbol{\eta}}$ on $\mathcal{D}$, we
also have the following conclusion.

######  Theorem 4.2.

Suppose ${\boldsymbol{\eta}}=\\{\eta_{n}\\}_{n=0}^{\infty}$ is a sequence of
complex numbers. Then the following conditions are equivalent.

  1. (i)

The Cesàro type operator $\mathcal{C}_{\boldsymbol{\eta}}$ is compact on
$\mathcal{D}$.

  2. (ii)

The reproducing kernel thesis holds; that is,

$\lim_{t\to 1^{-}}\|\mathcal{C}_{\boldsymbol{\eta}}k_{t}\|_{\mathcal{D}}=0,$

where $k_{t}$ is the normalized reproducing kernel of $\mathcal{D}$ at $t$ in
$[0,1)$.

  3. (iii)

The Widom type condition is true; that is,

$\sum^{\infty}_{n=m}n|\eta_{n}|^{2}=o\left(\frac{1}{\log(m+2)}\right).$

###### Proof.

$(i)\Rightarrow(ii)$. Note that $\|k_{t}\|_{\mathcal{D}}=1$ for each
$t\in[0,1)$ and $k_{t}$ tends to zero uniformly in compact subsets of
$\mathbb{D}$ as $t\to 1^{-}$. Then (ii) holds.

$(ii)\Rightarrow(iii)$. Checking the proof of Theorem 1.3, we see

$\displaystyle\log(m+1)\sum^{\infty}_{n=m}(n+1)|\eta_{n+1}|^{2}\lesssim\|\mathcal{C}_{\boldsymbol{\eta}}k_{t}\|^{2}_{\mathcal{D}}$
(4.1)

for all positive integers $m$ and all $t$ in $[0,1)$. Taking $t\to 1^{-}$ in
(4.1), we obtain (iii).

$(iii)\Rightarrow(i)$. Let $m$ be a positive integer. For
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ in $\mathcal{D}$, consider

$\mathcal{C}_{\boldsymbol{\eta}}^{(m)}(f)(z)=\sum_{n=0}^{m}\Big{(}\eta_{n}\sum_{k=0}^{n}a_{k}\Big{)}z^{n},\quad
z\in\mathbb{D}.$

Then $\mathcal{C}_{\boldsymbol{\eta}}^{(m)}$ is a finite rank operator. Thus,
$\mathcal{C}_{\boldsymbol{\eta}}^{(m)}$ is compact on $\mathcal{D}$. Because
of (iii), for every $\epsilon>0$, there exists a positive integer $N$ such
that

$\sum^{\infty}_{n=m}(n+1)|\eta_{n+1}|^{2}<\frac{\epsilon}{\log(m+2)}$

for $m>N$. Using the proof of Theorem 1.3, we have

$\displaystyle\sum^{\infty}_{n=m}(n+1)|\eta_{n+1}|^{2}\Big{(}\sum^{n+1}_{k=1}|a_{k}|\Big{)}^{2}$
$\displaystyle\lesssim$
$\displaystyle\sum^{\infty}_{k=1}\sum^{\infty}_{j=1}|a_{k}||a_{j}|\sum^{\infty}_{n=\max\\{k-1,j-1,m\\}}(n+1)|\eta_{n+1}|^{2}$
$\displaystyle\lesssim$
$\displaystyle\epsilon\sum^{\infty}_{k=1}\sum^{\infty}_{j=1}\frac{|a_{k}||a_{j}|}{\log(k+j+m)}$
$\displaystyle\lesssim$ $\displaystyle\epsilon\|f\|_{\mathcal{D}}^{2}$

for all $m>N$. Thus, for $m>N$,

$\displaystyle\|(\mathcal{C}_{\boldsymbol{\eta}}-\mathcal{C}_{\boldsymbol{\eta}}^{(m)})(f)\|^{2}_{\mathcal{D}}$
$\displaystyle\leq$
$\displaystyle|a_{0}|^{2}\sum^{\infty}_{n=m}(n+1)|\eta_{n+1}|^{2}+\sum^{\infty}_{n=m}(n+1)|\eta_{n+1}|^{2}\Big{(}\sum^{n+1}_{k=1}|a_{k}|\Big{)}^{2}$
$\displaystyle\leq$ $\displaystyle\epsilon\|f\|_{\mathcal{D}}^{2}.$

In other words,
$\|\mathcal{C}_{\boldsymbol{\eta}}-\mathcal{C}_{\boldsymbol{\eta}}^{(m)}\|\to
0$ as $m\to\infty$. Hence $\mathcal{C}_{\boldsymbol{\eta}}$ is compact on
$\mathcal{D}$. The proof is finished. ∎

The compact result corresponding to Theorem 3.1 also holds.

######  Theorem 4.3.

Suppose ${\boldsymbol{\lambda}}=\\{\lambda_{n}\\}^{\infty}_{n=0}$ is a
decreasing sequence of positive real numbers. Then the following conditions
are equivalent.

  1. (i)

The Hankel type operator $\mathcal{H}_{\boldsymbol{\lambda}}$ is compact on
$\mathcal{D}$.

  2. (ii)

The reproducing kernel thesis holds; that is,

$\lim_{t\to
1^{-}}\|\mathcal{H}_{\boldsymbol{\lambda}}k_{t}\|_{\mathcal{D}}<\infty,$

where $k_{t}$ is the normalized reproducing kernel of $\mathcal{D}$ at $t$ in
$[0,1)$.

  3. (iii)

The Widom type condition is true; that is,

$\sum^{\infty}_{n=m}n\lambda_{n}^{2}=o\left(\frac{1}{\log(m+2)}\right).$

###### Proof.

By the proof of Theorem 3.1, the arguments here are similar to that of Theorem
4.2. We omit it. ∎

##  5\. Random Hankel type operators on $\mathcal{D}$ and a result related to
Rudin’s $\Lambda(p)$ set

In this section, we prove Theorem 1.4 and Corollary 1.5. Many known real
random variables sequences $(X_{n})_{n\in\mathbb{N}}$ satisfy the conditions
in Theorem 1.4. First of all, all bounded mean zero random variables are
included in Thereom 1.4. The typical example is the Bernoulli random variables
sequence $(X_{n})_{n\in\mathbb{N}}$. In other words,
$(X_{n})_{n\in\mathbb{N}}$ is independent such that
$\mathbb{P}(X_{n}=1)=\mathbb{P}(X_{n}=-1)=1/2$. In addition, for the classical
independent normal distribution sequence $X_{n}=N(0,1)$, we have
$\mathbb{E}[X_{n}]=0$ and $\mathbb{E}[X_{n}^{4}]=3$ (cf. [30, p. 2]).
Therefore, Theorem 1.4 can be applied to the corresponding random Gaussian
analytic Dirichlet functions.

For the $\boldsymbol{\lambda}=\\{\lambda_{n}\\}_{n\in\mathbb{N}}$ complex
number sequence and a sequence of i.i.d. real random variables
$(X_{n})_{n\in\mathbb{N}}$ with $\mathbb{E}[X_{n}]=0$ and
$\mathbb{E}[X_{n}^{4}]<\infty$. If $h_{\boldsymbol{\lambda}}\in\mathcal{D}$,
then $\sum_{n=0}^{\infty}|\lambda_{n}|^{2}<\infty$. Moreover, by i.i.d. and
$\mathbb{E}[X_{n}^{2}]\leq\sqrt{\mathbb{E}[X_{n}^{4}]}$, we have

$\mathbb{E}\big{[}\sum_{n=0}^{\infty}|\lambda_{n}X_{n}|^{2}\big{]}=\sum_{n=0}^{\infty}|\lambda_{n}|^{2}\mathbb{E}[|X_{n}|^{2}]<\infty$

and almost surely

$h_{\boldsymbol{\lambda},\omega}(z)=\sum_{n=0}^{\infty}X_{n}(\omega)\lambda_{n}z^{n}$

is analytic on the unit disk.

For our proof, we need the following elementary inequalities which may be well
known. However, we cannot locate literature. For the sake of completeness, we
have included a brief proof.

######  Lemma 5.1.

Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of i.i.d. real random variables
with $\mathbb{E}[X_{n}]=0$ and $\mathbb{E}[X_{n}^{4}]=1$. Then for all finite
complex numbers $a_{1},\cdots,a_{n}$, we have

$\mathbb{E}\bigg{[}\bigg{|}\sum_{i=1}^{n}a_{i}X_{i}\bigg{|}^{4}\bigg{]}\leq
3\bigg{[}\sum_{i=1}^{n}|a_{i}|^{2}\bigg{]}^{2}.$

###### Proof.

For any fixed natural number $n\geq 1$, let $A_{n}=\sum_{i=1}^{n}a_{i}X_{i}.$
It is easy to see

$|A_{n}|^{2}=\sum_{i=1}^{n}|a_{i}|^{2}X_{i}^{2}+2\sum_{i<j}Re(a_{i}\bar{a}_{j})X_{i}X_{j}.$

Note that $(X_{n})_{n\in\mathbb{N}}$ is a sequence of i.i.d. real random
variables with $\mathbb{E}[X_{n}]=0$ for each $n$. Then

$\displaystyle\mathbb{E}\big{[}|A_{n}|^{4}\big{]}$
$\displaystyle=\mathbb{E}\left[\bigg{(}\sum_{i=1}^{n}|a_{i}|^{2}X_{i}^{2}\bigg{)}^{2}\right]+4\mathbb{E}\left[\bigg{(}\sum_{i<j}Re(a_{i}\bar{a}_{j})X_{i}X_{j}\bigg{)}^{2}\right]$
$\displaystyle=\sum_{i,j}|a_{i}|^{2}|a_{j}|^{2}\mathbb{E}[X_{i}^{2}X_{j}^{2}]+4\sum_{i<j}(Re(a_{i}\bar{a}_{j}))^{2}\mathbb{E}[X_{i}^{2}X_{j}^{2}]$
$\displaystyle\leq\sum_{i,j}|a_{i}|^{2}|a_{j}|^{2}\mathbb{E}[X_{i}^{2}X_{j}^{2}]+4\sum_{i<j}|a_{i}\bar{a}_{j}|^{2}\mathbb{E}[X_{i}^{2}X_{j}^{2}].$

Observe that
$\big{(}\mathbb{E}[X_{i}^{2}X_{j}^{2}]\big{)}^{2}\leq\mathbb{E}[X_{i}^{4}]\mathbb{E}[X_{j}^{4}]=1$,
hence

$\mathbb{E}\bigg{[}\bigg{|}\sum_{i=1}^{n}a_{i}X_{i}\bigg{|}^{4}\bigg{]}\leq
3\bigg{[}\sum_{i=1}^{n}|a_{i}|^{2}\bigg{]}^{2},$

and the proof is completed. ∎

Let $f\in H(\mathbb{D})$. For $0<p<\infty$ and $0\leq r<1$, let

$M_{p}(f,r)=\left(\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{i\theta})|^{p}d\theta\right)^{\frac{1}{p}},$

and

$M_{\infty}(f,r)=\max_{\theta\in[0,2\pi]}|f(re^{i\theta})|.$

Denote by $H^{p}$ the Hardy space consisting of those functions $f$ in
$H(\mathbb{D})$ with

$\|f\|_{H^{p}}=\sup_{0<r<1}M_{p}(f,r)<\infty.$

It is well known (cf. [19, p. 3]) that $\mathcal{D}$ is a subset of $H^{p}$
for any $p>0$.

From a result of L. Brown and A. Shields [12, p. 300], if an analytic function
$f$ satisfies $M_{p}(f^{\prime},r)\in L^{2}([0,1],dr)$ with $2<p\leq\infty$,
then $|f^{\prime}(z)|^{2}dA(z)$ is a Carleson measure for $\mathcal{D}$. We
strengthen this conclusion as follows.

######  Lemma 5.2.

Let $2<p\leq\infty$. Suppose $\phi$ is an analytic function in the open unit
disk $\mathbb{D}$ with $M_{p}(\phi^{\prime},r)\in L^{2}([0,1],dr)$. Then
$|\phi^{\prime}(z)|^{2}dA(z)$ is a vanishing Carleson measure for the
Dirichlet space.

###### Proof.

Let $2<p<\infty$. Suppose $\\{f_{m}\\}_{m=1}^{\infty}$ is a sequence in
$\mathcal{D}$ such that $\sup_{m}\|f_{m}\|_{\mathcal{D}}<\infty$ and functions
$f_{m}$ tend to zero uniformly in compact subsets of $\mathbb{D}$ as
$m\to\infty$. Due to $M_{p}(\phi^{\prime},r)\in L^{2}([0,1],dr)$, for any
$\epsilon>0$, there exists a $\delta$ in $(0,1)$ such that

$\int_{\delta}^{1}\left(\int_{0}^{2\pi}|\phi^{\prime}(re^{i\theta})|^{p}d\theta\right)^{\frac{2}{p}}dr<\epsilon.$

Then

$\displaystyle\int_{\\{z\in\mathbb{D}:|z|>\delta\\}}|f_{m}(z)|^{2}|\phi^{\prime}(z)|^{2}dA(z)$
$\displaystyle\approx$
$\displaystyle\int_{\delta}^{1}\int_{0}^{2\pi}|f_{m}(re^{i\theta})|^{2}|\phi^{\prime}(re^{i\theta})|^{2}d\theta
dr$ $\displaystyle\lesssim$
$\displaystyle\left(\int_{\delta}^{1}\left(\int_{0}^{2\pi}|\phi^{\prime}(re^{i\theta})|^{p}d\theta\right)^{\frac{2}{p}}dr\right)\left(\int_{\delta}^{1}\left(\int_{0}^{2\pi}|f_{m}(re^{i\theta})|^{\frac{2p}{p-2}}d\theta\right)^{\frac{p-2}{p}}dr\right)$
$\displaystyle\lesssim$ $\displaystyle\epsilon\
\|f_{m}\|_{H^{\frac{2p}{p-2}}}^{2}\lesssim\epsilon\
\|f_{m}\|_{\mathcal{D}}^{2}\lesssim\epsilon.$

Since functions $f_{m}$ tend to zero uniformly in
$\\{z\in\mathbb{D}:|z|\leq\delta\\}$, there is a positive integer $N$ such
that

$\int_{\\{z\in\mathbb{D}:|z|\leq\delta\\}}|f_{m}(z)\phi^{\prime}(z)|^{2}dA(z)\leq\epsilon$

for all $m>N$. Consequently,

$\lim_{m\to\infty}\int_{\mathbb{D}}|f_{m}(z)\phi^{\prime}(z)|^{2}dA(z)=0.$

Thus $|\phi^{\prime}(z)|^{2}dA(z)$ is a vanishing Carleson measure for
$\mathcal{D}$. The proof of the case of $p=\infty$ is similar. We omit it.

∎

Note that if $p=2$ and $\phi\in H(\mathbb{D})$, then
$M_{p}(\phi^{\prime},r)\in L^{2}([0,1],dr)$ if and only if
$\phi\in\mathcal{D}$. In general, the measure $|\phi^{\prime}(z)|^{2}dA(z)$
with $\phi\in\mathcal{D}$ is not a vanishing Carleson measure for the
Dirichlet space.

Now, we are ready to prove Theorem 1.4.

###### Proof.

If $\mathbb{E}[X_{n}^{4}]=0$, then $X_{n}=0$ almost surely. Hence without loss
of generality, we can assume that $\mathbb{E}[X_{n}^{4}]=1$. Recall that

$h_{\overline{\boldsymbol{\lambda}},\omega}(z)=\sum_{n=0}^{\infty}X_{n}(\omega)\overline{\lambda_{n}}z^{n}.$

By Theorem 1.1, we shall show that almost surely

$|h_{\overline{\boldsymbol{\lambda}},\omega}^{\prime}(z)|^{2}dA(z)$

is vanishing Carleson measure for $\mathcal{D}$. By Lemma 5.2, it is
sufficient to show that

$\mathbb{E}\bigg{[}\int_{0}^{1}\bigg{(}\int_{0}^{2\pi}|h_{\overline{\boldsymbol{\lambda}},\omega}^{\prime}(re^{i\theta})|^{4}\frac{d\theta}{2\pi}\bigg{)}^{1/2}dr\bigg{]}<\infty.$

It follows from Fubini’s theorem that

$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{1}\bigg{(}\int_{0}^{2\pi}|h_{\overline{\boldsymbol{\lambda}},\omega}^{\prime}(re^{i\theta})|^{4}\frac{d\theta}{2\pi}\bigg{)}^{1/2}dr\bigg{]}$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\int_{\Omega}\bigg{(}\int_{0}^{2\pi}|h_{\overline{\boldsymbol{\lambda}},\omega}^{\prime}(re^{i\theta})|^{4}\frac{d\theta}{2\pi}\bigg{)}^{1/2}d\mathbb{P}(w)dr.$

Applying the Hölder inequality with respect to the probability measure
$d\mathbb{P}$, we have

$\displaystyle\int_{0}^{1}\int_{\Omega}\bigg{(}\int_{0}^{2\pi}|h_{\overline{\boldsymbol{\lambda}},\omega}^{\prime}(re^{i\theta})|^{4}\frac{d\theta}{2\pi}\bigg{)}^{1/2}d\mathbb{P}(w)dr$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}\bigg{(}\int_{\Omega}\int_{0}^{2\pi}|h_{\overline{\boldsymbol{\lambda}},\omega}^{\prime}(re^{i\theta})|^{4}\frac{d\theta}{2\pi}d\mathbb{P}(w)\bigg{)}^{1/2}dr.$

Consequently,

$\int_{\Omega}\int_{0}^{2\pi}|h_{\overline{\boldsymbol{\lambda}},\omega}^{\prime}(re^{i\theta})|^{4}\frac{d\theta}{2\pi}d\mathbb{P}(w)=\int_{0}^{2\pi}\int_{\Omega}|h_{\overline{\boldsymbol{\lambda}},\omega}^{\prime}(re^{i\theta})|^{4}d\mathbb{P}(w)\frac{d\theta}{2\pi}.$

Recall that

$h_{\overline{\boldsymbol{\lambda}},\omega}^{\prime}(re^{i\theta})=\sum_{n=1}^{\infty}\overline{\lambda_{n}}nr^{n-1}e^{i(n-1)\theta}X_{n}(\omega)$

and Lemma 5.1, we get

$\displaystyle\int_{0}^{1}\left(\int_{\Omega}\bigg{|}\sum_{n=1}^{\infty}\overline{\lambda_{n}}nr^{n-1}e^{i(n-1)\theta}X_{n}\bigg{|}^{4}d\mathbb{P}(w)\right)^{\frac{1}{2}}dr$
$\displaystyle\leq$ $\displaystyle
3\sum_{n=1}^{\infty}\int_{0}^{1}|\overline{\lambda_{n}}nr^{n-1}|^{2}dr.$

Therefore, there is a constant $C$ such that

$\displaystyle\mathbb{E}\bigg{[}\int_{0}^{1}\bigg{(}\int_{0}^{2\pi}|h_{\boldsymbol{\lambda},\omega}^{\prime}(re^{i\theta})|^{4}\frac{d\theta}{2\pi}\bigg{)}^{1/2}dr\bigg{]}$
$\displaystyle\leq$ $\displaystyle
C\sum_{n=1}^{\infty}n|\lambda_{n}|^{2}<\infty.$

This completes the whole proof. ∎

To prove Corollary 1.5, we start with a simple observation.

######  Lemma 5.3.

Suppose $p>2$, $E$ is a Rudin’s $\Lambda(p)$ set and
$\boldsymbol{\lambda}=\\{\lambda_{n}:n\in E\\}$ is a sequence of complex
numbers. If $\sum_{n\in E}n|\lambda_{n}|^{2}<\infty$, then
$|h_{\overline{\boldsymbol{\lambda}}}^{\prime}(z)|^{2}dA(z)$ is a vanishing
Carleson measure for the Dirichlet space, where
$h_{\overline{\boldsymbol{\lambda}}}(z)=\sum_{n\in
E}\overline{\lambda_{n}}z^{n}.$

###### Proof.

Let $p>2$. For any $0<r<1$, observe that

$h_{\overline{\boldsymbol{\lambda}}}^{\prime}(re^{i\theta})=e^{-i\theta}\sum_{n\in
E}\overline{\lambda_{n}}nr^{n-1}e^{in\theta},$

we have

$\|h_{\overline{\boldsymbol{\lambda}}}^{\prime}(re^{i\theta})\|_{L^{p}([0,2\pi],d\theta)}=\bigg{\|}\sum_{n\in
E}\overline{\lambda_{n}}nr^{n-1}e^{in\theta}\bigg{\|}_{L^{p}([0,2\pi],d\theta)}.$

Since $E$ is a Rudin’s $\Lambda(p)$,

$\displaystyle\bigg{\|}\sum_{n\in
E}\overline{\lambda_{n}}nr^{n-1}e^{in\theta}\bigg{\|}^{2}_{L^{p}([0,2\pi],d\theta)}$
$\displaystyle\leq$ $\displaystyle C\bigg{\|}\sum_{n\in
E}\overline{\lambda_{n}}nr^{n-1}e^{in\theta}\bigg{\|}^{2}_{L^{2}([0,2\pi],d\theta)}$
$\displaystyle=$ $\displaystyle C\sum_{n\in
E}|\lambda_{n}|^{2}n^{2}r^{2(n-1)}.$

Recall that $\sum_{n\in E}n|\lambda_{n}|^{2}<\infty$, therefore,

$\displaystyle\int_{0}^{1}\bigg{\|}\sum_{n\in
E}\overline{\lambda_{n}}nr^{n-1}e^{in\theta}\bigg{\|}^{2}_{L^{p}([0,2\pi],d\theta)}dr$
$\displaystyle\leq$ $\displaystyle C\int_{0}^{1}\sum_{n\in
E}|\lambda_{n}|^{2}n^{2}r^{2(n-1)}dr$ $\displaystyle<$ $\displaystyle+\infty.$

The lemma then follows from Lemma 5.2. ∎

###### Proof of Corollary 1.5.

Suppose $\mathcal{H}_{\boldsymbol{\lambda}}$ is bounded on $\mathcal{D}$. By
Theorem 1.1, $h_{\overline{{\boldsymbol{\lambda}}}}\in\mathbb{D}$. By the
definition of Dirichlet norm, the condition (iii) holds. $(i)\Rightarrow(ii)$
is clear. The implication $(iii)\Rightarrow(i)$ follows from Lemma 5.3 and
Theorem 4.1. The proof is complete. ∎

##  6\. Final remarks

In this section, we give some remarks about some functions in $\mathcal{X}$
and the action of Hankel matrices on the Bergman space $A^{2}$.

As mentioned in [3, p. 16], very little is known about functions in
$\mathcal{X}$. Our results in this paper give a satisfactory description of
functions in $\mathcal{X}$ with decreasing Taylor’s sequences of positive
numbers. More precisely, suppose
${\boldsymbol{\lambda}}=\\{\lambda_{n}\\}^{\infty}_{n=0}$ is a decreasing
sequence of positive numbers. From Theorem 1.1 and Theorem 3.1, the measure
$|h^{\prime}_{\boldsymbol{\lambda}}(z)|^{2}dA(z)$ is a Carleson measure for
$\mathcal{D}$ (i.e. $h_{\boldsymbol{\lambda}}\in\mathcal{X}$) if and only if

$\sum^{\infty}_{n=m}n\lambda_{n}^{2}=O\left(\frac{1}{\log(m+2)}\right).$

By Theorem 4.1 and Theorem 4.3, the measure
$|h^{\prime}_{\boldsymbol{\lambda}}(z)|^{2}dA(z)$ is a vanishing Carleson
measure for $\mathcal{D}$ (i.e. $h_{\boldsymbol{\lambda}}\in\mathcal{X}_{0}$)
if and only if

$\sum^{\infty}_{n=m}n\lambda_{n}^{2}=o\left(\frac{1}{\log(m+2)}\right).$

Suppose $\boldsymbol{\lambda}=\\{\lambda_{n}\\}_{n\in\mathbb{N}}$ is a
sequence of complex numbers. Let $(X_{n})_{n\in\mathbb{N}}$ be a sequence of
i.i.d. real random variables with $\mathbb{E}[X_{n}]=0$ and
$\mathbb{E}[X_{n}^{4}]<\infty$. If $h_{\boldsymbol{\lambda}}\in\mathcal{D}$,
Theorem 1.4 and Theorem 4.1 yield that

$\mathbb{P}(h_{{\overline{{\boldsymbol{\lambda}}},\omega}}\in\mathcal{X})=\mathbb{P}(h_{\overline{{\boldsymbol{\lambda}}},\omega}\in\mathcal{X}_{0})=1,$

where
$h_{\overline{{\boldsymbol{\lambda}}},\omega}(z)=\sum_{n=0}^{\infty}X_{n}\overline{\lambda_{n}}z^{n}$.

Next we consider Hankel type operators $\mathcal{H}_{\boldsymbol{\lambda}}$ on
the Bergman space $A^{2}$ which is also a Hilbert space of analytic functions
on $\mathbb{D}$ and it is equipped with the inner product

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

From [15, p. 349], $(A^{2})^{*}\cong\mathcal{D}$ and $\mathcal{D}^{*}=A^{2}$
under the pairing

$\langle f,g\rangle=\sum^{\infty}_{k=0}a_{k}b_{k},$ (6.1)

where $f(z)=\sum^{\infty}_{k=0}a_{k}z^{k}$ and
$g(z)=\sum^{\infty}_{k=0}b_{k}z^{k}$. For a sequence
$\boldsymbol{\lambda}=\\{\lambda_{n}\\}_{n\in\mathbb{N}}$ of complex numbers,
it is easy to see

$\langle\mathcal{H}_{\boldsymbol{\lambda}}f,g\rangle=\langle
f,\mathcal{H}_{\boldsymbol{\lambda}}g\rangle.$

Consequently, $\mathcal{H}_{\boldsymbol{\lambda}}$ is bounded (resp. compact)
on $\mathcal{D}$ if and only if it is bounded (resp. compact) on $A^{2}$. If
we replace the pairing (6.1) by the Cauchy pairing

$(f,g)=\sum^{\infty}_{k=0}a_{k}\overline{b_{k}}.$

Then

$(\mathcal{H}_{\boldsymbol{\lambda}}f,g)=(f,\mathcal{H}_{\overline{\boldsymbol{\lambda}}}g).$

Hence we also get that $\mathcal{H}_{\boldsymbol{\lambda}}$ is bounded (resp.
compact) on $\mathcal{D}$ if and only if
$\mathcal{H}_{\overline{\boldsymbol{\lambda}}}$ is bounded (resp. compact) on
$A^{2}$.

Acknowledgement. The work was supported by National Natural Science Foundation
of China (No. 12231005 and No. 12271328), Guangdong Basic and Applied Basic
Research Foundation (No. 2022A1515012117), China Postdoctoral Science
Foundation (No. 2023TQ0074) and National Natural Science Foundation of of
Chongqing (No. CSTB2022BSXM-JCX0088 and 2022NSCQ-MSX0321).

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