# Assessing the Effectiveness of GPT-3 in Detecting False Political
Statements: A Case Study on the LIAR Dataset

Mars Gokturk Buchholz
Data Scientist and Software Engineer
<EMAIL_ADDRESS>
###### Abstract

The detection of political fake statements is crucial for maintaining
information integrity and preventing the spread of misinformation in society.
Historically, state-of-the-art machine learning models employed various
methods for detecting deceptive statements. These methods include the use of
metadata ([14]), n-grams analysis ([12]), and linguistic ([15]) and
stylometric ([5]) features. Recent advancements in large language models, such
as GPT-3 ([2]) have achieved state-of-the-art performance on a wide range of
tasks. In this study, we conducted experiments with GPT-3 on the LIAR dataset
([14]) and achieved higher accuracy than state-of-the-art models without using
any additional meta or linguistic features. Additionally, we experimented with
zero-shot learning using a carefully designed prompt and achieved near state-
of-the-art performance. An advantage of this approach is that the model
provided evidence for its decision, which adds transparency to the model’s
decision-making and offers a chance for users to verify the validity of the
evidence provided.

##  1 Introduction

The proliferation of fake news has become a major concern in today’s society,
with its potential to cause harm to individuals, organizations, and even
entire nations. A survey by Pew Research Center in 2016 ([1]) found that 64%
of American adults believe fabricated news stories cause confusion about basic
facts of current issues and events. About four-in-ten Americans (39%) feel
confident that they can recognize news that is fabricated, but 23% say they
have shared a made-up news story, with 14% admitting to sharing a story they
knew was fake at the time.

Organizations, such as PolitiFact111https://www.politifact.com, actively
investigate and rate the veracity of comments made by public figures,
journalists and organizations. The fact-checking process involves researching
the statement, consulting experts, and reviewing relevant documents.
Politifact uses a process called the Truth-O-Meter to judge the truth value of
statements made by politicians. The process involves fact-checking the
statement and assigning a rating on a scale from "True" to "Pants on Fire",
which indicates varying degrees of truth. Politifact also provides a detailed
explanation of their rating and sources used for each fact-check. We provide
the explanation for each label Politifact uses in Table 1.

Rating |  Explanation
---|---
True |  The statement is accurate and there is no significant omission of context.
Mostly True |  The statement is accurate but may require additional context or clarification.
Half True |  The statement is partially accurate but contains some misleading information or is taken out of context.
Mostly False |  The statement contains some element of truth but is mostly inaccurate.
False |  The statement is inaccurate and contains significant factual errors.
Pants on Fire |  The statement is not only inaccurate but also ridiculous or absurd.
Table 1: Politifact’s Truth-O-Meter labels and descriptions.

Automated detection of false political statements using natural language
processing (NLP) has the potential to streamline the fact-checking process,
prevent the spread of false statements on online media and social networks,
and increase public awareness of false statements. There has been a growing
interest in developing NLP methods to detect and combat false political
statements. Historically, some of these methods focused on analyzing and
extracting linguistic and stylistic features from statements, as well as
augmenting them with specialized knowledge bases or author and context
information. However, these approaches face challenges such as intentional
language use by bad actors to avoid detection and the potential irrelevance of
augmented author information in the long term.

In this paper, we investigate the application of GPT-3 on the task of graded
deception detection in political statements.

##  2 Related Work

Data Sample
---
id: 9616.json
label: false
statement: “Latinos now make up the majority population in Texas.”
subject: education, legal-issues, population, states
speaker: roberto-alonzo
speaker-job-title: Attorney
state-info: Texas
party-affiliation: democrat
barely-true-c: 0.0
half-true-c: 1.0
false-c: 0.0
mostly-true-c: 0.0
pantsonfire-c: 0.0
context: a press release
Table 2: A sample from the training dataset.

Several studies have investigated automated approaches to detect false
statements using the Politifact dataset. These studies have used various
features and machine learning techniques to build their models. Features used
in these studies include linguistic features such as unigrams, bigrams, and
trigrams ([13];[9]), sentiment scores ([9]; [7]; [3]), named entities
([9];[11]; [3]), and syntax features ([8];[4]). Some studies have also used
stylometric features such as readability scores ([7]) and punctuation features
([13]).

In this paper, we focus on two studies that utilized the Politifact data to
predict the truthfulness of a statement. We will summarize their approaches
below.

Lingustic and Stylometric features One example of using the linguistic and
stylometric properties of text to detect false statements is through the use
of LIWC. LIWC stands for "Linguistic Inquiry and Word Count," which is a text
analysis software developed by James W. Pennebaker and his colleagues at the
University of Texas at Austin. It is designed to analyze the psychological
content of natural language and has been widely used in social science
research. LIWC features are categories of words that represent specific
aspects of language use, such as emotions, social processes, and cognitive
processes. Examples of LIWC features include positive emotions like "love,"
"happy," and "amazing"; negative emotions like "hate," "angry," and "worried";
cognitive processes like "think," "know," and "consider"; and social processes
like "friend," "family," and "help". One study that utilized LIWC to detect
false statements is "Truth of Varying Shades: Analyzing Language in Fake News
and Political Fact-Checking" ([10]). The authors employed the PolitiFact data
to make predictions about the truthfulness of political statements. To extract
features for prediction, they utilized LIWC. The MaxEnt model, which utilized
both text and LIWC features, was found to be the best performing model. These
features were instrumental in providing insight into the language used in the
statements and the emotional and cognitive tone conveyed, thereby contributing
to the prediction. The best model achieved a 6-class prediction macro-averaged
F1 score of .22. Using LIWC features produced better performance for some
models compared to using only the text feature. However, one of the challenges
with using linguistic or stylometric features to detect false statements is
that actors may adapt to these detection methods, making it necessary to
evaluate the truth value of a statement solely based on the facts.

Meta-data augmentation The LIAR dataset, constructed by William Yang Wang
([14]) using Politifact data, served as the basis for the development of
several benchmark models. Both text and meta-information, including the
speaker, the speaker’s party affiliation, the location of the statement, and
its context, were employed as predictive features. The most successful model
was a hybrid-CNN model that utilized both text and meta-information, achieving
an accuracy of 0.274 on the test dataset. Meta-data played a significant role
in prediction, leading to substantial improvements when combined with text.
However, using author or context information introduces a challenge, as trends
related to the speaker and context may change over time, thereby affecting the
model’s long-term performance. We will use the same LIAR dataset and
benchmarks to evaluate our model’s performance.

##  3 Dataset

The experiments were conducted on the LIAR dataset ([14]). The dataset
contains 12,836 short statements extracted from PolitiFact, a well-known fact-
checking website in the United States. These statements were labeled by
PolitiFact staff as True, Mostly True, Half True, Barely True, False, or Pants
on Fire, according to their truthfulness. The distribution of the six labels
is relatively well-balanced, with the exception of the ’Pants on Fire’
category, which contains 1,050 instances. The label distribution is shown in
Figure 1. The dataset is divided into training, validation, and testing sets
using an 80/10/10 split. The average statement length, measured in tokens, is
17.9. The statements primarily date from 2007 to 2016.

TrueMostly TrueHalf TrueBarely TrueFalsePants on
Fire$0.1$$0.15$$0.2$%TrainValidTest Figure 1: Label distribution in training,
validation, and test datasets.

The dataset includes statements covering a wide range of topics, including
health care, taxes, education, and the economy. It also contains statements
made by politicians from both major political parties in the US. The
statements are accompanied by various metadata, such as the speaker’s name,
party affiliation, and job title, as well as the publication and date of the
statement.

##  4 Experiments

We conducted experiments in a single setting, using only the ’statement’
column from the LIAR dataset. The models were trained to predict one of the
six class labels.

###  4.1 Baseline

The LIAR dataset ([14]) has been widely used for fact-checking research, and
several models have been proposed to address the task of detecting fake news.
To the best of our knowledge, the best performing model on the LIAR dataset to
date is the CNN-hybrid model proposed by Wang ([14]), which achieved an
accuracy of 0.274 on the test set.

###  4.2 Metric

In our experiments, we evaluated the performance of our proposed models using
the accuracy metric, which is consistent with the LIAR dataset benchmarks.

$Accuracy=\frac{TP+TN}{TP+TN+FP+FN}$ (1)

where TP represents true positives, TN represents true negatives, FP
represents false positives, and FN represents false negatives.

###  4.3 Experiment 1: Fine-tuning GPT-3

We used the OpenAI Fine-tuning API222OpenAI Fine-tuning API documentation:
https://platform.openai.com/docs/guides/fine-tuning to adapt provided GPT-3
base models to the statement classification task. Our fine-tuning dataset
consisted of 10,240 training samples and 1,284 validation samples, which were
preprocessed to separate the statement and label columns using predefined
tokens.

Considering the rate limits set by OpenAI at the time of the study, we
initially fine-tuned the Ada model to identify the optimal hyperparameters,
such as the number of epochs and the learning rate multiplier. We experimented
with the number of epochs within the range of [2, 3, 4, 5] and the learning
rate multiplier within [0.02, 0.04, 0.08, 0.16, 0.2, 0.22]. To account for
possible variations in accuracy, each model underwent five training cycles.
The optimal hyperparameters were determined to be 5 epochs, a batch size of
256, and a learning rate multiplier of 0.2. We used these parameters to fine-
tune the Curie model with the training and validation datasets333Our datasets,
code, and experiment results are publicly available in our GitHub repository:
https://github.com/marsgokturk/gpt3-false-political-statement-detection.

We utilized the OpenAI fine-tuning API’s feature to procure classification
metrics on a specified dataset. In our case, we requested this option in the
API for our test dataset, which consists of 1,267 samples, to evaluate the
model’s performance.

The total cost of the fine-tuning process amounted to $111.27. There were no
additional costs associated with serving the fine-tuned model, as the OpenAI
API provides classification metrics within the API response for a given
dataset. This feature is available if requested as an option in the API
request.

Although superior performance might be achievable with different
hyperparameters or by using the Davinci model, we opted to fine-tune the Ada
model to remain within the rate limits established by OpenAI between March
25th and April 3rd.

Statement: "Medicaid spending declined by 1.9 percent in 2012, the second such
decline in 47 years."
---
Prompt: "Act as a research journalist doing fact-checking on the statements
given by politicians. You will label each statement as one of "true, mostly-
true, half-true, barely-true, false or pants-fire". Use below metric to label
each statement:
If the statement is accurate and there’s nothing significant missing, then
label as "true"
If the statement is accurate but needs clarification or additional
information, then label as "mostly-true"
If the statement is partially accurate but leaves out important details or
takes things out of context, then label as "half-true"
If the statement contains an element of truth but ignores critical facts that
would give a different impression, then label as "barely-true"
If the statement is not accurate, then label as "false"
If the statement is not accurate and makes a ridiculous claim, then label as
"pants-fire"
Use only factual data and reliable sources such as major news outlets and
fact-checking organizations in the USA. Provide evidence for the labels and
the name, date and author of the evidence for your answer. If you don’t have
any evidence, say "I don’t have any evidence". You answer should be in this
format: "Label: <statement label>, Evidence: <evidence>"
Output: "Label = mostly-true, evidence = According to a report by the Kaiser
Family Foundation, Medicaid spending did decline by 1.9 percent in 2012, which
was the second decline in 47 years. However, the report also notes that the
decline was largely due to a decrease in enrollment and utilization, rather
than a decrease in spending per enrollee. (source: Kaiser Family Foundation,
’Medicaid Spending and Enrollment Growth: FY 2012 & 2013, September 2014)"
Table 3: An example of the model’s output for an input statement and the
prompt.

###  4.4 Experiment II: Zero-shot learning with prompt-engineering

In this experiment, we utilized OpenAI Chat Completions API444OpenAI Chat
Completions API documentation:
https://platform.openai.com/docs/guides/gpt/chat-completions-api with the
GPT-3.5-turbo model to perform zero-shot learning through an engineered
prompt.

The prompt was meticulously designed to encompass descriptions for each label,
adhering to the criteria delineated in Politifact’s Truth-O-Meter, and
requested factual evidence to substantiate the predicted label. The model was
instructed to categorize each statement as "true", "mostly-true", "half-true",
"barely-true", "false", or "pants-fire", and to base its label solely on
factual information and reliable sources such as major news outlets and fact-
checking organizations in the USA.

To assess the model’s performance, we employed the same prompt for each test
sample, and the accuracy was computed by comparing the gold labels and the
extracted output labels. The evidence provided by the model was also
documented to gauge its usability and credibility.

We directed the model to generate only one token with a temperature of 0.0. An
example of the model’s output for an input statement can be observed in Table
3.

This experiment incurred a cost of $2.00. It is crucial to acknowledge that by
creating a more effective prompt, it is conceivable to attain superior
accuracy in the zero-shot learning setting.

##  5 Results

Model |  Test Accuracy
---|---
Baseline CNN using only the statement |  0.270
Baseline CNN-hybrid using the statement+meta-data |  0.274
Curie fine-tuned model using only the statement |  0.295
Zero-shot setting with the same prompt using only the statement |  0.266
Table 4: Comparison of baseline and GPT-3 models.

Our experiments show that fine-tuned GPT-3 models outperform the best baseline
CNN-hybrid model in political statement classification. Although the zero-shot
model’s performance is lower, it is still competitive with the baseline model
and provides transparency by giving evidence for predictions. Table 4
summarizes the test accuracy results of our experiments.

In the Appendix, we provide a few representative examples of the statement,
true label, predicted label, and the model’s evidence. These examples give a
deeper understanding of the model’s performance and reasoning.

##  6 Discussion and Conclusion

statement |  "Bernie Sanders voted against the Brady Bill – background checks and waiting periods."
---|---
Politifact label |  mostly-true
Predicted label;evidence |  false; "bernie sanders voted for the brady bill in 1993, which established background checks and waiting periods for firearm purchases. (source: politifact, "bernie sanders voted for brady bill, but also voted to shield gun makers," february 18, 2016)"
Table 5: Disagreement between Politifact label and Predicted label.

The training data for GPT-3, which only extends up to September 2021, might
limit its effectiveness in fact-checking tasks that require up-to-date
information. A potential solution for future research could be the use of
Retrieval-Augmented Generation (RAG; [6]). This method could provide the model
with relevant and current information for a given statement, enabling it to
predict the statement’s truth value and supply factual evidence to support its
prediction. Consequently, this could enhance the model’s ability to make
accurate predictions, especially when dealing with recent or current events.

In addition, future studies could explore few-shot learning scenarios. By
fine-tuning the model with a well-selected set of examples, performance might
surpass that of the zero-shot setting.

The GPT-3 model displayed a complex behavior by not consistently employing or
aligning with labels from Politifact data, which suggests that it isn’t simply
repeating previously absorbed information. Its utilization of Politifact data
for label generation sparks questions concerning potential data leaks in
contrast to its proficiency for information retrieval during truth-value
assessments. As demonstrated by an example in Table 5, an intriguing area for
additional exploration involves instances when models and fact-checkers
interpret the same evidence but reach different conclusions. A deeper
understanding of these conflicts and devising a robust framework to reconcile
these differences could be significantly impactful for future research.

While the fine-tuned GPT-3 model outperformed the state-of-the-art model in
classification accuracy, the utility of label-only models for political
statement classification remains debatable. The necessity of a six-label
classification system, as opposed to a simpler one, may also warrant
reconsideration.

## References

  * [1] Michael Barthel, Amy Mitchell and Jesse Holcomb  “Many Americans Believe Fake News is Sowing Confusion”  In _NUMBERS, FACTS AND TRENDS SHAPING THE WORLD_ , 2016  URL: https://www.pewresearch.org/journalism/2016/12/15/many-americans-believe-fake-news-is-sowing-confusion/
  * [2] Tom B. Brown et al.  “Language Models are Few-Shot Learners”, 2020  arXiv:2005.14165 [cs.CL]
  * [3] Wei-Fan Chen, Chien-Lun Liu, Yen-Chun Lin and Min Sun  “Multi-task learning for detecting check-worthy claims”  In _Proceedings of the 57th Conference of the Association for Computational Linguistics_ , 2019, pp. 5329–5339
  * [4] Benjamin D Horne, Jeppe Nørregaard and Sibel Adali  “Sampling the news producers: A large news and feature dataset for the study of the complex media landscape”  In _Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers)_ , 2018, pp. 2282–2292  Association for Computational Linguistics
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  * [6] Patrick S.. Lewis et al.  “Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks”  In _CoRR_ abs/2005.11401, 2020  arXiv: https://arxiv.org/abs/2005.11401
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  * [9] H. Rashkin et al.  “Truth of Varying Shades: Analyzing Language in Fake News and Political Fact-Checking”  In _Conference on Empirical Methods in Natural Language Processing_ , 2017
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  * [12] Sahil Singh, Manavpreet Kaur and Parul Gupta  “Automatic Detection of Fake News Using N-Gram Analysis and Machine Learning”  In _Journal of Artificial Intelligence and Data Science_ 3.1, 2021, pp. 1–9
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  * [15] Y. Wu, J. Zhang and X. Yu  “A Feature Selection Approach to Identify Linguistic Markers for Fake News Detection”  In _Information Processing & Management_ 59.1 Elsevier, 2022, pp. 102754

##  Appendix A Sample Predictions

The table below showcases representative examples of the model’s predictions.
It presents the statement being evaluated, the true label, the predicted
label, and the supporting evidence.

Statement |  True Label |  Predicted Label |  Zero-Shot Model’s Evidence
---|---|---|---
Says by rejecting an expansion of Medicaid under the new law, Georgia was depriving 25,000 veterans of health care coverage. |  True |  False |  according to politifact, "there’s no evidence that 25,000 veterans in Georgia would gain health care coverage if the state expanded Medicaid." Additionally, the statement ignores the fact that many veterans are already eligible for health care through the Department of Veterans Affairs. (source: https://www.politifact.com/factchecks/2014/mar/...)
Under Rosemary Lehmberg, the Travis County D.A.’s office convened the grand jury that indicted Rick Perry. |  True |  False |  "Under Lehmberg, the Travis County District Attorney’s office had convened a grand jury to investigate allegations that Perry had abused his power by vetoing funding for the state’s public integrity unit, which was housed in the Travis County District Attorney’s office." - politifact, "Rick Perry says he was right to veto funding for DA’s office," August 19, 2014.
Foreign Chinese prostitution money is allegedly behind the groups funding Congressman Sean Duffy’s Republican Majority. |  Pants-Fire |  Pants-Fire |  I don’t have any evidence.
Says the unemployment rate for college graduates is 4.4 percent and over 10 percent for noncollege-educated. |  True |  Mostly-True |  according to the Bureau of Labor Statistics, as of August 2021, the unemployment rate for individuals with a bachelor’s degree or higher is 3.2 percent, while the unemployment rate for those with less than a high school diploma is 9.0 percent and for those with a high school diploma but no college is 6.0 percent. However, the statement is mostly true as it is close to the actual figures and the difference between the two rates is significant. Source: https://www.bls.gov/web/empsit/cpsee_e16.htm
Building a wall on the U.S.-Mexico border will take literally years. |  True |  True |  "According to a report by the Government Accountability Office, it took an average of 4.5 years to complete the construction of the 654 miles of fencing already in place along the border as of 2015." (source: NPR, "How long would it take to build Trump’s wall?," January 25, 2017)


# subspace-hypercyclic conditional type operators on $L^{p}$-spaces

M. R. Azimi${}^{*^{1}}$ and Z. Naghdi2 M. R. Azimi
mhr.azimi@maragheh.ac.ir(for corresponding) Z. Naqdi<EMAIL_ADDRESS>$1,2$: department of mathematics, faculty of sciences, university of maragheh,
p. o. box: 55181-83111, Golshahr, maragheh, iran

###### Abstract.

A conditional weighted composition operator $T_{u}:L^{p}(\Sigma)\rightarrow
L^{p}(\mathcal{A})$ ($1\leq p<\infty$), is defined by
$T_{u}(f):=E^{\mathcal{A}}(uf\circ\varphi)$, where $\varphi:X\rightarrow X$ is
a measurable transformation, $u$ is a weight function on $X$ and
$E^{\mathcal{A}}$ is the conditional expectation operator with respect to
$\mathcal{A}$. In this paper, we study the subspace-hypercyclicity of $T_{u}$
with respect to $L^{p}(\mathcal{A})$. First, we show that if $\varphi$ is a
periodic nonsingular transformation, then $T_{u}$ is not
$L^{p}(\mathcal{A})$-hypercyclic. The necessary conditions for the subspace-
hypercyclicity of $T_{u}$ are obtained when $\varphi$ is non-singular and
finitely non-mixing. For the sufficient conditions, the normality of $\varphi$
is required. The subspace-weakly mixing and subspace-topologically mixing
concepts are also studied for $T_{u}$. Finally, we give an example which is
subspace-hypercyclic while is not hypercyclic.

###### Key words and phrases:

Subspace-hypercyclic, Orbit, Subspace-weakly mixing, Subspace-topologically
mixing, Measurable transformation, Normal, Radon-Nikodym derivative,
Conditional expectation, aperiodic.

###### 2000 Mathematics Subject Classification:

Primary 47A16, 46E30.

##  1\. Introduction and Preliminaries

Suppose that $T$ is a bounded linear operator on a topological vector space
$X$. If there is a vector $x\in X$ such that the orbit
$orb(T,x):=\\{T^{n}x:n=0,1,2,...\\}$ is dense in $X$, then $T$ will be
hypercyclic and $x$ is called a hypercyclic vector. Here, $T^{n}$ stands for
the $n$-th iterate of $T$ and $T^{0}$ is the identity map $I$. Let $M$ be a
closed and non-trivial subspace of $X$. An operator $T$ is subspace-
hypercyclic with respect to $M$ ($M$-_hypercyclic_), if there is a a vector
$x\in X$ such that $orb(T,x)\cap M$ is dense in $M$. Also an operator $T$ is
subspace-transitive with respect to $M$, if for any non-empty open set
$U,V\subseteq M$, there exists an $n\in\mathbb{N}$ such that $T^{-n}(U)\cap V$
contains an open non-empty subset of $M$. An operator $T$ is subspace-
topologically mixing with respect to $M$, if for any non-empty open set
$U,V\subseteq M$, there exists an $N\in\mathbb{N}$ such that $T^{-n}(U)\cap V$
contains an open non-empty subset of $M$ for each $n\geq N$. It is called
subspace-weakly mixing if $T\oplus T$ is subspace-hypercyclic with respect to
$M\oplus M$.

The study of subspace-hypercyclic linear operators was initiated by B. F.
Madore and R. A. Martínez-Avendaño [24]. They found out that subspace-
hypercyclicity like as hypercyclicity, can occur only on infinite-dimensional
spaces and even subspaces. Also, they proved an interesting Kitai’s type
_subspace-hypercyclicity criterion_ on a topological vector space as follows.
Assume that there exist $D_{1}$ and $D_{2}$, dense subsets of $M$, and an
increasing sequence of positive integers $(n_{k})$ such that

  * •

$T^{n_{k}}x\to 0$ for all $x\in D_{1}$;

  * •

for each $y\in D_{2}$, there exists a sequence $\\{x_{k}\\}$ in $M$ such that
$x_{k}\to 0$ and $T^{n_{k}}x_{k}\to y$;

  * •

$M$ is an invariant subspace for $T^{n_{k}}$ for all $k\in\mathbb{N}$.

Then $T$ is subspace-transitive and hence is subspace-hypercyclic [24, Theorem
3.6]. But the converse is not true, see [23, 28] for more details. Further, it
is showed that the compact or hyponormal operators are not subspace-
hypercyclic.

For the dynamics of linear operators the survey articles [29], [8], [27],
[31], [1], [24] and the books [6], [16] are useful.

Let $(X,\Sigma,\mu)$ be a complete $\sigma$-finite measure space and
$\mathcal{A}$ is a $\sigma$-finite subalgebra of $\Sigma$. For each $1\leq
p<\infty$, the Banach space $L^{p}(X,\mathcal{A},\mu_{|_{\mathcal{A}}})$ is
denoted by $L^{p}(\mathcal{A})$ simply. All comparisons between two functions
or two sets are to be interpreted as holding up to a $\mu$-null set. The
_support_ of any $\Sigma$-measurable function $f$ is defined by
$\sigma(f)=\\{x\in X:f(x)\neq 0\\}$. The _characteristic function_ of any set
$A$ and the class of all $\mathcal{A}$-measurable and simple functions on $X$
with finite supports will be denoted by $\chi_{A}$ and $S^{\mathcal{A}}(X)$,
respectively.

A $\Sigma$-measurable transformation $\varphi:X\rightarrow X$ is called _non-
singular_ whenever $\mu\circ\varphi^{-1}$ is absolutely continuous with
respect to $\mu$, which is symbolically shown by $\mu\circ\varphi^{-1}\ll\mu$.
In this case, _Radon-Nikodym property_ is denoted by
$h:=\frac{d\mu\circ\varphi^{-1}}{d\mu}$.

A $\Sigma$-measurable transformation $\varphi:X\rightarrow X$ is called
_periodic_ if $\varphi^{m}=I$ for some $m\in\mathbb{N}$. It is called
_aperiodic_ , if it is not periodic. Also, if for each subset $F\in\Sigma$
with finite measure, there exists an $N\in\mathbb{N}$ such that
$F\cap\varphi^{n}(F)=\emptyset$ for every $n>N$, then $\varphi$ is called
_finitely non-mixing_.

Set $\Sigma_{\infty}=\bigcap_{n=1}^{\infty}\varphi^{-n}(\Sigma)$ and suppose
that $h$ is $\Sigma_{\infty}$-measurable. The assumption
$\mu\circ\varphi^{-1}\ll\mu$ implies that $\mu\circ\varphi^{-n}\ll\mu$ for all
$n\in\mathbb{N}$ and then

$\displaystyle h_{n}:$
$\displaystyle=\frac{d\mu\circ\varphi^{-n}}{d\mu}=\frac{d\mu\circ\varphi^{-n}}{d\mu\circ\varphi^{-(n-1)}}\cdots\frac{d\mu\circ\varphi^{-1}}{d\mu}$
$\displaystyle=(h\circ\varphi^{-(n-1)})\cdots(h\circ\varphi^{0})=\prod_{i=0}^{n-1}h\circ\varphi^{-i}.$

Note that always $h\circ\varphi>0$ and $h_{n}=h^{n}$ whenever
$h\circ\varphi=h$. When it is restricted to a $\sigma$-subalgebra
$\mathcal{A}$, is denoted by
$h_{n}^{\mathcal{A}}=\frac{d(\mu\circ\varphi^{-n}|_{\mathcal{A}})}{d(\mu|_{\mathcal{A}})}$.

The _change of variable formula_

$\int_{\varphi^{-n}(A)}f\circ\varphi^{n}d\mu=\int_{A}h_{n}fd\mu,\ \ \ \
A\in\Sigma,\ f\in L^{1}(\Sigma),$

will be used frequently.

When $\varphi(\Sigma)\subseteq\Sigma$ and $\mu\circ\varphi\ll\mu$, then a
measure $\mu$ is called _normal_ with respect to $\varphi$ and in this case
$h^{\sharp}=\frac{d\mu\circ\varphi}{d\mu}$ is defined. Now, consider that

$h^{\sharp}=(\frac{d\mu}{d\mu\circ\varphi})^{-1}=(\frac{d\mu\circ\varphi^{-1}}{d\mu}\circ\varphi)^{-1}=\frac{1}{h\circ\varphi}$

and

$\displaystyle h_{n}^{\sharp}:$
$\displaystyle=\frac{d\mu\circ\varphi^{n}}{d\mu}=(h^{\sharp}\circ\varphi^{(n-1)})\cdots(h^{\sharp}\circ\varphi^{0})=\prod_{i=0}^{n-1}h^{\sharp}\circ\varphi^{i}=\prod_{i=1}^{n}(h\circ\varphi^{i})^{-1},$

$h^{\sharp}_{n}\circ\varphi>0$,
$h^{\sharp}_{n+1}=h^{\sharp}h^{\sharp}_{n}\circ\varphi$.
Let $1\leq p\leq\infty$. For any non-negative $\Sigma$-measurable functions
$f$ or for any $f\in L^{p}(\Sigma)$, Radon-Nikodym Theorem, ensures the
existence of a unique ${\mathcal{A}}$-measurable function $E^{\mathcal{A}}(f)$
such that

$\int_{A}E^{\mathcal{A}}(f)d\mu=\int_{A}fd\mu,\ \ \ \ \ \mbox{for all}\ \
A\in{\mathcal{A}}.$

A contractive projection $E^{\mathcal{A}}:L^{p}(\Sigma)\rightarrow
L^{p}({\mathcal{A}})$ is called a _conditional expectation operator_
associated with the $\sigma$-finite subalgebra ${\mathcal{A}}$.

Here, we list some useful properties of the conditional expectation operator:

  * •

$E^{\mathcal{A}}(1)=1$;

  * •

If $g$ is $\mathcal{A}$-measurable, then
$E^{\mathcal{A}}(fg)=E^{\mathcal{A}}(f)g$;

  * •

$|E^{\mathcal{A}}(f)|^{p}\leq E^{\mathcal{A}}(|f|^{p})$;

  * •

For each $f\geq 0,\sigma(f)\subseteq\sigma(E^{\mathcal{A}}(f))$;

  * •

Monotonicity: If $f$ and $g$ are real-valued with $f\leq g$, then
$E^{\mathcal{A}}f\leq E^{\mathcal{A}}g$;

  * •

For each $f\geq 0,E^{\mathcal{A}}(f)\geq 0$.

  * •

$h_{n+1}=hE^{\varphi^{-1}(\Sigma)}(h_{n})\circ\varphi^{-1}=h_{n}E^{\varphi^{-n}(\Sigma)}(h)\circ\varphi^{-1}$[19].

A detailed information of the condition expectation operator may be found in
[26, 25, 18, 22].

A _weighted composition operator_ $uC_{\varphi}:L^{p}(\Sigma)\rightarrow
L^{p}(\Sigma)$ defined by $f\mapsto uf\circ\varphi$ is bounded if and only if
$J\in L^{\infty}(\Sigma)$, where
$J:=hE^{\mathcal{A}}(|u|^{p})\circ\varphi^{-1}$, and in this case
$\|uC_{\varphi}\|^{p}=\|J\|_{\infty}$ (see [19, 30, 20]).

Now, we are ready to define a conditional weighted composition operator
$T_{u}$ by:

$T_{u}:L^{p}(\Sigma)\rightarrow L^{p}(\mathcal{A})$
$T_{u}f:=E^{\mathcal{A}}\circ
uC_{\varphi}(f)=E^{\mathcal{A}}(uf\circ\varphi).$

For the fundamental properties of the conditional type operators, the reader
is refereed to [12, 15, 13, 14].

The hypercyclicity of the well-known operators such as weighted shifts,
weighted translations, conditional weighted translations and weighted
composition operators in different settings has been studied in [29, 11, 7, 4,
5, 3, 1, 8, 31].

Separability and infinite-dimension are two essential objects for the
underlying space to admit a hypercyclic vector [6, 16]. To that end, it is
important to know that $L^{p}(X,\Sigma,\mu)$ is separable if and only if
$(X,\Sigma,\mu)$ is separable, i.e., there exists a countable
$\sigma$-subalgebra $\mathcal{F}\subseteq\Sigma$ such that for each
$\epsilon>0$ and $A\in\Sigma$ we have $\mu(A\Delta B)<\epsilon$ for some
$B\in\mathcal{F}$. For more details consult [25].

In this paper, we will survey the dynamics of a conditional weighted
composition operator $T_{u}=E^{\mathcal{A}}(uf\circ\varphi)$ on
$L^{p}(\Sigma)$ spaces. First, we prove that $T_{u}$ cannot be
$L^{p}(\mathcal{A})$-hypercyclic if $\varphi$ is a periodic non-singular
transformation. In addition, the necessary conditions for the subspace-
hypercyclicity of $T_{u}$ are then given provided that $\varphi$ is non-
singular and finitely non-mixing. For the sufficient conditions, we also
require that $\varphi$ is normal. The subspace-weakly mixing and subspace-
topologically mixing concepts are also studied for $T_{u}$. At the end, about
what we argued, an examples is given.

##  2\. Subspace-hypercyclicity of $T_{u}$ On $L^{p}(\Sigma)$

In this section, the $L^{p}(\mathcal{A})$-hypercyclicity of a conditional
weighted composition operator $T_{u}$ is studied. When $\varphi$ is periodic
transformation, it is seen that $T_{u}$ is not
$L^{p}(\mathcal{A})$-hypercyclic. But, when it is aperiodic, by Kitai’s
subspace-hypercyclicity criterion we obtain some necessary and then sufficient
conditions for $T_{u}$ to be subspace-hypercyclic. We are thankful to the
techniques used in [11, 29].

######  Theorem 2.1.

Let $\varphi$ be a periodic _non-singular_ transformation and
$\varphi^{-1}\mathcal{A}\subseteq\mathcal{A}$. Then a conditional weighted
composition operator $T_{u}:L^{p}(\Sigma)\rightarrow L^{p}(\mathcal{A})$ is
not subspace-hypercyclic with respect to $L^{p}(\mathcal{A})$, for each $1\leq
p<\infty$.

###### Proof.

Suppose that there exists an $m\in\mathbb{N}$ such that $\varphi^{m}=I$. Since
$\varphi^{-1}\mathcal{A}\subseteq\mathcal{A}$, the orbit of $T_{u}$ at each
$f\in L^{p}(\Sigma)$ is written as follows:

$\displaystyle
orb(T_{u},f)=\\{f,T_{u}f,\cdots,T_{u}^{m}f\\}\cup\\{T_{u}^{m+1}f,T_{u}^{m+2}f,\cdots,T_{u}^{2m}f\\}\cup\cdots$
$\displaystyle\cup\\{T_{u}^{km+1}f,T_{u}^{km+2}f,\cdots,T_{u}^{(k+1)m}f\\}\cup\cdots$
$\displaystyle=\big{\\{}f,E^{\mathcal{A}}(uf\circ\varphi),E^{\mathcal{A}}(u)E^{\mathcal{A}}(uf\circ\varphi)\circ\varphi,\cdots,\prod_{i=0}^{m-2}E^{\mathcal{A}}(u)\circ\varphi^{i}E^{\mathcal{A}}(uf\circ\varphi)\circ\varphi^{m-1}\big{\\}}$
$\displaystyle\cup\big{\\{}\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i}E^{\mathcal{A}}(uf\circ\varphi),\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i}E^{\mathcal{A}}(u)E^{\mathcal{A}}(uf\circ\varphi)\circ\varphi,\cdots,$
$\displaystyle\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i}\prod_{i=0}^{m-2}E^{\mathcal{A}}(u)\circ\varphi^{i}E^{\mathcal{A}}(uf\circ\varphi)\circ\varphi^{m-1}\big{\\}}$
$\displaystyle\cup\big{\\{}(\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i})^{2}E^{\mathcal{A}}(uf\circ\varphi),(\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i})^{2}E^{\mathcal{A}}(u)E^{\mathcal{A}}(uf\circ\varphi)\circ\varphi,\cdots,$
$\displaystyle(\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i})^{2}\prod_{i=0}^{m-2}E^{\mathcal{A}}(u)\circ\varphi^{i}E^{\mathcal{A}}(uf\circ\varphi)\circ\varphi^{m-1}\big{\\}}\cup$
$\displaystyle\vdots$

Now we consider that
$\|\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i}\|_{\infty}\leq 1$.
Since $\|T_{u}\|\leq\|J\|_{\infty}^{1/p}$,
$\|T_{u}^{n}\|\leq\|T_{u}\|^{n}\leq\|J\|_{\infty}^{n/p}$, and for each
$n\in\mathbb{N}$ we have

$\displaystyle\|T_{u}^{n}f\|_{p}$
$\displaystyle\leq\max\\{\|f\|_{p},\|E^{\mathcal{A}}(uf\circ\varphi)\|_{p},\|E^{\mathcal{A}}(u)E^{\mathcal{A}}(uf\circ\varphi)\circ\varphi\|_{p},\cdots,$
$\displaystyle\|\prod_{i=0}^{m-2}E^{\mathcal{A}}(u)\circ\varphi^{i}E^{\mathcal{A}}(uf\circ\varphi)\circ\varphi^{m-1}\|_{p}\\}$
$\displaystyle\leq\|f\|_{p}\max\\{1,\|J\|_{\infty}^{\frac{1}{p}},\|J\|_{\infty}^{\frac{2}{p}},\cdots,\|J\|_{\infty}^{\frac{m-1}{p}}\\}.$

Therefore, $orb(T_{u},f)$ is a bounded subset and cannot be dense in
$L^{p}(\mathcal{A})$.

Iv the second case
$\|\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i}\|_{\infty}>1$, assume
that $T_{u}$ is subspace-hypercyclic with respect to $L^{p}(\mathcal{A})$.
Then there exists a subset $F\in\mathcal{A}$ with $0<\mu(F)<\infty$ for each
$\varepsilon>0$, such that
$|\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i}|>1$. Then there is a
subspace-hypercyclic vector $f\in L^{p}(\mathcal{A})$ and $n\in\mathbb{N}$
such that

$\|f-2\chi_{F}\|_{p}<\varepsilon\quad\mbox{and}\quad\|(T_{u}^{m+1})^{n}f\|_{p}<\varepsilon.$

We set $S=\\{t\in F:|f(t)|<1\\}$ and note that
$\chi_{S}\leq\chi_{S}|f-2|\leq\chi_{S}|f-2\chi_{F}|$. Thus,
$\mu(S)<\varepsilon^{p}$. On the other hand,

$\displaystyle\varepsilon^{p}>\|(T_{u}^{m})^{n}f)\|_{p}^{p}$
$\displaystyle=\int_{X}|\prod_{i=0}^{mn-1}E^{\mathcal{A}}(u)\circ\varphi^{i}f\circ\varphi^{mn}|^{p}d\mu$
$\displaystyle=\int_{X}|\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i}|^{np}|f|^{p}d\mu\geq\int_{F-S}|f|^{p}d\mu\geq\mu(\chi_{F-S}).$

Therefore, $\mu(F)=\mu(S)+\mu(F-S)<2\varepsilon^{p}$, which is a
contradiction. ∎

######  Remark 2.2.

If $\varphi$ is a periodic _non-singular_ transformation,
$\varphi^{-1}\mathcal{A}\subseteq\mathcal{A}$ and $u=1$, then a conditional
composition operator $T_{u}f=E^{\mathcal{A}}(f\circ\varphi)$ is not subspace-
hypercyclic with respect to $L^{p}(\mathcal{A})$ either. Since its orbit at
$f\in L^{p}(\Sigma)$ i.e.,
$orb(T_{u},f)=\\{f,E^{\mathcal{A}}(f\circ\varphi),E^{\mathcal{A}}(f\circ\varphi)\circ\varphi,E^{\mathcal{A}}(f\circ\varphi)\circ\varphi^{2}\cdots,E^{\mathcal{A}}(f\circ\varphi)\circ\varphi^{m-1}\\}$
is a bounded subset. Indeed,

$\|T_{u}^{n}f\|_{p}\leq\|f\|_{p}\max\\{1,\|h\|_{\infty}^{\frac{1}{p}},\|h\|_{\infty}^{\frac{2}{p}},\cdots,\|h\|_{\infty}^{\frac{m-1}{p}}\\}.$

######  Corollary 2.3.

Suppose that $\mathcal{A}=\varphi^{-1}\Sigma$ and $\varphi$ is a periodic
_non-singular_ transformation. Then

$orb(T_{u},f)=\big{\\{}f,E^{\varphi^{-1}\Sigma}(u)f\circ\varphi,E^{\varphi^{-1}\Sigma}(u)E^{\varphi^{-1}\Sigma}(u)\circ\varphi
f\circ\varphi^{2},\cdots,\prod_{i=0}^{m-1}E^{\varphi^{-1}\Sigma}(u)\circ\varphi^{i}f\big{\\}}$

and hence $T_{u}$ is not subspace-hypercyclic with respect to
$L^{p}(\varphi^{-1}\Sigma)$, for each $1\leq p<\infty$.

######  Theorem 2.4.

Let $\varphi:X\rightarrow X$ be a non-singular and finitely non-mixing
transformation and $\varphi^{-1}\mathcal{A}\subseteq\mathcal{A}$. Suppose that
$T_{u}:L^{p}(\Sigma)\rightarrow L^{p}(\mathcal{A})$ is subspace-hypercyclic
with respect to $L^{p}(\mathcal{A})$. Then for each subset $F\in\mathcal{A}$
with $0<\mu(F)<\infty$, there exists a sequence of $\mathcal{A}$-measurable
sets $\\{V_{k}\\}\subseteq F$ such that $\mu(V_{k})\rightarrow\mu(F)$ as
$k\rightarrow\infty$, and there is a sequence of integers $(n_{k})$ such that

$\lim_{k\rightarrow\infty}\|(\prod_{i=0}^{{n_{k}}-1}E^{\mathcal{A}}(u)\circ\varphi^{i})^{-1}|_{V_{k}}\|_{\infty}=0$

and

$\lim_{k\rightarrow\infty}\|\sqrt[p]{h^{\mathcal{A}}_{n_{k}}}[E^{\varphi^{-n_{k}}(\mathcal{A})}(\prod_{i=0}^{{n_{k}}-1}E^{\mathcal{A}}(u)\circ\varphi^{i})]\circ\varphi^{-n_{k}}|_{V_{k}}\|_{\infty}=0.$

###### Proof.

Let $F\in\mathcal{A}$ be an arbitrary set with $0<\mu(F)<\infty$ and let
$\varepsilon>0$ be an arbitrary. A transformation $\varphi$ is finitely non-
mixing and hence, there is an $N\in\mathbb{N}$ such that
$F\cap\varphi^{n}(F)=\emptyset$ for each $n>N$. Choose $\varepsilon_{1}$ such
that $0<\varepsilon_{1}<\frac{\varepsilon}{1+\varepsilon}$. Since the set of
all subspace-hypercyclic vectors for $T_{u}$, is dense in
$L^{p}(\mathcal{A})$, there exist a subspace-hypercyclic vector $f\in
L^{p}(\mathcal{A})$ and $m\in\mathbb{N}$ with $m>N$ such that

$\|f-\chi_{F}\|_{p}<\varepsilon_{1}^{2}\quad\mbox{and}\quad\|T_{u}^{m}f-\chi_{F}\|_{p}<\varepsilon_{1}^{2}.$

Put $P_{\varepsilon_{1}}=\\{t\in F:|f(t)-1|\geq\varepsilon_{1}\\}$ and
$R_{\varepsilon_{1}}=\\{t\in X-F:|f(t)|\geq\varepsilon_{1}\\}$. Then we have

$\displaystyle\varepsilon_{1}^{2p}$ $\displaystyle>$
$\displaystyle\|f-\chi_{F}\|_{p}^{p}=\int_{X}|f-\chi_{F}|^{p}d\mu$
$\displaystyle\geq$
$\displaystyle\int_{P_{\varepsilon_{1}}}|f(x)-1|^{p}d\mu(x)+\int_{R_{\varepsilon_{1}}}|f(x)|^{p}d\mu(x)$
$\displaystyle\geq$
$\displaystyle\varepsilon_{1}^{p}(\mu(P_{\varepsilon_{1}})+\mu(R_{\varepsilon_{1}})).$

Then,
$\max\\{\mu(P_{\varepsilon_{1}}),\mu(R_{\varepsilon_{1}})\\}<\varepsilon_{1}^{p}$.
Set $S_{m,\varepsilon_{1}}=\\{t\in
F:|\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i}f\circ\varphi^{m}(t)-1|\geq\varepsilon_{1}\\}$
and now consider the following relationships:

$\displaystyle\varepsilon_{1}^{2p}$ $\displaystyle>$
$\displaystyle\|T_{u}^{m}f-\chi_{F}\|_{p}^{p}$ $\displaystyle=$
$\displaystyle\int_{X}|\prod_{i=0}^{m-2}E^{\mathcal{A}}(u)\circ\varphi^{i}E^{\mathcal{A}}(uf\circ\varphi)\circ\varphi^{m-1}-\chi_{F}|^{p}d\mu$
$\displaystyle\geq$
$\displaystyle\int_{S_{m,\varepsilon_{1}}}|\prod_{i=0}^{m-2}E^{\mathcal{A}}(u)\circ\varphi^{i}E^{\mathcal{A}}(uf\circ\varphi)\circ\varphi^{m-1}(t)-1|^{p}d\mu(t)$
$\displaystyle\geq$
$\displaystyle\int_{S_{m,\varepsilon_{1}}}|\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i}f\circ\varphi^{m}(t)-1|^{p}d\mu(t)$
$\displaystyle\geq$
$\displaystyle\varepsilon_{1}^{p}\mu(S_{m,\varepsilon_{1}})$

to deduce that $\mu(S_{m,\varepsilon_{1}})<\varepsilon_{1}^{p}$. But for an
arbitrary $t\in F$, it is readily seen that $\varphi^{m}(t)\notin F$ because
of $F\cap\varphi^{-m}(F)=\emptyset$. Hence, for each $t\in
F-(S_{m,\varepsilon_{1}}\cup\varphi^{-m}(R_{\varepsilon_{1}}))$, we have

$|(\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i})^{-1}(t)|<\frac{|f\circ\varphi^{m}(t)|}{1-\varepsilon_{1}}<\frac{\varepsilon_{1}}{1-\varepsilon_{1}}<\varepsilon.$

Now, let $U_{m,\varepsilon_{1}}=\varphi^{-m}(\\{t\in
F:\sqrt[p]{h^{\mathcal{A}}_{m}(t)}\
|E^{\varphi^{-m}(\mathcal{A})}(\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i})\circ\varphi^{-m}(t)f(t)|\geq\varepsilon_{1}\\})$.
Here, we remind that
$\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i}\circ\varphi^{-m}=\prod_{i=1}^{m}E^{\mathcal{A}}(u)\circ\varphi^{-i}$
on $\sigma(h_{m}^{\mathcal{A}})$. Use the change of variable formula to obtain
that

$\displaystyle\varepsilon_{1}^{2p}$ $\displaystyle>$
$\displaystyle\|T_{u}^{m}f-\chi_{F}\|_{p}^{p}$ $\displaystyle=$
$\displaystyle\int_{X}|\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i}f\circ\varphi^{m}-\chi_{F}|^{p}d\mu$
$\displaystyle\geq$
$\displaystyle\int_{X}|E^{\varphi^{-m}(\mathcal{A})}(\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i})f\circ\varphi^{m}-E^{\varphi^{-m}(\mathcal{A})}(\chi_{F})|^{p}d\mu$
$\displaystyle\geq$
$\displaystyle\int_{U_{m,\varepsilon_{1}}}|E^{\varphi^{-m}(\mathcal{A})}(\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i})f\circ\varphi^{m}|^{p}d\mu$
$\displaystyle\geq$
$\displaystyle\int_{\varphi^{m}(U_{m,\varepsilon_{1}})}|E^{\varphi^{-m}(\mathcal{A})}(\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i})\circ\varphi^{-m}f|^{p}h_{m}^{\mathcal{A}}d\mu$
$\displaystyle\geq$
$\displaystyle\varepsilon_{1}^{p}\mu(\varphi^{m}(U_{m,\varepsilon_{1}})),$

which implies in turn that
$\mu(\varphi^{m}(U_{m,\varepsilon_{1}}))<\varepsilon_{1}^{p}$. That
$E^{\varphi^{-m}(\mathcal{A})}(\chi_{F})=0$ is concluded of the fact that
$F\cap\varphi^{-m}(F)=\emptyset$. Note that for each $t\in
F-(\varphi^{m}(U_{m,\varepsilon_{1}})\cup P_{\varepsilon_{1}})$, we have

$\sqrt[p]{h_{m}(t)}\
|E^{\varphi^{-m}(\mathcal{A})}(\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i})\circ\varphi^{-m}(t)f(t)|<\frac{\varepsilon_{1}}{1-\varepsilon_{1}}<\varepsilon.$

Finally, put
$V_{m,\varepsilon_{1}}:=F-(P_{\varepsilon_{1}}\cup\varphi^{-m}(R_{m,\varepsilon_{1}})\cup
S_{m,\varepsilon_{1}}\cup\varphi^{m}(U_{m,\varepsilon_{1}}))$. Then, clearly
$\mu(F-V_{m,\varepsilon_{1}})<4\varepsilon_{1}^{p}$,
$\|(\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i})^{-1}|_{V_{m,\varepsilon_{1}}}\|_{\infty}<\varepsilon$
and

$\|\sqrt[p]{h_{m}}[E^{\varphi^{-m}(\mathcal{A})}(\prod_{i=0}^{m-1}E^{\mathcal{A}}(u)\circ\varphi^{i})]\circ\varphi^{-m}|_{V_{m,\varepsilon_{1}}}\|_{\infty}<\varepsilon.$

By induction, for each $k\in\mathbb{N}$ we get a measurable subset
$V_{k}\subseteq F$ and an increasing subsequence $(n_{k})$ such that
$\mu(F-V_{k})<4(\frac{1}{k})^{p}$,
$\|(\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i})^{-1}|_{V_{k}}\|_{\infty}<\varepsilon$
and
$\|\sqrt[p]{h^{\mathcal{A}}_{n_{k}}}[E^{\varphi^{-n_{k}}(\mathcal{A})}(\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i})]\circ\varphi^{-n_{k}}|_{V_{k}}\|_{\infty}<\varepsilon$.
∎

######  Theorem 2.5.

Let $T_{u}:L^{p}(\Sigma)\rightarrow L^{p}(\mathcal{A})$ be bounded with
$\sigma(u)=X$, and let $\varphi$ be a normal and finitely non-mixing
transformation provided that
$\varphi^{-1}\mathcal{A}\subseteq\mathcal{A}\subseteq\Sigma_{\infty}$ and
$\sup_{n}\|h_{n}^{\mathcal{A}\sharp}\|_{\infty}<\infty$. If for each subset
$F\in\mathcal{A}$ with $0<\mu(F)<\infty$, there exists a sequence of
$\mathcal{A}$-measurable sets $\\{V_{k}\\}\subseteq F$ such that
$\mu(V_{k})\rightarrow\mu(F)$ as $k\rightarrow\infty$, and there is a sequence
of integers $(n_{k})$ such that

$\lim_{k\rightarrow\infty}\|(\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i})^{-1}|_{V_{k}}\|_{\infty}=0$

and

$\lim_{k\rightarrow\infty}\|\sqrt[p]{h_{n_{k}}}\
[\prod_{i=0}^{{n_{k}}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}]\circ\varphi^{-n_{k}}|_{V_{k}}\|_{\infty}=0,$

then $T_{u}$ is subspace-hypercyclic with respect to $L^{p}(\mathcal{A})$.

###### Proof.

Since, $S^{\mathcal{A}}(X)$ is dense in $L^{p}(\mathcal{A})$, we may take
$D_{1}=D_{2}=S^{\mathcal{A}}(X)$ in the subspace-hypercyclicity’s criterion.
For an arbitrary $f\in S^{\mathcal{A}}(X)$, one can easily find
$\\{V_{k}\\}\subseteq\sigma(f)$ such that
$\mu(V_{k})\rightarrow\mu(\sigma(f))$ and finds an $N_{1}$ such that
$\sigma(f)\cap\varphi^{n}(\sigma(f))=\emptyset$ for each $n>N_{1}$. Now, for
each $n_{k}>N_{1}$ define the vector
$f_{k}=\frac{f\circ\varphi^{-n_{k}}}{[\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}]\circ\varphi^{-n_{k}}}$.
Since $\varphi^{-1}\mathcal{A}\subseteq\mathcal{A}\subseteq\Sigma_{\infty}$,
then $f_{k}\in L^{p}(\mathcal{A})$ and the simple computations show that
$T_{u}^{n_{k}}f_{k}=f$. Now, we will show that
$\|T_{u}^{n_{k}}f\|_{p}\rightarrow 0$ and $\|f_{k}\|_{p}\rightarrow 0$ as
$k\rightarrow\infty$. For an arbitrary $\varepsilon>0$, there exist
$M,N_{1}\in\mathbb{N}$, sufficiently large such that
$V_{N_{1}}\subseteq\sigma(f)$ and

$\mu(\sigma(f)-V_{N_{1}})<\frac{\varepsilon}{M\|f\|_{{}_{\infty}}^{p}}.$

By Egoroff’s theorem, there exists an $N_{2}$ such that for each
$n_{k}>N_{2}$,
$\|\sqrt[p]{h^{\mathcal{A}}_{n_{k}}}[\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}]\circ\varphi^{-n_{k}}\|^{p}_{\infty}<\frac{\varepsilon}{\|f\|^{p}_{\infty}}$
on $V_{N_{1}}$. So, there exists a non-negative real number $M$ such that
$\|\sqrt[p]{h^{\mathcal{A}}_{n_{k}}}[\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}]\circ\varphi^{-n_{k}}\|_{\infty}^{p}\leq
M<\infty$ on $\sigma(f)$. Now, by the change of variable formula, for each
$n_{k}>N=\max\\{N_{1},N_{2}\\}$ we have

$\displaystyle\|T_{u}^{n_{k}}f\|_{p}^{p}$
$\displaystyle=\int_{X}|\prod_{i=0}^{n_{k}-2}E^{\mathcal{A}}(u)\circ\varphi^{i}E^{\mathcal{A}}(uf\circ\varphi)\circ\varphi^{n_{k}-1}|^{p}d\mu$
$\displaystyle=\int_{X}|\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}f\circ\varphi^{n_{k}}|^{p}d\mu$
$\displaystyle=\int_{\sigma(f)}|\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}\circ\varphi^{-n_{k}}f|^{p}h_{n_{k}}d\mu$
$\displaystyle=\int_{{\sigma(f)-V_{N}}}|\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}\circ\varphi^{-n_{k}}f|^{p}h_{n_{k}}d\mu$
$\displaystyle+\int_{V_{N}}|\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}\circ\varphi^{-n_{k}}f|^{p}h_{n_{k}}d\mu$
$\displaystyle<\|\sqrt[p]{h_{n_{k}}}\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}\circ\varphi^{-n_{k}}\|_{\infty}^{p}\|f\|_{\infty}^{p}\
\mu(\sigma(f)-V_{N})$
$\displaystyle+\frac{\varepsilon}{\|f\|_{\infty}^{p}}\|f\|_{\infty}^{p}<2\varepsilon.$

By taking into account that
$\sup_{n}\|h_{n}^{\mathcal{A}\sharp}\|_{\infty}<\infty$, we have

$\displaystyle\lim_{k\rightarrow\infty}\|f_{k}\|_{p}^{p}$
$\displaystyle=\lim_{k\rightarrow\infty}\int_{X}|\frac{f\circ\varphi^{-n_{k}}}{\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}\circ\varphi^{-n_{k}}}|^{p}d\mu$
$\displaystyle=\lim_{k\rightarrow\infty}\int_{\sigma(f)}|\frac{f}{\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}}|^{p}h_{n_{k}}^{\sharp}d\mu$
$\displaystyle\leq\sup_{k}\|h_{n_{k}}^{\mathcal{A}\sharp}\|_{{}_{\infty}}(\lim_{k\rightarrow\infty}\int_{\sigma(f)-V_{N}}|\frac{f}{\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}}|^{p}d\mu$
$\displaystyle+\lim_{k\rightarrow\infty}\int_{V_{N}}|\frac{f}{\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}}|^{p}d\mu)$
$\displaystyle=0.$

Finally, it is clear that $T_{u}^{n_{k}}L^{p}(\mathcal{A})\subseteq
L^{p}(\mathcal{A})$ for all $k\in\mathbb{N}$, because of
$\varphi^{-1}\mathcal{A}\subseteq\mathcal{A}$ and hence $T_{u}$ satisfies in
the subspace-hypercyclicity criterion and is subspace-hypercyclic. ∎

######  Proposition 2.6.

Suppose that $\varphi:X\rightarrow X$ is a normal and finitely non-mixing
transformation with
$\varphi^{-1}(\mathcal{A})\subseteq\mathcal{A}\subseteq\Sigma_{\infty}$. Let
$\sup_{n}\|h_{n}^{\sharp}\|_{\infty}<\infty$ and $\sigma(u)=X$. Then the
following conditions are equivalent:

  * (i)

$T_{u}$ satisfies the subspace-hypercyclic criterion.

  * (ii)

$T_{u}$ is subspace-hypercyclic with respect to $L^{p}(\mathcal{A})$.

  * (iii)

$T_{u}\oplus T_{u}$ is subspace-hypercyclic with respect to
$L^{p}(\mathcal{A})\oplus L^{p}(\mathcal{A})$.

  * (iv)

$T_{u}$ is subspace-weakly mixing.

###### Proof.

$(i)\Rightarrow(ii)$. Note that if an operator satisfies the subspace-
hypercyclic criterion, then it is subspace-transitive and hence is subspace-
hypercyclic [24, Theorem 3.5]. For the implication $(ii)\Rightarrow(iii)$, we
show that $T_{u}\oplus T_{u}$ is subspace-topologically transitive, according
[24, Theorem 3.3]. To begin, pick two pairs of non-empty open sets
$(A_{1},B_{1})$ and $(A_{2},B_{2})$ in $L^{p}(\mathcal{A})\oplus
L^{p}(\mathcal{A})$ arbitrarily. For $j=1,2$, choose the functions
$f_{j},g_{j}\in S^{\mathcal{A}}(X)$ with $f_{j}\in A_{j}$ and $g_{j}\in
B_{j}$. Let
$F=\sigma(f_{1})\cup\sigma(f_{2})\cup\sigma(g_{1})\cup\sigma(g_{2})$. Then
$\mu(F)<\infty$. Assume that $\\{V_{k}\\}\subseteq F$,
$\\{(\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i})^{-1}\\}$ and
$\\{\sqrt[p]{h^{\mathcal{A}}_{n_{k}}}\
E^{\varphi^{-n_{k}}(\mathcal{A})}(\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i})\circ\varphi^{-n_{k}}\\}$
are as provided by Theorem 2.4. There is an $N_{1}\in\mathbb{N}$, such that
for all $n>N_{1}$, $F\cap\varphi^{n}(F)=\emptyset$. Moreover, for each
$\varepsilon>0$ there exists $N_{2}\in\mathbb{N}$, such that for each
$k>N_{2}$ and $n_{k}>N_{1}$, $\|\sqrt[p]{h^{\mathcal{A}}_{n_{k}}}\
E^{\varphi^{-n_{k}}(\mathcal{A})}(\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i})\circ\varphi^{-n_{k}}|_{V_{k}}\|_{\infty}^{p}<\frac{\varepsilon}{\|f_{j}\|_{p}^{p}}$
on $V_{k}$. Hence, for $k>N_{2}$, we get that

$\displaystyle\|T_{u}^{n_{k}}(f_{j}\chi_{V_{k}})\|_{p}^{p}$
$\displaystyle=\int_{X}|T_{u}^{n_{k}}(f_{j}\chi_{V_{k}})|^{p}d\mu$
$\displaystyle=\int_{X}|\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}(f_{j}\chi_{V_{k}})\circ\varphi^{n_{k}})|^{p}\
d\mu$
$\displaystyle=\int_{V_{k}}|[\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}]\circ\varphi^{-n_{k}}f_{j}|^{p}h_{n_{k}}d\mu<\varepsilon.$

Now, define a map
$D_{\varphi}(f)=\frac{f\circ\varphi^{-1}}{E^{\mathcal{A}}(u)\circ\varphi^{-1}}$
on the subspace $S^{\mathcal{A}}(X)$. Then for each $f\in S^{\mathcal{A}}(X)$,
$T_{u}^{n_{k}}D_{\varphi}^{n_{k}}(f)=f$. Again, we may find an
$N_{3}\in\mathbb{N}$ such that for each $k>N_{3}$ and $n_{k}>N_{1}$,
$\|(\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i})^{-1}\|_{\infty}^{p}<\frac{\varepsilon}{M\|g_{j}\|_{\infty}^{p}}$
on $V_{k}$, where $M=\sup_{n}\|h_{n}^{\mathcal{A}\sharp}\|_{\infty}<\infty$.
On the other hand, for each $k>N_{3}$ note that

$\displaystyle\|D_{\varphi}^{n_{k}}(g_{j}\chi_{V_{k}})\|_{p}^{p}$
$\displaystyle=\int_{\varphi^{n_{k}}(V_{k})}|\frac{g_{j}\circ\varphi^{-n_{k}}}{[\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}]\circ\varphi^{-n_{k}}}|^{p}d\mu$
$\displaystyle=\int_{V_{k}}|\frac{g_{j}}{\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}}|^{p}\
h_{n}^{\sharp}d\mu<\varepsilon.$

For each $k\in\mathbb{N}$, let
$f^{\natural}_{j,k}=f_{j}\chi_{V_{k}}+D_{\varphi}^{n_{k}}(g_{j}\chi_{V_{k}})$.
Then we have $f^{\natural}_{j,k}\in L^{p}(\mathcal{A})$,

$\|f^{\natural}_{j,k}-f_{j}\|_{p}^{p}\leq\|f_{j}\|_{\infty}^{p}\
\mu(F-V_{k})+\|D_{\varphi}^{n_{k}}(g_{j}\chi_{V_{k}})\|_{p}^{p}$

and

$\|T_{u}^{n_{k}}f^{\natural}_{j,k}-g_{j}\|_{p}^{p}\leq\|g_{j}\|_{\infty}^{p}\
\mu(F-V_{k})+\|T_{u}^{n_{k}}(f_{j}\chi_{V_{k}})\|_{p}^{p}.$

Hence, $\lim_{k\rightarrow\infty}f^{\natural}_{j,k}=f_{j}$,
$\lim_{k\rightarrow\infty}T_{u}^{n_{k}}f^{\natural}_{j,k}=g_{j}$ and
$T_{u}^{n_{k}}(A_{j})\cap B_{j}\neq\emptyset$ for some $k\in\mathbb{N}$.
Moreover, since $\varphi^{-1}(\mathcal{A})\subseteq\mathcal{A}$ then
$T_{u}^{n_{k}}(L^{p}(\mathcal{A}))\subseteq L^{p}(\mathcal{A})$. So
$T_{u}\oplus T_{u}$ is subspace-hypercyclic on $L^{p}(\mathcal{A})\oplus
L^{p}(\mathcal{A})$.

To prove the implication $(iv)\Rightarrow(i)$, we use B$\grave{e}$s-Peris’s
approach stated in [6, Theorem 4.2]. Assume that $T_{u}\oplus T_{u}$ is
subspace-hypercyclic on $L^{p}(\mathcal{A})\oplus L^{p}(\mathcal{A})$ with
subspace-hypercyclic vector $f\oplus g$. Note that for each $n\in\mathbb{N}$,
the operator $I\oplus T_{u}^{n}$ has dense range and commutes with
$T_{u}\oplus T_{u}$, therefore $orb(I\oplus T_{u}^{n},f\oplus g)=(I\oplus
T_{u}^{n})orb(T_{u}\oplus T_{u},f\oplus g)$. Eventually $f\oplus T_{u}^{n}g$
is subspace-hypercyclic vector as well. We show that the subspace-hypercyclic
criterion is satisfied by $D_{1}=D_{2}=orb(T_{u}\oplus T_{u},f\oplus g)$. Let
$U$ be an arbitrary open neighborhood of $0$ in $L^{p}(\mathcal{A})$. Hence,
one can find a sequence $(g_{k})\subseteq U$ and an increasing sequence of
integers $(n_{k})$ such that $T_{u}^{n_{k}}f\oplus
T_{u}^{n_{k}}g_{k}\rightarrow 0\oplus g$ and $g_{k}\rightarrow 0$. Clearly,
$T_{u}^{n_{k}}(L^{p}(\mathcal{A}))\subseteq L^{p}(\mathcal{A})$. ∎

######  Corollary 2.7.

Under the assumptions of Proposition 2.6, the following conditions are
equivalent:

  * (i)

$T_{u}$ is subspace-topologically mixing on $L^{p}(\mathcal{A})$.

  * (ii)

For each $\mathcal{A}$-measurable subset $F\subseteq X$ with
$0<\mu(F)<\infty$, there exists a sequence of $\mathcal{A}$-measurable sets
$\\{V_{n}\\}\subseteq F$ such that $\mu(V_{n})\rightarrow\mu(F)$ as
$n\rightarrow\infty$ and
$\lim_{n\rightarrow\infty}\|(\prod_{i=0}^{n-1}E^{\mathcal{A}}(u)\circ\varphi^{i})^{-1}|_{V_{n}}\|_{\infty}=\lim_{n\rightarrow\infty}\|\sqrt[p]{h^{\mathcal{A}}_{n}}(\prod_{i=0}^{n-1}E^{\mathcal{A}}(u)\circ\varphi^{i}\circ\varphi^{-{n}})|_{V_{n}}\|_{\infty}=0$.

###### Proof.

By Theorem 2.5 and Proposition 2.6 the implication $(ii)\Rightarrow(i)$ is
established, just use the full sequences instead of subsequences. For the
implication $(i)\Rightarrow(ii)$, let $\varepsilon>0$ and $F\in\mathcal{A}$
with $0<\mu(F)<\infty$ be arbitrary. Consider a non-empty and open subset
$U=\\{f\in L^{p}(\mathcal{A}):\|f-\chi_{F}\|_{p}<\varepsilon\\}$. Since
$T_{u}$ is subspace-topologically mixing and $\varphi$ is finitely non-mixing,
one may find $N\in\mathbb{N}$ such that for all $n>N$, $T_{u}^{n}(U)\cap
U\neq\emptyset$ and $F\cap\varphi^{n}(F)=\emptyset$. Hence, for each $n>N$, we
can choose a function $f_{n}\in U$ such that $T_{u}^{n}f_{n}\in U$. Then
$\|f_{n}-\chi_{F}\|_{p}<\varepsilon$ and
$\|T_{u}^{n}f_{n}-\chi_{F}\|_{p}<\varepsilon$. The rest of the proof can be
proceed like as Theorem 2.4. ∎

######  Example 2.8.

Let $X=\mathbb{R}$ be the real line with Lebesgue measure $\mu$ on the
$\sigma$-algebra $\Sigma$ of all Lebesgue measurable subsets of $\mathbb{R}$.
Let $\mathcal{A}$ be the $\sigma$-subalgebra generated by the symmetric
intervals about the origin. For a positive real number $t$ define the
transformation $\varphi:\mathbb{R}\rightarrow\mathbb{R}$ by
$\varphi(x)=x+t,~{}~{}x\in\mathbb{R}$. Clearly,
$\varphi^{-1}\mathcal{A}\subseteq\mathcal{A}\subseteq\Sigma_{\infty}$ and in
this setting, $E^{\mathcal{A}}(f)=\frac{f(x)+f(-x)}{2}$, which is the even
part of $f\in L^{p}(\Sigma)$. Fix $r>1$ and define the weight function $u$ on
$\mathbb{R}$ by

$u(x)=\left\\{\begin{array}[]{ll}2x+r,&{1\leq x},\\\
-x^{2}-\frac{x}{2}+2,&{-1<x<1},\\\
x^{3}+\frac{1}{r},&{x\leq-1}.\end{array}\right.$

Then, we have

$E^{\mathcal{A}}(u)(x)=\left\\{\begin{array}[]{lll}r,&1\leq x,\\\
-\frac{x^{2}}{2}+2,&-1<x<1,\\\ \frac{1}{r},&x\leq-1.\end{array}\right.$

For an arbitrary $F=[-a,a]$, take $V_{k}=(-a+\frac{1}{k},a-\frac{1}{k})$. In
this case, one may easily find a sequence $(n_{k})$ such that both quantities
$\|(\prod_{i=0}^{n_{k}-1}E^{\mathcal{A}}(u)\circ\varphi^{i})^{-1}|_{V_{k}}\|_{\infty}$
and $\|\sqrt[p]{h^{\mathcal{A}}_{n_{k}}}\
[\prod_{i=0}^{{n_{k}}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}]\circ\varphi^{-n_{k}}|_{V_{k}}\|_{\infty}$
tend zero as $k\rightarrow\infty$. Because,
$h^{\mathcal{A}}_{n_{k}}=h^{\mathcal{A}^{\sharp}}_{n_{k}}=1$ and
$[\prod_{i=0}^{{n_{k}}-1}E^{\mathcal{A}}(u)\circ\varphi^{i}]\circ\varphi^{-n_{k}}=\prod_{i=1}^{{n_{k}}}E^{\mathcal{A}}(u)\circ\varphi^{-i}$,
since $\varphi$ is onto (or $\sigma(h^{\mathcal{A}}_{n_{k}})=\mathbb{R}$).
Therefore, by Theorem 2.5, $T_{u}$ is subspace-hypercyclic with respect to
$L^{p}(\mathcal{A})$ while it is not hypercyclic on $L^{p}(\Sigma)$ [5,
Theorem 2.3]. For this, just consider that
$\|\sqrt[p]{h_{n_{k}}}[E_{n_{k}}(\prod_{i=0}^{{n_{k}}-1}u\circ\varphi^{i})]\circ\varphi^{-n_{k}}|_{V_{k}}\|_{\infty}=\|\prod_{i=1}^{{n_{k}}}u\circ\varphi^{-i}|_{V_{k}}\|_{\infty}\nrightarrow
0$.

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