# Detection of fake news on CoViD-19
on Web Search Engines

V. Mazzeo
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>Department of Physics and Astronomy “E. Majorana”,
University of Catania, Italy A. Rapisarda
<EMAIL_ADDRESS>Department of Physics and Astronomy “E. Majorana”,
University of Catania and INFN Sezione di Catania, Italy; Complexity Science
Hub Vienna (CSH), Austria; G. Giuffrida
<EMAIL_ADDRESS>Department of Political and Social Sciences
(DSPS), University of Catania, Italy

###### Abstract

In early January 2020, after China reported the first cases of the new
coronavirus (SARS-CoV-2) in the city of Wuhan, unreliable and not fully
accurate information has started spreading faster than the virus itself.
Alongside this pandemic, people have experienced a parallel infodemic, i.e.,
an overabundance of information, some of which misleading or even harmful,
that has widely spread around the globe. Although Social Media are
increasingly being used as information source, Web Search Engines, like Google
or Yahoo!, still represent a powerful and trustworthy resource for finding
information on the Web. This is due to their capability to capture the largest
amount of information, helping users quickly identify the most relevant,
useful, although not always the most reliable, results for their search
queries.
This study aims to detect potential misleading and fake contents by capturing
and analysing textual information, which flow through Search Engines. By using
a real-world dataset associated with recent CoViD-19 pandemic, we first apply
re-sampling techniques for class imbalance, then we use existing Machine
Learning algorithms for classification of not reliable news. By extracting
lexical and host-based features of associated Uniform Resource Locators (URLs)
for news articles, we show that the proposed methods, so common in phishing
and malicious URLs detection, can improve the efficiency and performance of
classifiers.
Based on these findings, we suggest that the use of both textual and URLs
features can improve the effectiveness of fake news detection methods.

Keywords: fake news, covid-19, search engine, machine learning, class
imbalance, phishing

##  1 Introduction

The reliability and credibility of both information source and information
itself have emerged as a global issue in contemporary society [1; 2].
Undoubtedly, in the last decades Social Media have revolutionised the way in
which information spreads across the Web and, more generally, the world [3;
4], by allowing users to freely share content faster than traditional news
sources. The fact that content spreads so quickly and easily across platforms
suggests that people (and algorithms behind the platforms) are potentially
vulnerable to misinformation, hoaxes, biases and low-credibility contents
which are daily shared, accidentally or intentionally.
The problem of spreading misinformation, however, affects not only the Social
Media platforms, but also the World Wide Web. In fact, every time people enter
a search query on Web Search Engines (WSE), like Google or Bing, they can view
and potentially access hundreds, or thousands, of web pages with helpful
information, sometimes potentially misleading. Meta title tags displayed on
Search Engine Results Pages (SERPs) [5] represent then a crucial factor in
helping user understand pages’ content, being the user’s first experience of a
website.
Although people tend more likely to view and click on the first results on the
first page because of Search Engines’ rank algorithms, e.g., PageRank (Page,
1998), which show the most relevant information in response to a specific
query, a good title can be the make-or-break factor which brings users to
click on that link and read on [6; 7].
Despite the systematic and significant response efforts and fact-checking
against misinformation mobilised by both Social Media and Media companies,
fake news still persist due to the vast volume of online content, which leads
people to see and share information that is partly, or completely, misleading.
Previous and recent studies have almost exclusively focused on data from
Social Media (e.g., Twitter)[8], fact-checking or reliable websites (e.g.,
Snopes.com, PolitiFact.com)[9], or existing datasets[10] which have the
benefit to be cost-efficient.
Due to the current difficult and unprecedented situation with the CoViD-19
pandemic, never seen in the modern era [11], people have asked many questions
about the novel coronavirus, such as the origin of the disease, treatment,
prevention, cure, and transmission from or to pets, to face these challenges,
while staying informed and safe. In this study, we focus on news displayed by
Web Search Engines, since they represent the best tools for bringing up
answers to people’s current questions, extracting information related to
CoViD-19 outbreak, and proposing an approach based on both textual and Uniform
Resource Locator (URL) features to analyse and detect whether news are
fake/misleading or reliable (real). The contribution of our work can be
summarised as follows:

  * •

we use real-world data from WSE, analysing both textual data (meta titles and
descriptions) and URLs information, by extracting features representations;

  * •

since most of the previous works on fake news detection were focused on
classifier enhancements, not engaging in feature engineering, in this document
we want to provide a new direction for the classification of fake news,
proposing an integration of the most commonly used features in fake news
detection and features that play an important role in the malicious URL
detection. The purpose of feature engineering is indeed to feed the original
data and provide new and meaningful feature representations to improve machine
learning (ML) algorithms for classification. Currently, the problem of
detecting fake news via URL has not been well and sufficiently addressed.
Several studies focused on fake news detection via ML in Social Networks [12]
have looked at the presence of URLs in the user’s published content [13],
without generally performing further analysis on the source of information or
extracting other potential relevant URL-based features (features that are,
indeed, more common in malicious URLs/phishing detection classifiers).
Although, in the past, the usage of URLs in a post/news could have represented
a useful parameter for enhancing and improving ML classifiers’ performance,
nowadays this could result not enough for differentiating a good source from a
bad one in terms of information credibility without a URL-based feature
engineering approach. In fact, the more ML techniques have evolved over time,
the more schemes for spreading fake news have changed;

  * •

we apply re-sampling techniques, such as under-sampling and over-sampling, due
to the class imbalance of the real-world dataset [14; 15]. Disproportion
between classes still represents an open issue and challenge for researchers
focused on classification problems. In a typical news dataset, the number of
fake news are likely to be very few compared to the number of real ones, and
this fact makes the positive class (fake news) very small compared to the
negative class (real news). This imbalance between the two classes would
likely make classifiers biased towards the majority class leading to classify
all the instances in the dataset as belonging to the majority class;

  * •

we compare different ML algorithms (Support Vector Machine, Stochastic
Gradient Descent, Logistic Regression, Naïve Bayes, and Random Forest), based
on their performance. Since we deal with imbalanced data, we evaluate the
models looking at $F_{1}$ score and Recall metrics, and not at predictive
accuracy, as the latter represents a misleading indicator which reflects the
underlying class distributions [16; 17].

The paper is structured as follows: Section 2 introduces our material and
methodology; Section 3 describes the results of our experimentation along with
their evaluation. In Section 4 we summarise our key findings and give an
interpretation of them, also by discussing the implications. Finally, Section
5 draws the conclusions, giving some prospective points for future work.

##  2 Material and Methods

###  2.1 Data collection

We submitted several CoViD-19-related queries (Table 1) on a WSE. For each
search result, we extracted metadata information, i.e., URL, title, meta
description (or snippet), and date (Figure 1(a)).

Search Query | Example of Title
---|---
italy+travel+coronavirus | Italy will PAY half the price of your hotel_Travel Tourism News
japan+travel+price | Japan Foreign Arrivals Down 99.9% In April & To Cover 50 …
phones+missing | A Thread from @ApostleKom: \lq21 Million Chinese Cellphone …
event 201+gates | QAnon Supporters, Anti-Vaxxers Spread A Hoax Bill Gates …
coronavirus+size+diameter | Carona Virus Updates ! Unicef Corona… - Jodhbir Singh …
people+collapsing+china | China Corona Virus Horror: Hospital Corridor of the Dead and Dying …
5g+coronavirus | New Study Suggests 5G Could Create Coronavirus Type … - lbry.tv
Table 1:  Search queries submitted to Web Search Engine and meta information
results. The results of a search query are often returned as a list of meta
data and they may consist of web pages, images, and other types of files with
helpful information.

The final dataset consisted of a collection of approximately 3350 news results
(fake/misleading and trusted/‘real’), gathered from 2084 different urls. All
the news were published within a 7-month time interval, between January
$20^{th}$ and July $28^{th}$, 2020. We chose this time interval as it covered
the first CoViD-19 pandemic lockdowns proclaimed in Italy and in other
countries [18]. Queries were selected based on topics (e.g., generic
information on the new virus; pseudo-scientific therapies; conspiracy
theories; travels; etc.) [19] that we were monitoring both on the Web (online
newspapers) and Social Media, during the first lockdown period. We also looked
at fact-checking websites, such as politifact.com or poynter.com, to check
news and information credibility [20].

(

(a)

(

(b)

(

(c)  Figure 1: Example of search results on some of the most popular Web
Search Engines: Google, Bing, and Yahoo!. The most important meta tags
displayed by the Search Engine Results Pages, in response to user queries, are
title, description (or snippet), date and cached URL.The sub-figure (A) is
scaled up in order to highlight the meta tags (in red rectangles) extracted
during data collection.

A) B) C)

In order to reduce potential bias due to Search Engine optimisation, we had
carefully planned our data collection as follows:

  * •

we used a VPN to be more consistent with the WSE domain inspected and its
results;

  * •

in order to browse the Internet and query the WSE, we used a Private/Incognito
window. This allowed us to prevent our browsing history from being stored and
from biasing our results. By using Incognito/Private mode, we did not build a
detailed picture of our online activity: in this way, all cookies were removed
at the end of each session, i.e., we did not save any information about the
pages we were visiting, avoiding to create customised results based on our
search history. This process was repeated for each query and for each day of
collection;

  * •

we collected all the results, and not some of them (e.g., the first 2 pages or
the top 10 results).

Even if we dramatically reduced bias during our data collection, the results
from WSE might be automatically biased by the WSEs we were querying because of
their ranking systems, which sort results by relevance. We did not have any
control on that but we tried to address this potential bias comparing results
from different WSEs, also across different days.
Once we collected data, the labelling procedure was done manually and it
consisted of assigning a binary class label indicating whether the news was
real ($0$) or fake/misleading ($1$). In the binary fake news detection
problem, fake news is usually associated with the positive class, since these
are the news detected by the classifier. Data labelling process for training
ML algorithms is not only critical but also time consuming. Because of the
limited resources, we considered a limited sample size in our study, but big
enough to be considered reliable and sufficiently large for binary detection
[21].
The ML workflow proposed in this study was implemented in Python 3.8. Its
schematic representation is illustrated in Figure 2.

Figure 2: Machine Learning workflow for training algorithms and classification
of results from Web Search Engines (WSE). A KNN imputation algorithm and
MinMaxScalar were used to rescale variables. However, the percentage of
missing values in the dataset was very low ($<1\%$). Since high correlation
among features leads to redundancy of features and instability of the model,
statistical tests, like Chi-Squared and Pearson’s correlation coefficient,
were used for feature selection. The corpus collected from WSEs was,
therefore, pre-processed before being used as an input for training the
models. Due to the class imbalance in the dataset, re-sampling techniques were
applied to the training set only, during cross-validation (k-fold=5).
Different classification models were then evaluated by scoring the
classification outcomes from a testing set, in terms of the following
performance metrics: $F_{1}$-score, Recall, Accuracy and Precision.

###  2.2 Data pre-processing and Feature Engineering

In order to observe the most meaningful context words and to improve the
performance of the classifiers, in the data pre-processing stage we removed
all parts that were irrelevant, redundant and not related to the content:
punctuation (with a few exceptions of symbols, like exclamation mark, question
mark, and quotation marks) and extra delimiters; symbols; dashes from both
titles and descriptions; stopwords [22].
By following guidance and advice given by fact-checking websites (e.g.,
factcheck.org) and reputable outlet sources (e.g., bbc.com) on how to spot
fake news, we looked at the presence of words in capital letters and at the
excessive use of punctuation marks in both titles and descriptions. Figure 3
shows the frequency of specific punctuation characters (‘!’, ‘?’, ‘ “ ’, ‘:’)
and upper case words in titles and descriptions for news labelled as fake (1)
and real (0). It is notable that fake news differs much more for real one by
the excessive use of punctuation, quotes, interrogatives, words in all capital
letters, and exclamation mark to alert and urge people to read the news.

(A)
---

(B)
---

Figure 3: Frequency of: (A) punctuation characters and upper case words in
meta titles and descriptions; (B) some specific punctuation characters
(exclamation mark, question mark, quotes, colons). With the exception of upper
case words that were used more frequently in meta descriptions, fake news has
shown a higher percentage of punctuation in both title and description,
probably by ‘over-dramatising’ events; also, the use of upper case characters
in titles appears more evident in misleading content.

The frequency distributions in Figure 4 illustrate the top 20 uppercase words
in the fake news and in the real news datasets. From the two histograms, we
can derive an important information regarding the use of various uppercase
words in the two news sets. It can be noticed, in fact, that in the real news
dataset all uppercase words are more related to abbreviations (e.g., US, UK),
acronyms (e.g., UNICEF), or organisations’ name (NBC, NCBI), while in fake
news dataset the use of uppercase letters highlights potential warnings (e.g.,
CONTROL, CREATE), capitalising on coronavirus fears and conspiracy theories.
This highlights the different use of capitalising all characters in a word, an
unusual habit for reporters working for trustworthy websites, who generally
follow style-guidelines and journalistic convention.

(A)
---

(B)
---

(C)
---

(D)
---

Figure 4: The 20 mostly used uppercase words in fake news (A) and in real news
(C) datasets. In the sub-figures (B) and (D), respectively, the word clouds
show words sized according to their weights in the datasets. The use of
uppercase words is different between the two datasets: in real news, the use
of uppercase words is more frequent to indicate acronyms, brands,
organisations, while in fake news uppercase words emphasise feelings, creating
alerts and potential warnings.

In the feature engineering stage, which typically includes feature creation,
transformation, extraction and selection), we used pre-training algorithms,
such as Bag-of-Words (BoW) [23] and Term Frequency–Inverse Document Frequency
(TF-IDF) [24; 25], for mapping cleaned texts (titles and descriptions) into
numeric representations. Further features (length, counting, binary) were also
extracted from URLs [26].

Information on the age of domain names was gathered from both Wayback Machine
and WHOIS [27], two tools that are crucial resources in the fight against fake
news, as they allow users to see how a website has changed and evolved through
time, gathering information on when the website was founded, on its country
code top-level domain, and contact information.

###  2.3 Re-sampling

Although various ML methods assume that the target classes have same or
similar distribution, in real conditions this does not happen as data is
unbalanced [28], with nearly most of the instances labelled with one class,
while a few instances are labelled as the other one. Since we worked with
real-world data [29; 30], our dataset presented a high class imbalance with
significantly less samples of fake news than real one. To address poor
performance in case of unbalanced dataset, we used:

  * •

minority class random over-sampling technique, which consists in over-sizing
the minority class by adding observations;

  * •

majority class random under-sampling technique, which consists in down-sizing
the majority class by randomly removing observations from the training
dataset.

The re-sampling algorithms chosen depend on the nature of the data,
specifically on the ratio between the two classes, fake/real. Although we had
a class imbalance skewed (90:10), we could not treat our case as a problem of
anomaly (or outlier) detection. In fact, in order to be considered such a
case, we would have had a very skewed distribution (100:1) between the normal
(real) and rare (fake) classes.
Although the choice of the number of folds is still an open problem, generally
researchers choose a number of folds equal to 3 (less common), 5 or 10. We
used a 5-fold cross validation due to the small size of our dataset, but
enough to contain sufficient variation [31]. Each fold was used once as a
validation, while the k - 1 remaining folds formed the training set. This
process repeatedly ran until each fold of the 5 folds were used as the testing
set.

##  3 Results

In this section we discuss features from URLs, the metrics used for evaluating
models’ performance and we report the classification results.

###  3.1 URLs analysis

We analysed lexical and host-based features from 2084 distinct URLs. To
implement lexical features, we used a Bag-of-Words of tokens in URL, where ‘/’
, ‘?’, ‘.’, ‘=’, ‘_’, and ‘-’ are delimiters. We distinguished tokens that
appear in the host name, path, top-level domain, using also the lengths of the
host name and the URL as features [32; 33]. In Table 2 we show all the
features extracted from URLs. Word-based features were introduced as well, as
URLs were found to contain several suggestive word tokens. An example of URL
structure is shown indeed in Figure 5, where it is possible to distinguish the
following parts:

  * •

scheme: it refers to the protocol, i.e., a set method for exchanging or
transferring data, that the browser should use to retrieve any resource on the
web. https is the most secured version;

  * •

third-level domain: it is the next highest level following the second-level
domain in the domain name hierarchy. The most commonly used third-domain is
www;

  * •

second-level domain: it is the level directly before the top-level domain. It
is generally the part of a URL that identifies the website’s domain name;

  * •

top-level domain: it is the domain’s extension. The most used TLD is .com. The
TLD can also give about geographic of a website, since each country has a
unique domain suffix (e.g., .co.uk for UK websites).

We used the Chi-Squared (${\chi}^{2}$) statistical test to assess the
alternate hypothesis that the association we observed in the data between the
independent variables (URL-feature) and the dependent variable (fake/not fake)
was significant; specifically:

  * •

null hypothesis ($H_{0}$): there is no significant association between the
variables and the dependent variable (fake/not fake);

  * •

alternate hypothesis ($H_{1}$): there is an association between the variables
and the dependent variable (fake/not fake).

We set a significance level of 0.05 [34]:

  * •

if the p-value was less than the significance level, then we rejected the null
hypothesis and concluded that there was a statistically significant
association between the variables;

  * •

if the p-value was greater than or equal to the significance level, we failed
to reject the null hypothesis because there was not enough evidence to
conclude that the variables were associated.

The correlation-based feature selection (CFS) algorithm was used for
evaluating the worth or merit of a subset of features, taking into account the
usefulness of individual features for predicting the class label.
In order to check high correlations among independent variables, we also
performed a multicollinearity test. Multicollinearity is indeed a common
problem when estimating models such as logistic regression. In general, to
simulate predictor variables with different degree of collinearity, the
Pearson pairwise correlation coefficients were varied: an absolute correlation
of greater than or equal to $0.7$ can be considered an appropriate indicator
for strong correlation [35].
To measure the increase in the prediction error of the model, permutation
importance feature was employed. The method is most suitable when the number
of features is not huge as it is resource-intensive. This method can be also
used for feature selection. In fact, it allows to select features based on
their importance on the model. If there are features correlated, then the
permutation importance will be low for all the correlated features. The choice
of permutation importance as extra method for feature selection was justified
also by the use of different models, tree and not-tree based, respectively
[36].
These feature selection methods allowed us to select a small number of highly
predictive features in order to avoid over-fitting.

Figure 5: URL’s structure. A URL may contain mandatory (scheme, third-level, second-level, top-level domain) and optional (path) parts. Feature | Feature Name | Type | Fake News | Real News
---|---|---|---|---
Domain | URL length | continuous | $95.2\pm 2.0$ | $82.1\pm 0.7$
| if domain starts with numbers | binary | 1.2% | 0.2%
| if domain is an IP address | binary | . | .
| if .com | categorical | 85.0% | 73.3%
| if .org | categorical | 8.4% | 15.0%
| if .gov | categorical | 0.3% | 4.6%
| if .net | categorical | 2.3% | 2.3%
| age | continuous | $2010\pm 0.5$ | $2006\pm 0.2$
Word-based | if blogspot, blog, wordpress, blogger, … is contained in the URL | binary | 4.4% | 1.7%
| if facebook, twitter, instagram, … is contained in the URL | binary | 8.7% | 1.0%
| if news press, journal, publisher, … is contained in the URL | binary | 10.2% | 7.1%
| if coronavirus, virus, covid is contained in the URL | binary | 0.6% | 0.1%
Host-based | if http | binary | 6.1% | 3.8%
| if https | binary | 63.7% | 57.5%
Lexical | dots count | continuous | $1.73\pm 0.04$ | $2.07\pm 0.02$
| semicolons count | continuous | . | $0.002\pm 0.002$
| ampersand count | continuous | . | 0.031
| slash count | continuous | $5.19\pm 0.08$ | $5.195\pm 0.029$
| hyphen count | continuous | $7.8\pm 0.3$ | $5.4\pm 0.1$
| underscore count | continuous | $0.3$ | $0.2$
| equal count | continuous | $0.085\pm 0.015$ | $0.081\pm 0.009$
| question mark count | continuous | $0.085\pm 0.015$ | $0.053\pm 0.004$
| @ count | continuous | $0.003\pm 0.003$ | $0.001\pm 0.001$
| digit count | continuous | $7.6\pm 0.4$ | $5.7\pm 0.1$
Table 2: List of domain, host-based, and lexical features, extracted from
URLs. There are: 8 domain features; 4 word-based features, which show the
presence of specific words in a URL; 2 host-based features; and the remaining
10 features are lexical-based and include special characters count or show the
presence of digits in a URL. The purpose of feature engineering was to find
and explore new features to improve model performance for fake news detection.
We used Chi-square statistics and a correlation-based feature selection (CFS)
approach. If reported, the error is meant the standard error of the mean.

In our fake news dataset, the average length of URL was 95.2 characters. This
URL length is greater than the values found by [37; 38; 39] for phishing URLs.
The number of hyphens (-) in the URL was found to be greater than 7 on average
for fake news URLs. The most surprising result is in the number of dots in
URL: fake news URLs in our dataset do not contain more than 2 dots on average.
In Table 3 we can observe that websites publishing fake news have generally
newer domain name’s age than websites publishing reliable news (Table 3 and
Figure 6).

| Total | Mean | Median | 10th percentile | 90th percentile
---|---|---|---|---|---
Websites publishing Fake news | 265 | 2010 | 2011 | 1998 | 2019
Websites publishing Real news | 1865 | 2006 | 2004 | 1996 | 2017
Table 3:  Summary statistics of domain name’s age (Wayback Machine) of
distinct websites in our dataset.

For validating the results shown in Table 3, we used Welch’s t-test, which is
usable independently of the data distribution thanks to the central limit
theorem. The p-value ($4.195e^{-17}$) we got is less than the chosen
significance level (0.05), therefore we reject the null hypothesis in support
of the alternative.

(A)
---

(B)
---

Figure 6: Density distribution of domain name’s age by class. The resulting
distributions show a very clear negatively skew (fake news) versus a
positively skew (real news). Age of domain can be extracted from Wayback or
WHOIS external services. Based on the analysis, we observed that fake news or
misleading content is published or shared more likely by newer websites.

###  3.2 Performance metrics

In terms of model performance measurement, the decision made by the classifier
can be represented as a $2\times 2$ confusion matrix having the following four
categories:

  * •

True Positives (TP), i.e., number of positive instances that are correctly
classified;

  * •

False Positives (FP), i.e., number of misclassified positive instances;

  * •

True Negatives (TN), i.e., number of negative instances that are correctly
classified;

  * •

False Negatives (FN), i.e., number of misclassified negative instances.

To evaluate the effectiveness of models, we used the metrics shown in Table 4
[40]:

Metric | Formula | Description
---|---|---
Precision | $\frac{TP}{TF+FP}$ | $\frac{\textrm{number of correctly classified positive instances}}{\textrm{total number of classified instances as positive}}$
Accuracy | $\frac{TP+TN}{TP+FP+TN+FN}$ | $\frac{\textrm{number of correctly classified instances}}{\textrm{total number of classified instances}}$
Recall | $\frac{TP}{TP+FN}$ | $\frac{\textrm{number of correctly classified positive instances}}{\textrm{actual number of positive instances}}$
$F_{1}$-Score | $2\times\frac{Precision\times Recall}{Precision+Recall}$ | harmonic mean of precision and recall
Table 4:  Evaluation metrics based on confusion matrix. Both $F_{1}$ score and
Recall are good metrics for the evaluation of imbalanced data.

Both $F_{1}$ score and Recall are good metrics for the evaluation of
imbalanced data.

###  3.3 Model Evaluation

Since we are dealing with imbalanced data, the predictive, accuracy represents
a misleading indicator, as it reflects the underlying class distributions,
making it difficult for a classifier to perform well on the minority class
[24]. For this reason, we used $F_{1}$ score [28; 41] and Recall metrics, as
the higher the value assumed by these metrics, the better the class of
interest is classified.
Table 5 shows the evaluation metrics for all the classifiers we considered. It
can be noticed that the classification metrics depend on the type of
classifier and on the extracted features used for the classification. Logistic
Regression with BoW model was the most effective classifier when we
oversampled the data, reaching the highest $F_{1}$-score (71%), followed by
Naïve Bayes with BoW model (70%), and SVM with TF-IDF (69%).
When we used the under-sampling technique and removed instances from the
majority class, the score of the classifier models was very poor compared to
over-sampling technique. SGD with TF-IDF and Naïve Bayes with TF-IDF and BoW
came out the worst with $F_{1}$ scores of 34%, 35%, and 37%, respectively.
From Table 5, only Random Forest classifier got a $F_{1}$-score greater than
50%, unlike the other classifiers when the under-sampling algorithm was
applied, though the Precision metric results very poor.

Re-Sampling | Classifier | Pre-Processing | Precision | Accuracy | Recall |  $F_{1}$-Score
---|---|---|---|---|---|---
| Naïve Bayes | BoW | 0.64 | 0.92 | 0.78 | 0.70
| Logistic Regression | BoW | 0.73 | 0.93 | 0.68 | 0.71
| SVM | BoW | 0.73 | 0.93 | 0.63 | 0.68
| SGD | BoW | 0.70 | 0.92 | 0.62 | 0.66
Over | Random Forest | BoW | 0.82 | 0.92 | 0.42 | 0.56
| Naïve Bayes | TF-IDF | 0.38 | 0.82 | 0.88 | 0.53
| Logistic Regression | TF-IDF | 0.68 | 0.92 | 0.67 | 0.67
| SVM | TF-IDF | 0.79 | 0.94 | 0.60 | 0.69
| SGD | TF-IDF | 0.60 | 0.91 | 0.64 | 0.62
| Random Forest | TF-IDF | 0.76 | 0.93 | 0.58 | 0.66
| Naïve Bayes | BoW | 0.23 | 0.64 | 0.91 | 0.37
| Logistic Regression | BoW | 0.38 | 0.83 | 0.79 | 0.51
| SVM | BoW | 0.34 | 0.80 | 0.79 | 0.47
| SGD | BoW | 0.26 | 0.71 | 0.83 | 0.39
Under | Random Forest | BoW | 0.46 | 0.87 | 0.74 | 0.57
| Naïve Bayes | TF-IDF | 0.22 | 0.62 | 0.89 | 0.35
| Logistic Regression | TF-IDF | 0.34 | 0.79 | 0.84 | 0.48
| SVM | TF-IDF | 0.30 | 0.75 | 0.85 | 0.44
| SGD | TF-IDF | 0.22 | 0.64 | 0.83 | 0.34
| Random Forest | TF-IDF | 0.48 | 0.88 | 0.68 | 0.57
Table 5:  Evaluation metrics (Precision, Accuracy, Recall, and $F_{1}$ score)
for all the classifiers, considering only textual features selection. Bold
values represent the highest performance results based on $F_{1}$ score. We
used different metrics for evaluating ML classifiers with BoW and with TF-IDF.
As shown in this table, Logistic Regression classifier performed better with
$F_{1}$ score $0.71$ compared to other classifiers using BoW in case of over-
sampled data. Similarly, SVM classifier performed better using TF-IDF in case
of over-sampled data. Due to the small size of the dataset, under-sampling
technique was not well-suited for the classifiers, as the results showed that
the classifiers performed very poorly.

Figure 7 shows a comparison of the classifiers using different feature
extraction techniques (BoW and TF-IDF) based on $F_{1}$-score metric (Table
5).

(A)
---

(B)
---

Figure 7: Comparison of classification $F_{1}$ score metric for the models
obtained by the two feature extraction methods (BoW and TF-IDF) for under-
sampled (A) and over-sampled (B) data. $F_{1}$ score is used in the case of
imbalanced datasets.

Based on the analysis we performed on Section 3.1, we observed a positive
influence on the $F_{1}$-score and Recall metrics (Figure 8) in some ML
classifiers, after including the most relevant features extracted from URLs.
As shown in Table 6, the implementation of new features extracted from URLs
successfully assisted the classifiers, by improving their performance.

Re-Sampling | Classifier | Pre-Processing | Precision | Accuracy | Recall |  $F_{1}$-Score
---|---|---|---|---|---|---
| Naïve Bayes | BoW | 0.76 | 0.96 | 0.86 | 0.81
| Logistic Regression | BoW | 0.77 | 0.94 | 0.67 | 0.72
| SVM | BoW | 0.81 | 0.94 | 0.64 | 0.71
| SGD | BoW | 0.79 | 0.94 | 0.64 | 0.71
Over | Random Forest | BoW | 0.90 | 0.94 | 0.50 | 0.64
| Naïve Bayes | TF-IDF | 0.34 | 0.79 | 0.97 | 0.50
| Logistic Regression | TF-IDF | 0.72 | 0.94 | 0.78 | 0.75
| SVM | TF-IDF | 0.80 | 0.95 | 0.78 | 0.79
| SGD | TF-IDF | 0.82 | 0.95 | 0.71 | 0.76
| Random Forest | TF-IDF | 0.80 | 0.94 | 0.65 | 0.72
Table 6:  Evaluation metrics (Precision, Accuracy, Recall, and $F_{1}$ score)
for all the classifiers after URL features selection. Bold values represent
the highest performance results based on $F_{1}$-score. As shown in this
table, Naïve Bayes classifier performed better with $F_{1}$ score $0.81$
compared to other classifiers using BoW in case of over-sampled data.
Similarly, SVM classifier performed better using TF-IDF with $F_{1}$ score
$0.79$. It is evident that adding the URL features improved the model’s
performance.

A visual inspection of metrics by model, before and after adding URL features
in our ML classifiers, is illustrated in Figure 8. The results verify the
effectiveness of introducing URL features, with values approximately above
0.70 for the two types of pre-processing. Before URL features selection, the
highest $F_{1}$-score was 0.71.

Figure 8: Comparison of metrics (precision, accuracy, recall, and
$F_{1}$-score) by model, before and after adding URL features in our ML
classifiers. Overall, the results verify the effectiveness of introducing URL
features. This is more evident in the classifiers that use TF-IDF.

##  4 Discussion

In binary classification problems, class imbalance represents an open
challenge as real-word datasets are usually skewed. One issue involves the
determination of the most suitable metrics for evaluating model performance.
$F_{1}$ score, defined as the harmonic mean of Precision and Recall (Section
3.2), has been commonly used to measure the level of imbalance. Our data had a
significantly high level of imbalance (majority class, i.e., real news, was
approximately 90% of our dataset, and minority class, i.e., fake news,
represented only 10% of the dataset). A way to address and mitigate class
imbalance problem was data re-sampling, which consists of either over-sampling
or under-sampling the dataset. Over-sampling the dataset is based on
rebalancing distributions by supplementing artificially generated instances of
the minor class (i.e., fake news). On the other hand, under-sampling method is
based on rebalancing distributions by removing instances of the majority class
(i.e., real news). By under-sampling the majority class, we had to reduce the
sample size, which resulted too small for training models, causing poor
performance. By over-sampling data, we instead noticed better results in terms
of both Recall and $F_{1}$ score metrics, boosting up the model performance.
We compared models based on popular feature representations, such as BoW and
TF-IDF. After over-sampling data, the evaluation metrics returned results with
$F_{1}$-score over 70% for both Logistic Regression and Naïve Bayes
classifiers with BoW.
In order to further improve the results, we decided to focus on news sources
as well, exploring and selecting URL features that have displayed high impact
in various studies [42; 43; 27].
In fact, just like phishing attacks (e.g., suspicious e-mails or malicious
links), fake news continues to be a top concern, as they still spread across
the Web and will continue to spread until everyone understands how to spot
them.
A comparison between phishing websites and websites that deliberately have
published fake news is shown in Table 7. It is evident that websites that
publish and share misleading content have generally URLs with identifiable
features (Table 2), like malicious URLs.

Type | Not-reliable website | Reliable website | Registration Date - NR | Registration Date - R
---|---|---|---|---
phishing | paypal–accounts.com | paypal.com | NA | 1999
phishing | fr.facebok.com | facebook.com | NA | 1997
fake news | cnn-trending.com | cnn.com | 2017 | 1993
fake news | fox-news24.com | foxnews.com | 2018 | 1995
fake news | abcnews.com.co | abcnews.com | 2016 | 1995
fake news | ilfattoquotidaino.it | ilfattoquotidiano.it | 2016 | 2009
Table 7:  Comparison of not-reliable (NR) and reliable (R) websites. Alongside
lexical features, domain name’s registration date (extracted from WHOIS) can
be relevant to spot fake websites.

As shown in Table 7, phishing carries out also by typosquatting domain, i.e.,
by registering a domain name that is extremely similar to that of an existing
popular one. In the past few years, various online websites have been created
to imitate trustworthy websites in order to publish misleading and fake
content: for example, abcnews.com (registered on 1995) and abcnews.com.co
(registered ahead of the 2016 US election); or ilfattoquotidiano.it
(registered on 2009) and ilfattoquotidaino.it (registered on 2016).
One of the most relevant URL features was certainly registration date. In our
dataset, the average age of domain name of websites publishing fake news was
2008, while that one of websites publishing real news was 2004 (Table 3). Most
of websites publishing fake news are, therefore, newer than websites which
spread reliable news. This was in line with our expectation, i.e., that
websites publishing reliable news are typically older, having more time to
build reputation, while those ones that publish fake news and misleading
content are likely unknown websites created more recently.
The effects on the other features extracted from URLs had also a positive
impact on the detection problem. By using correlation matrix heatmap and
looking at findings from other research works, we selected features that most
affected the target variable. Like in phishing, websites or blogs that publish
and share fake news may contain special symbols (such as @ and &) to obfuscate
links and trick readers into thinking that the URL leads to a legitimate
website. For example, abcnews.com.co is a fake website, where the use of dots
is for adding an extension (i.e., .co). On the other hand, the proportion of
http and https did not provide relevant information, as https secured protocol
now is commonly used. News by TLD showed that the most popular TLDs are .com
(85% in fake news dataset; 73.3% in real news dataset) and .org. (8.4% in fake
news dataset; 15% in real news dataset) (Table 2). Furthermore, large numbers
of digits and hyphens (greater than 7 on average) were found within URLs in
the fake news dataset, making it possible to compare with results obtained
from the analysis of malicious URLs [44; 45].
By entering the selected URL features the model, Naïve Bayes classifier with
BoW achieved $F_{1}$ score of 81%, while SVM with TF-IDF got 79%,
significantly exceeded results based on features built from lexical
representations of the text (titles and descriptions) only.

Based on the achieved result, we concluded that the use of URL features
increased the performance of models.
In terms of challenges, the class imbalance of real-world data and the limited
accessibility of high-quality labelled dataset are two of the major ones. The
use of ML classification models in fake news detection still appears more
challenging in realistic situations, especially on Web Search Engines, where
metadata information from thousands websites are collected.

Date | URL | Title | Description
---|---|---|---
2020-01-27 | prepareforchange.net |  The “Event” is the moment of the “Compression Breakthrough” on earth |  It is a cosmic EVENT HORIZON created by big solar waves reaching the Earth from the Galactic Central causing the activation of “The Compression …
2020-03-04 | www.drugtodayonline.com |  Sun exposure, washing hands kills corona virus: Unicef |  The Unicef has said that corona virus is large in size where the cell diameter is 400-500 micro and for this reason any mask prevents its entry. The virus does not settle in the air but is grounded, so it is not transmitted by air.
2020-03-20 | politicalfilm.wordpress.com |  Event 201: October 18, 2019 | Political Film Blog |  John Hopkins / Bill Gates Foundation ran a CORONAVIRUS event in October. How prescient???They claim it originated in pigs, contradicting the wild animal narrative that currently dominates. The bioweapons question is nowhere to be seen.
2020-03-22 | www.theepochtimes.com |  21 Million Fewer Cellphone Users in China May Suggest a High CCP Virus Death Toll |  The number of Chinese cellphone users dropped by 21 million in the past three months, Beijing authorities announced on March 19.
2020-03-28 | beforeitsnews.com |  Coronavirus: Follow The Money… and the Players! |  Coronavirus: Follow the Money … and the Players! – The problem with the Deep State Shadow Government – America, contracted out!
2020-04-04 | productivityhub.org |  BILL GATES CALLS FOR A “DIGITAL CERTIFICATE” TO IDENTIFY WHO RECEIVED COVID-19 VACCINE |  In October 2019 (only a few months before the apparition of COVID-19) the Bill and Melinda Gates Foundation (in cooperation with the World Economic Forum) hosted Event 201, a 3.5-hour table-top simulation of a global pandemic.
Table 8:  Example of fake news in our dataset. CoViD-19 pandemic has been
resulted in misleading information, false claims and pseudo-scientific
therapies, regarding the diagnosis, treatment, prevention, origin and spread
of the virus. Conspiracy theories have been widespread too, mostly online
through blogs, forums and social media.

Furthermore, as in phishing attacks, who writes fake news and misleading
content constantly looks for new and creative ways to fool users into
believing their stories involve a trustworthy source. This makes necessary to
keep models continuously updated as fake news is becoming more and more
sophisticated and difficult to spot. Also, misleading contents vary greatly
and change over time: therefore, it is essential to investigate new features.

##  5 Conclusion

In this study, we analysed meta data information extracted from Web Search
Engines, after submitting specific search queries related to the CoViD-19
outbreak, simulating a normal user’s activity. By using both textual and URLs
properties of data, we trained different Machine Learning algorithms with pre-
processing methods, such as Bag-of-Words and TF-IDF. In order to deal with
class imbalance due to real-world data, we applied re-sampling techniques,
i.e., over-sampling of fake news and under-sampling of real news. While over-
sampling technique allowed us to get satisfactory results, the under-sampling
method was not able to increase model performance, showing very poor results
due to the small sample size.
Although news has some specific textual properties which can be used for its
classification as fake or real, when we look at search results (titles,
snippets, and links), some additional pre-processing can be used to obtain
some specific extra features for fake news detection on WSEs. While text
features are related to news content, gathered from both titles and snippets,
URL features are based on the source websites returned as search results on
WSEs.
While most previous studies focused on fake news detection in Social Media,
relying on data which can be directly gathered from the text (e.g., tweets)
and from the usage of URLs for improving source credibility, our proposed
approach goes further and analyse URL-features of the source of information
itself.
We believe indeed that URL pattern analysis via phishing detection techniques
can enhance ML algorithms ability to detect and mitigate the spread of fake
news across the World Wide Web. Checking the source is, indeed, one of the
most common advice that fact-checking websites give to online readers [46].
The results from this study suggest that information on URLs, extracted by
using phishing techniques (e.g., number of digits, dots and length of the
URL), could provide indications to researchers regarding a number of
potentially useful features that future fake news detection algorithms might
have or develop in order to bring out further valuable information on websites
containing mostly false content and improve the model performance.
The analysis of fake news which spreads on the Web might have, however, a
potential limitation, due to Search Engine optimisation. In this study we
proposed a possible solution to address it. In fact, although Search Engine
results might be customised based on online user location and user’s search
history, in order to reduce bias due to prior searching on the WSEs, it would
be helpful to change settings preferences, delete cache, cookies, search
history or use Incognito/Private windows. Furthermore, the use of proxies (or
VPN) could allow to search queries on WSEs being location independent.
In terms of future research on fake news detection, we believe that techniques
commonly used for malicious URLs detection should also be considered for fake
news detection: this would mean building classifiers based not only on
traditional lexical and semantic features of texts, but also on lexical and
host-based features of the URL.
As future work, we therefore plan to construct more discriminative features to
detect fake content, by profiling malicious sources of information based on
domains, investigating in more detail, with additional performance metrics
such as Net Reclassification Index (NRI), the improvement in prediction
performance gained by adding a marker to the set of baseline predictors, in
order to facilitate designing even better classification models for fake news
detection.

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# Group of isometries of Hilbert ball equipped with the
Carath$\mathbf{\acute{E}}$odory metric

Mukund Madhav Mishra and Rachna Aggarwal  Department of Mathematics, Hansraj
College, University of Delhi, Delhi, India<EMAIL_ADDRESS>Department
of Mathematics, University of Delhi, Delhi, India<EMAIL_ADDRESS>
(Date: May 2021)

###### Abstract.

In this article, we study the geometry of an infinite dimensional Hyperbolic
space. We will consider the group of isometries of the Hilbert ball equipped
with the Carath$\acute{e}$odory metric and learn about some special subclasses
of this group. We will also find some unitary equivalence condition and
compute some cardinalities.

###### Key words and phrases:

Hyperbolic space ; isometry group ; Carathéodory metric ; dynamical types

###### 2020 Mathematics Subject Classification:

51M10; 51F25

##  1\. Introduction

Groups of isometries of finite dimensional hyperbolic spaces have been studied
by a number of mathematicians; to name a few, Anderson [1], Chen and Greenberg
[5], and Parker [16]. Hyperbolic spaces can largely be classified into four
classes: real, complex, quaternionic hyperbolic spaces and octonionic
hyperbolic plane. The respective groups of isometries are $PO(n,1),\,PU(n,1),$
$PSp(n,1)$ and $F_{4(-20)}$. Real and complex cases are standard and have been
discussed at various places. For example Anderson [1] and Parker [16].
Quaternionic spaces have been studied by Cao and Parker [4] and Kim and Parker
[11]. For octonionic hyperbolic plane, one may refer to Baez [2] and Markham
and Parker [15]. The basic hyperbolic plane model is half plane model usually
given by $H^{2}$ and it is well connected with the famous Poincar$\acute{e}$
disc model in two dimensions [18]. The distance in the Poincar$\acute{e}$ disc
model is given by the Poincar$\acute{e}$ metric and one of the crucial
properties of this metric is that the holomorphic self maps on the
Poincar$\acute{e}$ disc satisfy Schwarz-Pick lemma, see [13, Sec.2.3.5,
Theorem 2.3.22]. Now in an attempt to generalize this lemma to higher
dimensions, Carath$\acute{e}$odory and Kobayashi metrics were discovered which
formed one of the ways to discuss hyperbolic structure on domains in
$\mathbb{C}^{n}$. It is therefore natural to look for an infinite dimensional
counterpart of the finite dimensional hyperbolic spaces. In late 20th century,
Franzoni and Vesentini studied infinite dimensional Hilbert ball equipped with
the Carathéodory metric. A description of the group of isometries of the
hyperbolic ball has been given in [7]. Motivational sources behind studies
carried out in this article are [3], [8] and [14].

In this article, we intend to study dynamical aspects of the group of
isometries of infinite dimensional hyperbolic ball. To this end, we consider
this group, focus on some special subclasses of this group and explore their
properties. Sectional detail is as follows. Section 1 is a brief literature
survey. Section 2 gives the description of group of holomorphic isometries and
its linear representation. In section 3, we will study a class of isometries
having a two dimensional reducing subspace, class of normal isometries, self
adjoint isometries and involutory isometries. In section 4, we will discuss
the condition on an isometry to be unitarily equivalent to its inverse and
compute cardinality of the group of isometries, class of self adjoint elements
and set of unitary equivalence classes of normal operators on
$H\oplus\mathbb{C}$.

##  2\. The group of isometries

We begin with the notion of holomorphicity in infinite dimensional set up,
give a brief introduction of the Carath$\acute{e}$odory metric and describe
the form of corresponding isometries on the infinite dimensional hyperbolic
space which has been taken from [7].

We denote the space of bounded linear operators from a Banach space $V$ to a
Banach space $W$ by $B(V,W)$ and space of bounded linear operators on a
Hilbert space $H$ by $B(H)$. Let $V$ and $W$ be two complex Banach spaces. Let
$U$ be an open subset of $V$. A mapping $f:U\longrightarrow W$ is called a
holomorphic function if it is Fr$\acute{e}$chet differentiable on $U$, i.e.
for every $a\in U$, there exists $A\in B(V,W)$ such that
$\lim\limits_{x\rightarrow a}\dfrac{f(x)-f(a)-A(x-a)}{\|x-a\|}=0$ [9, Page 68,
Definition 147]. Let $H$ be an infinite dimensional complex Hilbert space and
B be the open unit ball in $H$ equipped with the Carath$\acute{e}$odory metric
defined as follows. Let $D$ be a domain in a complex normed space $E$ and
$\Delta$ denote the open unit ball in $\mathbb{C}$ equipped with the
Poincar$\acute{e}$ metric $\rho$. Let Hol$(D,\Delta)$ denote the set of all
holomorphic mappings $f:D\rightarrow\Delta$. Then

$C_{D}(x,y)=\sup_{f}{\rho(f(x),f(y))}\,\,\,\,\,\,\text{for\,\,all}\,\,x,\,y\,\in
D$

defines the Carath$\acute{e}$odory pseudo-distance on $D$. $C_{D}$ becomes a
complete metric on every bounded homogeneous domain in a complex Banach space.
This metric is considered a generalization of $\rho$ as it coincides with the
latter on $\Delta$. For a detailed study of this metric, refer [7], [12]. A
holomorphic map on B is distance decreasing for the Carath$\acute{e}$odory
metric. So, every bi-holomorphic surjection is an isometry for this metric.
Let $Aut(B)$ denote the group of all bi-holomorphic surjections on B. This is
the group of isometries we will be working with. A general element of $Aut(B)$
is of the form $F=U\circ f_{x_{0}}$, $x_{0}\in B$, (see [7, Theorem VI.1.3])
where $U$ is a unitary operator on $H$ and $f_{x_{0}}:B_{x_{0}}\longrightarrow
H$ defined by
$f_{x_{0}}(x)=T_{x_{0}}\left(\dfrac{x-{x_{0}}}{1-\big{<}x,x_{0}\big{>}}\right)$
is a holomorphic map, $B_{x_{0}}=\left\\{x\in
H\,\,:\,\,\|x\|<\dfrac{1}{\|x_{0}\|}\right\\}$. $f_{x_{0}}$ restricted to B is
a bi-holomorphic surjection. $T_{x_{0}}:H\longrightarrow H$ is a linear map
expressed as
$T_{x_{0}}(x)=\dfrac{\left<x,x_{0}\right>}{1+\sqrt{1-\|x_{0}\|^{2}}}x_{0}+\sqrt{1-\|x_{0}\|^{2}}\,\,x.$
$Aut(B)$ is known to act transitively on B.

A linear representation of the elements of $Aut(B)$ is as follows. Let
$\mathcal{A}$ be the sesquilinear form on $H\oplus\mathbb{C}$ defined by
$\mathcal{A}((x,\lambda),\,(y,\mu))=\left<x,y\right>-\lambda\overline{\mu}$.
Consider the collection of all linear operators on $H\oplus\mathbb{C}$ leaving
$\mathcal{A}$ invariant. Such linear operators are bounded and injective by
definition. Let $G$ be the group consisting of all the bijective linear
operators on $H\oplus\mathbb{C}$ leaving $\mathcal{A}$ invariant. General form
of an element of $G$ is $T=\left[{\begin{array}[]{cc}A&\xi\\\
\left<\cdot,\dfrac{A^{*}(\xi)}{a}\right>&a\\\ \end{array}}\right]$ where $A\in
B(H),\,\,\xi\in H$ and $a\in\mathbb{C}$ satisfying

$\displaystyle
A^{*}A=I+\dfrac{1}{|a|^{2}}\left<\cdot,{A^{*}(\xi)}\right>A^{*}(\xi)$

and

$\displaystyle|a|^{2}=1+\|\xi\|^{2}.$

The center $Z(G)$ of $G$ is $\\{e^{i\theta}I,\,\,\,\,\theta\in\mathbb{R}\\}$,
see [7, Lemma VI.3.4].

Let $\sigma(T)$ denote the spectrum of a bounded linear operator $T$.

######  Theorem 2.1 ([7] Theorem VI.3.5).

The map $\phi:G\rightarrow\text{Aut(B)}$ defined by $\phi(T)=\widetilde{T}$ is
an onto homomorphism where $T=\left[{\begin{array}[]{cc}A&\xi\\\
\left<\cdot,\dfrac{A^{*}(\xi)}{a}\right>&a\\\ \end{array}}\right]$ and
$\widetilde{T}=\dfrac{A(\cdot)+\xi}{\left<\cdot,\frac{A^{*}(\xi)}{a}\right>+a}$.

###### Proof.

Well definedness and homomorphic nature of $\phi$ can easily be verified. We
will show that $\phi$ is an onto map. For $x_{0}\in B$, let $U\circ
f_{{x_{0}}}$ be a general element of $Aut(B)$. We will find preimages of
$f_{{x_{0}}}$ and $U$ separately. For a unitary element $U\in Aut(B)$,
$V=\left[{\begin{array}[]{cc}U&0\\\ 0&1\\\ \end{array}}\right]$ will work. Let
$x_{0}\in B\setminus\\{0\\}$. To find suitable $A,\,\xi$ and $a$ such that
$\dfrac{A(x)+\xi}{\left<x,\frac{A^{*}(\xi)}{a}\right>+a}=f_{{x_{0}}}(x)$ for
all $x\in B$. Suppose such $A,\xi$ and $a$ exist. This gives
$\widetilde{T}(0)=\dfrac{\xi}{a}=f_{{x_{0}}}(0)=-x_{0}$. Choose
$a=\sqrt{1+\|\xi\|^{2}}$. Then $\|\xi\|^{2}=(1+\|\xi\|^{2})\|x_{0}\|^{2}$ and
$\|\xi\|=\dfrac{\|x_{0}\|}{\sqrt{1-\|x_{0}\|^{2}}}$. Hence, choose
$\xi=-ax_{0}$. Now, we will choose a suitable self adjoint operator $A$.
Construct an operator $S\in B(H)$ such that $S(\xi)=a^{2}\xi$ and $S=I$ on
$\left<\xi\right>^{\perp}$. Then $S=I+\left<\cdot,\xi\right>\xi$. For,
$x=k\xi+y\in H$, for some $k\in\mathbb{C}$ and $y\in\left<\xi\right>^{\perp}$,
$S(x)=a^{2}k\xi+y+k\xi-k\xi=x+(1+\|\xi\|^{2}-1)k\xi=x+\|\xi\|^{2}k\xi+\left<y,\xi\right>\xi=x+\left<k\xi,\xi\right>\xi+\left<y,\xi\right>\xi=x+\left<x,\xi\right>\xi$.
Also, $S$ is a positive operator such that $\sigma(S)=\\{a^{2},\,1\\}$ making
$S$ invertible. Choose $A$ to be the positive square root of $S$. Then
$A(\xi)=a\xi$ and $A=I$ on $\left<\xi\right>^{\perp}$. It now follows that
with $a$, $\xi$ and $A$ as chosen above,
$T_{1}=\left[{\begin{array}[]{cc}A&\xi\\\ \left<\cdot,\xi\right>&a\\\
\end{array}}\right]\in G$ satisfies $\widetilde{T_{1}}(x)=f_{{x_{0}}}(x)$.
Finally, concluding the detailed working, we have for every element $U\circ
f_{x_{0}}\in Aut(B)$ there exists an isometry of the form $V\circ T_{1}\in G$
such that $\phi(V\circ T_{1})=\widetilde{V\circ T_{1}}=U\circ f_{{x_{0}}}$. ∎∎

######  Remark 1.

Since, $\mathrm{ker}(\phi)=Z(G)$, the map
$\widetilde{\phi}:G/{Z(G)}\rightarrow Aut(B)$ is an onto isomorphism.

##  3\. Some special subclasses of $G$

We begin by describing a simplified form of elements of $G$.

######  Proposition 3.1.

A general element of $G$ is of the form
$e^{i\theta}\left[{\begin{array}[]{cc}UA&U(\xi)\\\ \left<\cdot,\xi\right>&a\\\
\end{array}}\right],\,\,\,\theta\in\mathbb{R}$, where $\xi\in H$,
$a=\sqrt{1+\|\xi\|^{2}}$, $U$ is a unitary operator on $H$ and A is a positive
operator on $H$ such that $A=I$ on $\left<\xi\right>^{\perp}$ and
$A(\xi)=a\xi$.

###### Proof.

From the proof of Theorem 1, we see that that for every element $U\circ
f_{x_{0}}\in Aut(B)$, there exists an isometry of the form $V\circ T_{1}$
where $V$ is given by $\left[{\begin{array}[]{cc}U&0\\\ 0&1\\\
\end{array}}\right]$ and $T_{1}$ is given by
$\left[{\begin{array}[]{cc}A&\xi\\\ \left<\cdot,\xi\right>&a\\\
\end{array}}\right]$. So, $V\circ T_{1}=\left[{\begin{array}[]{cc}UA&U(\xi)\\\
\left<\cdot,\xi\right>&a\\\ \end{array}}\right]$ satisfying the desired
properties and $\phi(V\circ T_{1})=U\circ f_{x_{0}}$. Also, Remark 1 tells
that the pre image of a general element $U\circ f_{x_{0}}$ of $Aut(B)$ is the
set $\left\\{e^{i\theta}\left[{\begin{array}[]{cc}UA&U(\xi)\\\
\left<\cdot,\xi\right>&a\\\
\end{array}}\right],\,\,\,\theta\in\mathbb{R}\right\\}$ and hence every
element of $G$ is of the form
$e^{i\theta}\left[{\begin{array}[]{cc}UA&U(\xi)\\\ \left<\cdot,\xi\right>&a\\\
\end{array}}\right]$. ∎∎

From now onwards, $T\in G$ will denote an element of the form
$e^{i\theta}\left[{\begin{array}[]{cc}UA&U(\xi)\\\ \left<\cdot,\xi\right>&a\\\
\end{array}}\right].$
The sesquilinear form $\mathcal{A}$ defined above is identified with the
linear operator $A^{\prime}=\left[{\begin{array}[]{cc}I&0\\\ 0&-1\\\
\end{array}}\right]$, i.e.
$\mathcal{A}((x,\lambda),(y,\mu))=\left<A^{\prime}(x,\lambda),(y,\mu)\right>$
for all $(x,\lambda),\,(y,\mu)\in H\oplus\mathbb{C}$. This gives
$T^{*}=e^{-i\theta}\left[{\begin{array}[]{cc}(UA)^{*}&\xi\\\
\left<\cdot,U(\xi)\right>&\overline{a}\\\ \end{array}}\right]$. Also, for $T$
of the above form,
$T^{-1}=e^{-i\theta}\left[{\begin{array}[]{cc}(UA)^{*}&-\xi\\\
-\left<\cdot,U(\xi)\right>&\overline{a}\\\ \end{array}}\right]$. Notice that
$\overline{a}=a$ as $a\in\mathbb{R}$.
Also observe that in Theorem 1, the operator $V$ is unitary, i.e.
$V^{*}=V^{-1}$ and the operator $T_{1}$ is self adjoint, i.e.
${T_{1}}^{*}=T_{1}$. Hence every element of $G$ is a composition of a unitary
and a self adjoint element.

The following theorem by Hayden and Suffridge gives the fixed point
classification of isometries in $Aut(B)$.

Let $\partial{B}$ denote the boundary of the open unit ball $B$ and
$\overline{B}=B\cup\partial{B}$.

######  Theorem 3.2 ( [10]).

If $g\in\text{Aut(B)}$ has no fixed point in B, then the fixed point set in
$\overline{\text{B}}$ consists of one or two points.

We call an isometry in $Aut(B)$ elliptic if has a fixed point in B, hyperbolic
or parabolic if it is not elliptic and has two or one fixed points on
$\partial{B}$ respectively.

We say an element of $G$ is elliptic (resp. hyperbolic or parabolic) if it is
in the pre image of an elliptic (resp. hyperbolic or parabolic) isometry of
$Aut(B)$ defined by the homomorphism $\phi$.

Observe that for $x\in\overline{B}$, $x$ is a fixed point for an isometry
$\dfrac{UA+U(\xi)}{\left<\cdot,\xi\right>+a}\in Aut(B)$ if and only if $(x,1)$
is an eigenvector for the corresponding element in $G$ having eigenvalue
$\left<x,\xi\right>+a$.

Here from, we will assume $\xi\neq 0$, unless stated otherwise.

We will now investigate a subclass of isometries of $G$ which decomposes
$H\oplus\mathbb{C}$ into a two dimensional subspace containing $\mathbb{C}$
and its orthogonal complement.

We know that a general element $T$ of $G$ is of the form
$e^{i\theta}\left[{\begin{array}[]{cc}UA&U(\xi)\\\ \left<\cdot,\xi\right>&a\\\
\end{array}}\right]$ where $A$ decomposes $H$ into $\left<\xi\right>$ and
$\left<\xi\right>^{\perp}$. If the unitary operator $U$ is chosen in a manner
so as to leave $\left<\xi\right>$ invariant, then all the isometries of $G$
comprising of such unitary operators are precisely the ones decomposing
$H\oplus\mathbb{C}$ into $\left<\xi\right>\oplus\mathbb{C}$ and its orthogonal
complement.

Thus we have the following.

######  Proposition 3.3.

Let $T=\left[{\begin{array}[]{cc}UA&U(\xi)\\\ \left<\cdot,\xi\right>&a\\\
\end{array}}\right]\in G$. Then $T=T_{1}\oplus T_{2}$ where
$T_{1}=T\restriction_{\left<\xi\right>\oplus\mathbb{C}}$ and
$T_{2}=U\restriction_{\left<\xi\right>^{\perp}}$ if and only if $U(\xi)=r\xi$,
$|r|=1$.

###### Proof.

Write
$H\oplus\mathbb{C}=\left<\xi\right>\oplus\left<\xi\right>^{\perp}\oplus\mathbb{C}$.
Let $U(\xi)=r\xi$, $|r|=1$. Then for any element
$(\xi,z)\in\left<\xi\right>\oplus\mathbb{C}$,
$\left[{\begin{array}[]{cc}UA&U(\xi)\\\ \left<\cdot,\xi\right>&a\\\
\end{array}}\right]\left[{\begin{array}[]{c}\xi\\\ z\\\
\end{array}}\right]=\left[{\begin{array}[]{c}r(a+z)\xi\\\ \|\xi\|^{2}+az\\\
\end{array}}\right]\in\left<\xi\right>\oplus\mathbb{C}$. Also for
$x\in\left<\xi\right>^{\perp}$, $\left[{\begin{array}[]{cc}UA&U(\xi)\\\
\left<\cdot,\xi\right>&a\\\ \end{array}}\right]\left[{\begin{array}[]{c}x\\\
0\\\ \end{array}}\right]=\left[{\begin{array}[]{c}U(x)\\\ 0\\\
\end{array}}\right]\in\left(\left<\xi\right>\oplus\mathbb{C}\right)^{\perp}$.
Thereby showing $\left<\xi\right>\oplus\mathbb{C}$ is a reducing subspace for
$T$ and
$T\restriction_{({\left<\xi\right>\oplus\mathbb{C}})^{\perp}}=U\restriction_{\left<\xi\right>^{\perp}}$.
Conversely, if $T=T_{1}\oplus T_{2}$, then invariance of
$\left<\xi\right>\oplus\mathbb{C}$ under $T$ yields invariance of
$\left<\xi\right>$ under $U$. ∎

######  Proposition 3.4.

Let $T=\left[{\begin{array}[]{cc}UA&r\xi\\\ \left<\cdot,\xi\right>&a\\\
\end{array}}\right]\in G$, $|r|=1$. Then

  1. (1)

$\sigma(T)=\\{\lambda_{1},\,\lambda_{2}\\}\cup\sigma(U\restriction_{\left<\xi\right>^{\perp}})$
where $\lambda_{1},\,\lambda_{2}=\dfrac{a(r+1)\pm\sqrt{a^{2}(r+1)^{2}-4r}}{2}$
respectively. The eigenspaces corresponding to the eigenvalues $\lambda_{1}$
and $\lambda_{2}$ are generated by the eigenvectors $\left(k_{1}\xi,1\right)$
and $(k_{2}\xi,1)$ where

$k_{1},\,k_{2}=\dfrac{a(r-1)\pm\sqrt{a^{2}(r+1)^{2}-4r}}{2\|\xi\|^{2}}$
respectively.

  2. (2)

$|\lambda_{1}|=\dfrac{1}{|\lambda_{2}|}$ and
$\|k_{1}\xi\|=\dfrac{1}{\|k_{2}\xi\|}$.

###### Proof.

(1) As $T=T_{1}\oplus T_{2}$ (from Proposition 2),
$\sigma(T)=\sigma(T_{1})\cup\sigma(T_{2})$. Clearly,
$\sigma(T_{2})=\sigma(U\restriction_{\left<\xi\right>^{\perp}})$. Also,
$T_{1}$ is an operator on a 2 dimensional space
$\left<\xi\right>\oplus\mathbb{C}$. So, $\lambda$ is an eigenvalue of $T_{1}$
with eigenvector $(k\xi,1)$ if $\left[{\begin{array}[]{cc}UA&r\xi\\\
\left<\cdot,\xi\right>&a\\\
\end{array}}\right]\left[{\begin{array}[]{c}k\xi\\\ 1\\\
\end{array}}\right]=\lambda\left[{\begin{array}[]{c}k\xi\\\ 1\\\
\end{array}}\right]$ which gives $r(ak+1)\xi=\lambda k\xi$ and
$\lambda=k\|\xi\|^{2}+a$. Simplification of these two expressions yield
$k^{2}\|\xi\|^{2}+a(1-r)k-r=0$. Hence
$k=\dfrac{a(r-1)\pm\sqrt{a^{2}(r+1)^{2}-4r}}{2\|\xi\|^{2}}$ (using
$a^{2}=1+\|\xi\|^{2}$) and
$\lambda=\dfrac{a(r+1)\pm\sqrt{a^{2}(r+1)^{2}-4r}}{2}$. Note that $(\xi,0)$
cannot be an eigenvector for $T_{1}$.

(2) Observe that $\lambda_{1}\lambda_{2}=r$ and
$k_{1}k_{2}=\dfrac{-r}{\|\xi\|^{2}}$. This gives
$|\lambda_{1}|=\dfrac{1}{|\lambda_{2}|}$ and
$\|k_{1}\xi\|=\dfrac{1}{\|k_{2}\xi\|}$ respectively. ∎∎

In general, we have

######  Proposition 3.5.

Let $T=\left[{\begin{array}[]{cc}UA&U(\xi)\\\ \left<\cdot,\xi\right>&a\\\
\end{array}}\right]\in G$ and $K$ be a closed subspace of $H$ containing $\xi$
such that $K$ is invariant under $U$. Then $T$ decomposes $H\oplus\mathbb{C}$
into $K\oplus\mathbb{C}$ and its orthogonal complement.

###### Proof.

Represent a general element $y\in K$ as $y=k\xi+x$, where $k\in\mathbb{C}$ and
$x\in\left<\xi\right>^{\perp}\cap K$. Let $(k\xi+x,z)\in K\oplus\mathbb{C}$,
$z\in\mathbb{C}$. Then $\left[{\begin{array}[]{cc}UA&U(\xi)\\\
\left<\cdot,\xi\right>&a\\\
\end{array}}\right]\left[{\begin{array}[]{c}k\xi+x\\\ z\\\
\end{array}}\right]=\left[{\begin{array}[]{c}U((ak+z)\xi+x)\\\
k\|\xi\|^{2}+az\\\ \end{array}}\right]\in K\oplus\mathbb{C}$ as $K$ is
invariant under $U$ thereby making $K\oplus\mathbb{C}$ invariant under $T$.
Also, for $y\in K^{\perp},\,\left[{\begin{array}[]{cc}UA&U(\xi)\\\
\left<\cdot,\xi\right>&a\\\ \end{array}}\right]\left[{\begin{array}[]{c}y\\\
0\\\ \end{array}}\right]=\left[{\begin{array}[]{c}U(y)\\\ 0\\\
\end{array}}\right]\in(K\oplus\mathbb{C})^{\perp}$. ∎∎

An operator $T\in{H}$ is said to be _normal_ if $T^{*}T=TT^{*}$. We are now
ready to describe the normal isometries of $G$. Class of normal isometries
forms a subpiece of the above defined subclass of $G$. In exact terms, if
$\left<\xi\right>\oplus\mathbb{C}$ reduces $T$, then $\left<\xi\right>$
reduces $U$. In particular, the class of all such isometries where $U$ acts as
identity on $\left<\xi\right>$ precisely forms the class of normal isometries
in $G$.

######  Theorem 3.6 (Normal elements of $G$).

Let $S\in G$ be given by $e^{i\theta}\left[{\begin{array}[]{cc}UA&U(\xi)\\\
\left<\cdot,\xi\right>&a\\\ \end{array}}\right]$, $\theta\in\mathbb{R}$. Then

  1. (1)

$S$ is normal if and only if $S=e^{i\theta}\left[{\begin{array}[]{cc}UA&\xi\\\
\left<\cdot,\xi\right>&a\\\ \end{array}}\right]$, i.e. $U(\xi)=\xi$.

  2. (2)

$S$ is unitary if and only if $S=e^{i\theta}\left[{\begin{array}[]{cc}U&0\\\
0&1\\\ \end{array}}\right]$, i.e. $\xi=0$.

  3. (3)

If $S$ is normal, then
$\sigma(S)=\\{a\pm\|\xi\|\\}\cup\sigma\left(U\big{|}_{\left<\xi\right>^{\perp}}\right)$,
where $a\pm\|\xi\|$ are both positive, non-unit modulus and inverses of each
other. Eigenspaces corresponding to the eigenvalues $a\pm\|\xi\|$ are spanned
by the eigenvectors $\left(\pm\dfrac{\xi}{\|\xi\|},1\right)$ respectively.

  4. (4)

Normal isometries are hyperbolic in nature.

###### Proof.

(1) For an element $S\in G$, direct computation shows that $SS^{*}=S^{*}S$ if
and only if $U(\xi)=\xi$.

(2) $S^{*}=S^{-1}$ if and only if $\xi=0$ is straightforward. Notice that if
$\xi=0$, then $A$ reduces to $I$ on $H$.

(3) follows from Proposition 3 for $r=1$.

(4) $S$ has two fixed points $\left\\{\pm\dfrac{\xi}{\|\xi\|}\right\\}$ lying
on the unit sphere. Also $S$ can’t be elliptic as it does not have an
eigenvector of the form $(x,1),\,\|x\|<1$, for eigenvectors of S either come
from $\left<\xi\right>\oplus\mathbb{C}$ or
$(\left<\xi\right>\oplus\mathbb{C})^{\perp}$ and $a\pm||\xi||\neq 1$. ∎∎

Next we explore the self adjoint elements of $G$. Prior to that, let us make a
simple observation. In the construction of $T$, $A=I$ on
$\left<\xi\right>^{\perp}$. This implies that every unitary operator $U$ which
decomposes $H$ into $\left<\xi\right>$ and $\left<\xi\right>^{\perp}$ commutes
with $A$.

######  Proposition 3.7 (Self adjoint elements of $G$).

Let $T$ be a self adjoint element in $G$. Then

  1. (1)

$T$ is of the form $\pm\left[{\begin{array}[]{cc}UA&\xi\\\
\left<\cdot,\xi\right>&a\\\ \end{array}}\right]$, where $U$ is a unitary and
involutory operator.

  2. (2)

$\sigma(T)=\\{a\pm\|\xi\|\\}\cup\\{\pm 1\\}$.

###### Proof.

(1) $T\in G$ is self adjoint if and only if $T=T^{*}$, i.e
$e^{i\theta}\left[{\begin{array}[]{cc}UA&U(\xi)\\\ \left<\cdot,\xi\right>&a\\\
\end{array}}\right]=e^{-i\theta}\left[{\begin{array}[]{cc}(UA)^{*}&\xi\\\
\left<\cdot,U(\xi)\right>&a\\\ \end{array}}\right]$ which gives $UA$ is self
adjoint, $U(\xi)=\xi$ and $\theta=n\pi$, $n\in\mathbb{Z}$. Now, $UA$ is self
adjoint, $A$ is self adjoint and $U$ commutes with $A$ (by the above
observation) gives $U$ is self adjoint. Hence $U$ is involutory.

(2) Clearly, $T=T_{1}\oplus
T_{2},\,\sigma(T_{2})=\sigma(U\restriction_{\left<\xi\right>^{\perp}})=\\{\pm
1\\}$. ∎∎

Let us now discuss the form of involutory elements of $G$. It is easy to see
that unitary and involutory elements of $G$ are of the form
$\left[{\begin{array}[]{cc}V&0\\\ 0&\pm 1\\\ \end{array}}\right]$ where $V$ is
unitary and involutory.

######  Proposition 3.8 (Involutory elements of $G$).

Let $T$ be an involutory element in $G$. Then

  1. (1)

$T=\pm\left[{\begin{array}[]{cc}UA&-\xi\\\ \left<\cdot,\xi\right>&a\\\
\end{array}}\right]$ where $U$ is involutory.

  2. (2)

$T=T_{1}\oplus T_{2}$ where
$T_{1}=T\restriction_{\left<\xi\right>\oplus\mathbb{C}}$ and
$T_{2}=U\restriction_{{\left<\xi\right>}^{\perp}}$.

  3. (3)

$\sigma(T)=\\{\pm 1\\}$, where $\sigma(T_{1})=\pm 1$ with eigenspace spanned
by
$\left(\dfrac{-a\pm 1}{\|\xi\|^{2}}\,\xi\,,1\right)$.

  4. (4)

Involutory elements are elliptic in nature.

###### Proof.

(1) $T$ is involutory if and only if $T=T^{-1}$. On comparing respective
entries of $T$ and $T^{-1}$, we get $UA$ is self adjoint, $U(\xi)=-\xi$ and
$\theta=n\pi$. Following the same argument as in part (1) of the previous
proposition, we get $U$ is involutory.

(2) and (3) follow from Proposition 3 for $r=-1$.

(4) Since $\dfrac{|-a+1|}{\|\xi\|}=\dfrac{a-1}{\|\xi\|}<1$, $T$ is elliptic in
nature. ∎∎

##  4\. Unitarily equivalence condition and cardinality

Next we will learn about the condition on an isometry to be unitarily
equivalent to its inverse.

######  Proposition 4.1.

An isometry $T=e^{i\theta}\left[{\begin{array}[]{cc}UA&U(\xi)\\\
\left<\cdot,\xi\right>&a\\\ \end{array}}\right]$ and its inverse are unitarily
equivalent if and only if there exists a unitary operator $V$ such that
$(UA)^{*}=V^{-1}UAV$, $V(\xi)=V^{-1}(\xi)=-U(\xi)$ and $\theta=n\pi$.

###### Proof.

An isometry $T$ and its inverse are unitarily equivalent to each other if and
only if there exists a unitary operator $V$ satisfying

$\left[{\begin{array}[]{cc}V^{-1}&0\\\ 0&1\\\
\end{array}}\right]\left\\{e^{i\theta}\left[{\begin{array}[]{cc}UA&U(\xi)\\\
\left<\cdot,\xi\right>&a\\\
\end{array}}\right]\right\\}\left[{\begin{array}[]{cc}V&0\\\ 0&1\\\
\end{array}}\right]=e^{-i\theta}\left[{\begin{array}[]{cc}(UA)^{*}&-\xi\\\
-\left<\cdot,U(\xi)\right>&a\\\ \end{array}}\right]$

i.e.

$e^{i\theta}\left[{\begin{array}[]{cc}V^{-1}UAV&V^{-1}U(\xi)\\\
\left<\cdot,V^{-1}(\xi)\right>&a\\\
\end{array}}\right]=e^{-i\theta}\left[{\begin{array}[]{cc}(UA)^{*}&-\xi\\\
-\left<\cdot,U(\xi)\right>&a\\\ \end{array}}\right]$

if and only if $(UA)^{*}=V^{-1}UAV$, $V(\xi)=V^{-1}(\xi)=-U(\xi)$ and
$\theta=n\pi$. ∎∎

Example of an operator which is unitarily equivalent to its inverse is
$\left[{\begin{array}[]{cc}A&\xi\\\ \left<\cdot,\xi\right>&a\\\
\end{array}}\right]$ where the conjugating operator $V$ can be taken $-I$.
Notice that the way $A$ is defined, $A$ clearly commutes with $V$.

In finite dimensions, the number of conjugacy classes of centralizers (called
$z$-classes) in a group of isometries uses to have implications on the
dynamical types of the action of the group on the underlying space, see [[3],
[8], [14]]. In infinite dimensional setup, this number should better be
represented by cardinality of the set of $z$-classes. Computation of the size
(in terms of cardinality) of the set of conjugacy classes happens to be the
initial attempt towards the computation of number of $z$-classes. In what
follows, we use the spectral decomposition of normal operators to find
cardinality of the set of conjugacy classes of normal operators.

In what follows, we will assume $H$ to be a separable Hilbert space.

We know that $|H|=\mathfrak{c}$. Let $U(H)$ denote the group of unitary
operators on $H$ and $G_{s}$ denote the collection of self adjoint isometries
in $G$. Then

######  Proposition 4.2.

$|U(H)|=|G|=|G_{s}|=\mathfrak{c}$.

###### Proof.

As $e^{i\theta}I\in U(H)$, $\theta\in\mathbb{R}$, we have
$|U(H)|\geq\mathfrak{c}$. Also, $|U(H)|\leq|L(H)|$, where $L(H)$ is the space
of linear operators on $H$. Now, each linear operator has an infinite matrix
representation. This gives $|L(H)|$ is same as the cardinality of the set of
all complex sequences which is same as $\mathfrak{c}$. Hence
$|U(H)|=\mathfrak{c}$.

Also, by the construction of elements of $G$, it can be seen that
$|G|=|H|\times|U(H)|\times|\mathbb{R}|=\mathfrak{c}\times\mathfrak{c}\times\mathfrak{c}=\mathfrak{c}$.

For the collection $G_{s}$, elements of the form
$\left[{\begin{array}[]{cc}A&\xi\\\ \left<\cdot,\xi\right>&a\\\
\end{array}}\right]$ where $\xi\in H$ lie in $G_{s}$ and cardinality of such
elements is $|H|=\mathfrak{c}$. This gives $|G_{s}|\geq\mathfrak{c}$. Also,
$|G_{s}|<|G|=\mathfrak{c}$. So $|G_{s}|=\mathfrak{c}$. ∎∎

Next we will find out the cardinality of the set of unitary equivalence
classes of normal operators on $H$. For this, we will use the following result
from Conway [6, Chapter 9, Corollary 10.12].

######  Theorem 4.3.

If $N$ is a normal operator on a separable Hilbert space $H$ with scalar-
valued spectral measure $\mu$, then there is a decreasing sequence
$\\{\Delta_{n}\\}$ of Borel subsets of $\sigma(N)$ such that
$\Delta_{1}=\sigma(N)$ and

$N\cong N_{\mu}\oplus N_{\mu\restriction_{\Delta_{2}}}\oplus
N_{\mu\restriction_{\Delta_{3}}}\oplus...\,.$

Let us introduce some quick notations for the sake of next proposition.

Let the set of all normal operators be denoted by $\mathcal{N}$, set of all
unitary equivalence classes of normal operators by $\mathcal{N_{U}}$, set of
all compact subsets of $\mathbb{C}$ by $\mathcal{K}$, Borel $\sigma$-algebra
of a set $A\subset\mathbb{C}$ by $\mathcal{B}(A)$ and the set of all sequences
of all the Borel subsets of a set $A$ by $\mathcal{B}_{s}(A)$.

The above theorem gives
$|\mathcal{N}|\leq|\underset{K\in\mathcal{K}}{\bigcup}\mathcal{B}_{s}(K)|$.

######  Proposition 4.4.

$|\mathcal{N_{U}}|=\mathfrak{c}$.

###### Proof.

Let $K\subset\mathbb{C}$ be a compact set. Then there exists a normal operator
with $K$ as its spectrum. Also, unitarily equivalent normal operators have the
same spectrum. This gives $|\mathcal{N_{U}}|\geq|\mathcal{K}|$. We know that
$|\mathcal{B}(\mathbb{C})|=\mathfrak{c}$ ([17, page no. 53]). Now, every
singleton is a compact set in $\mathbb{C}$ and every compact set is a Borel
subset of $\mathbb{C}$ giving $|\mathcal{K}|=\mathfrak{c}$ and hence
$|\mathcal{N_{U}}|\geq\mathfrak{c}$.

We will now show that $|\mathcal{N_{U}}|\leq\mathfrak{c}$.
$|\mathcal{B}(K)|\leq|\mathcal{B}(\mathbb{C})|=\mathfrak{c}$ which gives
$|\mathcal{B}_{s}(K)|\leq|\mathcal{B}_{s}(\mathbb{C})|=\mathfrak{c}$,
therefore
$|\underset{K\in\mathcal{K}}{\bigcup}\mathcal{B}_{s}(K)|\leq\mathfrak{c}\times\mathfrak{c}=\mathfrak{c}$.
Finally,
$|\mathcal{N_{U}}|\leq|\mathcal{N}|\leq|\underset{K\in\mathcal{K}}{\bigcup}\mathcal{B}_{s}(K)|\leq\mathfrak{c}$.
Hence the result. ∎∎

Acknowledgement The research is supported by Council of Scientic and
Industrial Research, India (File no. 09/045(1668)/2019-EMR-I).

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# Doping Dependence of the in-Plane Transition in Co3Sn2S2

Ivica Živković<EMAIL_ADDRESS>Laboratory for Quantum Magnetism,
Institute of Physics, École Polytechnique Fédérale de Lausanne, CH-1015
Lausanne, Switzerland  Mohamed A. Kassem  Department of Materials Science and
Engineering, Kyoto University, Kyoto 606-8501, Japan  Department of Physics,
Faculty of Science, Assiut University, 71516 Assiut, Egypt  Yoshikazu Tabata
Department of Materials Science and Engineering, Kyoto University, Kyoto
606-8501, Japan  Takeshi Waki  Department of Materials Science and
Engineering, Kyoto University, Kyoto 606-8501, Japan  Hiroyuki Nakamura
Department of Materials Science and Engineering, Kyoto University, Kyoto
606-8501, Japan

###### Abstract

In Co3Sn2S2two transitions are observed, the main one to a ferromagnetic state
at $T_{C}=174$ K and the second one, involving in-plane components at
$T_{P}=127$K̇. We follow their doping dependence as Sn is replaced with In,
which causes a reduction of $T_{C}$ and $T_{P}$. Importantly, both transitions
follow the same doping dependence, indicating a single energy scale involved
with both processes.

Co3Sn2S2 is a candidate Weyl semimetal material, shown to exhibit anomalous
Hall effect [1] and surface Fermi arcs [2]. Weyl semimetals are characterized
by the existence of Weyl nodes, acting as sources and sinks of Berry curvature
[3], which when found close to the Fermi energy are contributing to anomalous
transport properties. Their number and the location in the k-space are
intimately linked to the material’s magnetic order. In Co3Sn2S2 it has been
demonstrated that the separation of Weyl nodes is linked to the value of the
magnetic moment [4]. On the other hand, applying the magnetic field
perpendicular to the hexagonal c-axis shifts, creates and annihilates Weyl
nodes as the moment is canted towards the kagome plane [5].

The properties of the magnetic order in Co3Sn2S2 have been under scrutiny for
some time. Although established as an itinerant ferromagnet [6], successive
reports have indicated a more complicated picture emerging below $T_{C}=174$
K. Antiferromagnetic [7] and spin-glass [8] phases have been invoked, with
later reports showing inconsistencies with these proposals [9, 10]. The
existence of an additional transition has been proposed [11], with a detailed
angular dependence showing that it is linked with the in-plane projection of
moments, which are slightly tilted away from the c-axis and form an umbrella
structure [10].

In this short note we present experimental evidence of the intrinsic origin of
the in-plane transition in Co3Sn2S2 revealed by its dependence on In-doping.
By replacing Sn with In the number of electrons is reduced which leads to a
suppression of the density of states at the Fermi level and a subsequent
reduction of the main ordering temperature. Previous studies on single
crystals [12] have pointed out that the critical concentration is around 0.8,
with larger doping levels resulting in a non-magnetic ground state. Here we
extend those studies by focusing also on the second transition recently
revealed to involve in-plane components of magnetic moments [10].

The preparation of single crystals has been described previously [12]. Samples
were prepared by flux and Bridgman methods and their precise composition has
been determined by wave length dispersive X-ray spectroscopy (WDX). We did not
observe any systematic difference in the properties of samples related to the
growth method. The specimens of an appropriate size were mounted on a sample
holder which permits the rotation around an axis perpendicular to the
direction of magnetic field $B$ with a resolution better than 0.10. Both ac
and dc signals were measured on a superconducting quantum interference device
(SQUID)-based magnetometer (MPMS3, Quantum Design) with respect to temperature
in zero-field-cooled (ZFC)/field-cooled (FC) regimes. Zero-field state has
been ensured by resetting the superconducting magnet.

Figure 1: (Color online) Magnetization of Co3Sn2S2 parallel to ((a), (c), (e)
and (g)) and perpendicular to the c-axis ((b), (d), (f) and (h)) measured in
$B=1$ mT.

ZFC/FC temperature scans were measured with magnetic field $B=1$ mT parallel
and perpendicular to the c-axis. Figure 1 displays the evolution of the
magnetization with respect to the doping level of indium. The parent compound
($x=0$) exhibits the main transition at $T_{C}=174$ K and the second
transition at $T_{P}=127$ K, in accordance with previous results [10]. Both
transitions are characterized by a sharp increase of magnetization and a
splitting of ZFC and FC measurements. With In doping those features are
shifted to lower temperatures, reflecting the decrease of electron density of
states. At larger doping levels the splitting below $T_{P}$ for $B\perp c$ is
significantly decreased, which we attribute to the decrease of energy barriers
pinning the in-plane moment projections. One should note that the splitting of
the in-plane component remains more than two orders of magnitude smaller
compared to the c-axis splitting across the doping levels.

Alongside the ZFC/FC splitting, each transition is accompanied by a sharp peak
in ac magnetic susceptibility. In Figure 2 we show the real component of the
susceptibility across transitions for four doping levels grown by the flux
method ($x=0,0.14,0.34,0.69$, their M(T) curves are presented in Figure 1) and
two doping levels grown by the Bridgman method ($x=0.28,0.47$). Magnetization
curves from Bridgman-grown samples are qualitatively the same as those of
flux-grown samples.

Figure 2: (Color online) Real part of ac susceptibility across two transitions
for several doping levels measured with alternating magnetic field
perpendicular to the c-axis.

We can now extract the temperatures of peaks and construct a phase diagram. In
Figure 3 dark circles represent the main transition $T_{C}$ while light
diamonds stand for the in-plane transition at $T_{P}$. The solid line that
follows the doping dependence of the main transition is a simple second-order
polynomial which hints that around $x\sim 0.8$ a quantum phase transition
occurs between the itinerant ferromagnetic phase and a nonmagnetic metal. This
is in excellent agreement with previous findings [12] where for $x>0.8$ it has
been indeed shown that the ground state is nonmagnetic.

If we now renormalize the second order polynomial on the temperature scale
given by $T_{P}$, we obtain the dashed line in Figure 3. It is found that
there is a very good agreement with experimental points for a wide doping
level. It is only the highest doping level that deviates somewhat, which can
potentially be assigned to a disorder effect on a fragile state of in-plane
components.

The successful description of $T_{P}$ vs $x$ with a simple rescaling of
$T_{C}$ vs $x$ dependence indicates that a single energy scale is responsible
for both features. Most importantly, it proves that the feature observed at
$T_{P}$ and the array of properties characterizing it, like strictly in-plane
dynamics and orders of magnitude weaker signal, are intrinsic to Co3Sn2S2.

Figure 3: (Color online) Doping dependence of the main transition $T_{C}$ and
the in-plane transition $T_{P}$.

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