11institutetext: University College London 11email<EMAIL_ADDRESS>
# A Filtering-based General Approach to Learning Rational Constraints of
Epistemic Graphs

Xiao Chi

###### Abstract

Epistemic graphs generalize the epistemic approach to probabilistic
argumentation and tackle the uncertainties in and between arguments. A
framework was proposed to generate epistemic constraints from data using a
two-way generalization method in the perspective of only considering the
beliefs of participants without considering the nature of relations
represented in an epistemic graph. The deficiency of original framework is
that it is unable to learn rules using tighter constraints, and the learnt
rules might be counter-intuitive. Meanwhile, when dealing with more restricted
values, the filtering computational complexity will increase sharply, and the
time performance would become unreasonable. This paper introduces a filtering-
based approach using multiple-way generalization step to generate a set of
rational rules based on both the beliefs of each agent on different arguments
and the epistemic graph corresponding to the epistemic constraints. This
approach is able to generate rational rules with multiple restricted values in
higher efficiency. Meanwhile, we have proposed a standard to analyze the
rationality of a dataset based on the postulates of deciding rational rules.
We evaluate the filtering-based approach on two suitable databases. The
empirical results show that the filtering-based approach performs well with a
better efficiency compared to the original framework, and rules generated from
the improved approach are ensured to be rational.

###### Keywords:

Epistemic constraint rule learning abstract argumentation probabilistic
argumentation.

##  1 Introduction

Argumentation plays an important role in daily life when there exist conflicts
containing inconsistent and incomplete information [10]. Meanwhile,
argumentation is often pervaded by uncertainty either within an argument or
between arguments. Researchers are working on dealing with the uncertainty and
one solution is to quantify the uncertainty of arguments by assigning
probability values to each argument to represent the degree of beliefs an
agent has in the argument [9, 14, 20, 8]. The rise of the epistemic graphs
[12] presents a new formalism that further develops the fundamental
probabilistic argumentation by tackling the problem of only modelling attack
relations since the epistemic graph models support, attack and neither support
nor attack relations as well as how a specified argument is influenced by
other arguments using epistemic constraints. A framework proposed by Anthony
Hunter [11] provides a pathway for learning epistemic constraints from data.
Here, each constraint is in the form of a rule. Note that rules generated from
the framework are based completely on the tabular data after knowing the
influence tuples so that they only represent the beliefs of agents. However,
people often require knowledge for learning rules not only from tabular data
but also from natural language data since the former is to generate epistemic
constraints based on the beliefs of agents while the latter can reflect the
nature of relations represented in an epistemic graph. Examples for the
natural language data and the tabular data are represented in Figure 1 and
Table 1 respectively. In addition, the framework is primary since it can only
be implemented to generate constraints for the restricted value 0.5 based on
two-way generalization step. An example of generating epistemic constraints
using the two-way generalization step is stated in Example 1. However, in
reality, it is essential for epistemic graphs to represent various degrees of
beliefs for an argument instead of just knowing whether an argument is
believed or not. Meanwhile, the problem that follows is that when dealing with
more restricted values, the complexity of generating rules and filtering the
best and simplest rules will increase sharply, and the time performance would
become unacceptable.

Figure 1: The epistemic graph corresponding to Table 1 with the influence
tuple $(\\{Dw2,Dw5,Dw3\\},Dw6)$. Here, $Dw6$ denotes "The society has become
more lawless and bestial nowadays", $Dw5$ denotes "The society isn’t dangerous
nowadays", $Dw2$ denotes "Chaos and anarchy can erupt around us nowadays",
$Dw3$ denotes "There exists dangerous people in current society". The $+$
label denotes support while the $-$ label denotes attack. We call the
relations containing in the epistemic graph the natural language data.

| Dw6 | Dw2 | Dw5 | Dw3
---|---|---|---|---
004 | 0.2 | 0.3 | 0.3 | 0.3
026 | 0.4 | 0.6 | 0.3 | 0.6
111 | 0.6 | 0.1 | 0.6 | 0.2

Table 1: Containing columns and rows of data obtaining from the Italian study
after changing the Likert values to probability values. The probability values
represent the beliefs of agents on statements. The epistemic graph for the
data is stated in Figure 1. We call the data in the table the tabular data.

###### Example 1

Consider Table 1. The epistemic constraint generated from row $026$ for the
influence tuple $(\\{Dw2,Dw5,Dw3\\},Dw6)$ is presented by the following
formula: $p(Dw2)>0.5\wedge p(Dw5)<0.5\wedge p(Dw3)>0.5\rightarrow p(Dw6)<0.5.$
Note that this constraint is counter-intuitive in terms of the support and
attack relations between the arguments in Figure 1. It can only be identified
by considering the argument graph that is obtained from the natural language
data.

The above analysis gives rise to the following three research questions.

  * 1.

How to develop a method without generating counter-intuitive rules?

  * 2.

How to make the method more general such that various degrees of beliefs can
be represented?

  * 3.

How to improve the efficiency of computation?

To realize these three research questions, we set forward a filtering-based
approach using a multiple-way generalization step. The filtering algorithm for
rational rules ensures the resulting rules to be rational. After adding the
filtering algorithm, to decide whether a rule is suitable, one not only
considers parameters Lift, Support and Confidence which are measurements to
evaluate how well a rule performs, but also the rational postulates that will
be proposed later. Another benefit of the filtering algorithm is that it
increases the efficiency of generating rules since fewer data enter the
process of generating rules based on the restricted values. The multiple-way
generalization step allows us to generate constraints based on a general
restricted value set, i.e., the number of restricted values is not limited.
Besides generating rational rules from a dataset, we discover a standard to
decide the rationality of a dataset based on the rationality of the resulting
rules generated from the original framework. We evaluate our approach on the
dataset about political attitudes in Italy [19] and the dataset about the
appropriateness of using Wikipedia in teaching in Higher Education [16]. We
then compare the results generated from two different rule generalization
frameworks and the performances of two datasets.

After the introduction, the remaining part of this paper is proceeded as
follows: Section 2 reviews the preliminary knowledge required for this paper.
Section 3 introduces a filtering-based approach using a multiple-way
generalization step method by providing new definitions. Section 4 evaluates
the improved approach on two datasets by comparing it with the ordinary
framework and the performances of the two datasets. Section 5 discusses the
contributions, limitations and future work of this paper.

##  2 Preliminaries

Dung developed a theory of argumentation with only the notions of arguments
and attacks [8]. This theory is based on the idea that regards the
acceptability of arguments as the central notion. Here, we introduce the
definition of extensions of an abstract argumentation framework (AF for
short), in terms of the notions of acceptability and conflict-freeness.

###### Definition 1

An AF is a tuple $AF=\langle A,R_{att}\rangle$, where $A$ is a set of
arguments, and $R_{att}\subseteq A\times A$ is the attack relation over $A$.

###### Definition 2

An argument $a\in A$ is acceptable w.r.t a set $S\subseteq A$ of arguments if
and only if for each argument $b\in A$: if $b$ attacks $a$ then $b$ is
attacked by one argument in $S$.

###### Definition 3

A set of arguments S is conflict-free if for all $a,b\in S$, $a$ does not
attack $b$.

###### Definition 4

Let $AF=\langle A,R_{att}\rangle$ be an AF, $S\subseteq A$ be a set of
arguments. $S$ is admissible iff $S$ is conflict-free and each argument in $S$
is acceptable w.r.t $S$. $S$ is a complete extension of $AF$ iff $S$ is
admissible and each argument in $A$ that is acceptable w.r.t $S$ is in $S$.
$S$ is a preferred extension of $AF$ iff $S$ is a maximal complete extension
(w.r.t set inclusion). $S$ is the grounded extension of $AF$ iff $S$ is the
minimal complete extension (w.r.t set inclusion).

According to Definition 4, we may define a function satisfying some
principles, called argumentation semantics, mapping each AF to a set of
extensions.

One of the principles called reinstatement principle is satisfied by complete
semantics. This principle says that for each AF and for each extension $E$ of
this AF, if an argument $a$ in this AF is acceptable w.r.t $E$, then $a$
should be in $E$. For more information about argumentation semantics and
principles, the readers are referred to [2].

For Dung’s abstract argumentation framework, interactions between two
arguments are limited to conflicts. The bipolar framework proposed by Cayrol
and Lagasquie-Schiex [4] and Amgoud et al. [1] deals with the limitation by
modelling a positive interaction called the support relation. Let us look at
two proposals that specialize the support relation, which are deductive
support [3] and necessary support [17, 18].

###### Definition 5

Let $A$ be a finite and non-empty set of arguments, $a,b\in A$ be two
arguments, $R_{att}$ be a binary relation over $A$ called the attack relation
and $R_{sup}$ be a binary relation over $A$ called the support relation.
$aR_{sup}b$ is defined as $a$ supports $b$ and $aR_{att}b$ is defined as $a$
attacks $b$.

The deductive supports need to satisfy the following constraint. Let a, b be
two arguments. If $a$ supports $b$, then the acceptance of $a$ implies the
acceptance of $b$, and the non-acceptance of $b$ implies the non-acceptance of
$a$.

There is another important property that can be derived from the fact that the
introduction of the supported attacks can be justified by the support being
interpreted deductively [5]. This is presented as follows. Let a, b and c be
three arguments. If $aR_{sup}c$ and $cR_{att}b$, then the acceptance of $a$
implies the acceptance of $c$ while the acceptance of $c$ implies the non-
acceptance of $b$. In summary, the acceptance of $a$ implies the non-
acceptance of $b$.

The necessary supports can be expressed as the following interpretation [6]:

  * •

If $a$ supports $b$, then the acceptance of $a$ is necessary to obtain the
acceptance of $b$, i.e., the acceptance of $b$ implies the acceptance of $a$.

  * •

If $aR_{att}c$ and $cR_{sup}b$, then the acceptance of $a$ leads to the non-
acceptance of $c$ and therefore leads to the non-acceptance of $b$.

  * •

If $cR_{sup}a$ and $cR_{att}b$, then the acceptance of $a$ implies the
acceptance of $c$ while the acceptance of $c$ implies the non-acceptance of
$b$. In summary, the acceptance of $a$ implies the non-acceptance of $b$.

We have introduced two kinds of abstract frameworks. Various recent studies
are working on enhancing the abstract argumentation by improving the
evaluation of argument graphs [13]. We will now focus on introducing the
probabilistic argumentation in the aspect of epistemic extensions since it
puts forward a practical framework that generates a fine-grained way to
represent uncertainties in arguments and acts as the foundation of our
framework. Hunter and Thimm [13] proposed an approach to denote the belief of
an argument such that the premises of an argument and the claim derived from
the premises are both valid. The following is how the approach decides whether
to believe an argument or not. We denote $p$ as a probability function and $a$
as an argument.

  * •

If $p(a)>0.5$, then the argument $a$ is believed.

  * •

If $p(a)<0.5$, then the argument $a$ is disbelieved.

  * •

If $p(a)=0.5$, then the argument $a$ is neither believed nor disbelieved.

Hunter and Thimm [13] introduced epistemic extensions from two different
views, the standard and non-standard views. We will look at the coherent
constraint which is consistent with definitions raised by Dung [7]. The
definition for the coherent property is defined below.

###### Definition 6

Let $AF=\langle A,R_{att}\rangle$ be an AF and $P:2^{A}\rightarrow[0,1]$. P is
coherent (COH) with respect to $AF$ if for all arguments $a,b\in A$ such that
$a$ attacks $b$, then $P(a)\leq 1-P(b)$.

Hunter et al. [12] made a progress by introducing epistemic graphs as the
generalization of the epistemic approach to probabilistic argumentation.
Epistemic graphs are labelled graphs that are associated with a set of
epistemic constraints [12]. In many situations, instead of focusing on the
full power of the epistemic language, people tend to focus on a restricted
language since the previous one might be redundant when handling a finite set
of probability values [12]. According to Hunter [11], the restricted epistemic
language not only makes the presentation and evaluation simpler but also
generates a finite set of distributions. The following are two main
definitions related to the restricted language.

###### Definition 7

Let $\Pi$ be a unit interval and $1\in\Pi$. A finite set of rational numbers
from $\Pi$ is a restricted value set iff for every $x,y\in\Pi$, it satisfies
the following constraints: 1) If $x+y\leq 1$, then $x+y\in\Pi$. 2) If $x-y\geq
0$, then $x-y\in\Pi$.

###### Definition 8

Let $G$ be a graph, $Nodes(G)$ be arguments in $G$.
$\\{\alpha_{1},...,\alpha_{n}\\}\subseteq Nodes(G)\backslash\\{\beta\\}$ and
$\beta\in Nodes(G)$. An influence tuple is defined as
$({\alpha_{1},...,\alpha_{n}},\beta)$ where each $\alpha_{i}$ influences
$\beta$. $\alpha_{i}$ is defined as an influencer and $\beta$ is defined as an
influence target.

###### Definition 9

The restricted epistemic language based on the directed graph $G$ and a
restricted set $\Pi$ is defined as follows:

  * •

An epistemic atom is defined as $P(\alpha)$ $\\#$ $x$, where
$\\#\in\\{=,\neq,\geq,\leq,>,<\\}$, $x\in\Pi$ and $\alpha\in Nodes(G)$.

  * •

An epistemic formula is defined as a Boolean combination of epistemic atoms.

Hunter [11] put forward an approach to filter a set of simplest and best rules
using parameters Support, Confidence and Lift after obtaining a set of rules
from data using the 2-way generalization method. The following are definitions
for the three parameters.

###### Definition 10

Let $d\in D$ be a data item, $G$ be a graph, and $\Pi$ be a restricted set.
Let $R=\phi_{1}\wedge...\wedge\phi_{n}\rightarrow\phi_{n+1}$ be a rule where
$\phi_{i}$ is of the form $P(\phi_{i})\\#_{i}v_{i}$ for $i\in\\{1,...,n+1\\}$,
$v_{i}\in\Pi$ and $\alpha_{i}\in Nodes(G)$. $R$ is fired by $d$ iff for every
$\phi_{i}$ such that $i\leq n$, $d(\alpha_{i})\in
Values(P(\alpha_{i})\\#_{i}v_{i},\Pi)$. $R$ agrees with $d$ iff
$d(\alpha_{i+1})\in Values(\phi_{n+1},\Pi)$. $R$ is correct w.r.t $d$ iff $R$
satisfies both fired and agrees conditions. Then,
$Support(R,D)=\frac{1}{|D|}\times|\\{d\in D\mid R\ is\ fired\ by\ d\\}|$,
$Confidence(R,D)=\frac{1}{|D|}\times|\\{d\in D\mid R\ is\ correct\ w.r.t.\
d\\}|$ and $Lift(R,D)=\frac{|\\{d\in D\mid R\ is\ correct\ w.r.t.\
d\\}|}{|\\{d\in D\mid R\ is\ fired\ by\ d\\}|\times|\\{d\in D\mid R\ agrees\
with\ d\\}|}$.

##  3 Rule Learning Framework Revisit

We now introduce an improved efficient rule learning framework based on the
framework generated by Hunter[11] to obtain a set of rational epistemic
constraints such that the restricted values do not confine to 0.5, but also
other values satisfying the conditions of the restricted value set. The
framework will be presented as follows. 1) Introduce the inputs for the
framework. 2) Introduce a filtering algorithm that filters columns and rows of
data corresponding to specific influence tuples out of a data set based on the
knowledge from both tabular data and natural language data, and we call the
data set obtained from the filtering algorithm the rational data set. 3)
Introduce the generating algorithm using a multiple-way generalization step
method to generate a set of rational simplest and best rules out of the
rational data set.

###  3.1 Input for the framework

Same as [11], we use a data item to represent the response of a participant
towards an argument.

###### Definition 11

Let A be a set of arguments. A data item can be defined as
$d:A\rightarrow\\{0,0.1,0.2,...,0.9,1\\}.$ So for $a\in A$,
$d(a)\in\\{0,0.1,0.2,...,0.9,1\\}.$

We will use the dataset about political attitudes in Italy [19] and the
dataset about the appropriateness of using Wikipedia in teaching in Higher
Education [16] to illustrate how we apply our 11-point scale to map each data
to a probability value and how we transfer the relations between arguments to
the inputs for the algorithm. Note that both studies used Likert scales to
record the responses of participants towards various arguments. Pellegrini et
al. [19] used 7-point, 8-point and 10-point Likert-type scales to specify the
agreement while Meseguer-Artola et al. [16] applied the 5-point Likert scale.
The difference between the dataset obtained using the Likert scale and the
dataset which matches the requirements of our framework is that the Likert
scale used integers from one to ten to express the beliefs while our framework
requires probabilities. So in this paper, as what Hunter et al. [12] did, we
used 11-point scale to map each value of the Likert scale to a probability
value. The following explains how this scale works for different point scales.

For a 5-point scale, the mapping is as follows:

$\displaystyle 1\mapsto 0.1,2\mapsto 0.3,3\mapsto 0.5,4\mapsto 0.7,5\mapsto
0.9$

For a 7-point scale, the mapping is as follows:

$\displaystyle 1\mapsto 0,2\mapsto 0.1,3\mapsto 0.3,4\mapsto 0.5,5\mapsto
0.7,6\mapsto 0.9,7\mapsto 1.0$

For a 8-point scale, the mapping is as follows:

$\displaystyle 1\mapsto 0,2\mapsto 0.2,3\mapsto 0.3,4\mapsto 0.4,5\mapsto
0.6,6\mapsto 0.7,7\mapsto 0.8,8\mapsto 1.0$

For a 10-point scale, the mapping is as follows:

$\displaystyle 1\mapsto 0.1,2\mapsto 0.2,3\mapsto 0.3,4\mapsto 0.4,5\mapsto
0.5,$ $\displaystyle 6\mapsto 0.6,7\mapsto 0.7,8\mapsto 0.8,9\mapsto
0.9,10\mapsto 1.0$

The relations between two arguments can be either support, attack or neither
support nor attack. We use $`0^{\prime}$ to represent the attack relation and
$`1^{\prime}$ to represent the support relation for our inputs of relation.

###### Definition 12

Let $A$ be a set of arguments. A relation item can be defined as $r:A\times
A\rightarrow\\{0,1\\}.$ So for $\alpha,\beta\in A$, $r(a,b)\in\\{0,1\\}.$ Let
$I$ be an influence tuple, we use $rel(I)$ to denote the set containing all
the relation items between the influencers and the influence target of the
influence tuple $I$.

We now present an example of how to decide the input relations.

###### Example 2

The following shows some of the arguments from the politics database.
(dw2) Chaos and anarchy can erupt around us nowadays.
(dw3) There exists dangerous people in current society.
(dw4) There isn’t any more "crime in the street" nowadays.
(dw5) The society isn’t dangerous nowadays.

If the influence tuple is $(\\{dw2,dw3,dw4\\},dw5)$, then we regard arguments
$dw2$ and $dw3$ attack $dw5$ while $dw4$ supports $dw5$. Then $r(dw2,dw5)=0$,
$r(dw3,dw5)=0$ and $r(dw4,dw5)=1$. The $rel((\\{dw2,dw3,dw4\\},dw5))$ for this
influence tuple is $\\{0,0,1\\}$

For this influence tuple, .

In summary, inputs for the filtering algorithm are a data tuple, an influence
tuple, $rel(I)$, and a restricted value set.

###  3.2 A filtering algorithm to generate rational rules

Now we present three standards to generate rational rules in this section,
respectively corresponding to the situation when there only exist attackers,
the situation when there only exist supporters, and the situation when there
exist both supporters and attackers. The new definitions which act as criteria
to filter rational rules are obtained according to Dung’s argumentation theory
for attack relations and the deductive and necessary supports for support
relations. For each situation, we will give an example based on data sets of
Spain and Italy.

####  3.2.1 Filtering algorithm for rules containing only attack relations

###### Definition 13

Let $P:2^{\mathrm{A}}\rightarrow[0,1]$, argument $a$ be an influence target
and $Att(a)$ be a set of influencers that attack $a$.

  1. (a)

$P$ is incoherent if any argument in $Att(a)$ is believed and $a$ is believed
simultaneously.

  2. (b)

$P$ is non-reinstated if all arguments in $Att(a)$ are disbelieved and $a$ is
also disbelieved.

Figure 2: The epistemic graph corresponding to Table 2 with the influence
tuple $((Sys7,Dw6),Sys4)$. Here, $Sys4$ denotes "Italy is the best place to
live", $Sys7$ denotes "Our society gets worse year by year", $Dw6$ denotes
"The society has become more lawless and bestial nowadays". The $-$ label
denotes attack.

| Sys4 | Sys7 | Dw6
---|---|---|---
000 | 0.2 | 0.3 | 0.3
001 | 0.6 | 0.3 | 0.6
003 | 0.3 | 0.7 | 0.1

Table 2: Containing columns and rows of data obtaining from the Italian study
after changing the Likert values to probability values. Note that the
epistemic graph for the data is stated in Figure 2.

In this definition, the first constraint is derived from the coherent
constraint and the second constraint is derived from the definition of
reinstatement. Any rule satisfying any one of the constraints is regarded as
irrational.

In this paper, we identify the relations between arguments by reading and
analyzing the statements of two arguments by hand. Methods of automatically
identifying relations between arguments from the natural language text are
left to future work.

###### Example 3

Consider Table 2. Here, the influence tuple is $(\\{Sys7,Dw6\\},Sys4)$, where
$Sys4$ is the influence target and $Sys7$, $Dw6$ are attackers for $Sys4$. The
epistemic graph for $(\\{Sys7,Dw6\\},Sys4)$ can be referred to Figure 2. For
row $001$, we can see that $P(Sys4)>0.5$ , i.e., $Sys4$ is believed.
Meanwhile, $Dw6$ is believed. This result satisfies the incoherent constraint,
and rules obtained from $(\\{Sys7,Dw6\\},Sys4)$ for row $001$ are regarded as
irrational and should be deleted. For row $000$, we can see that $P(Sys4)<0.5$
, i.e., $Sys4$ is disbelieved. Meanwhile, $Sys7$ and $Dw6$ are both
disbelieved. This result satisfies the non-reinstated constraint, and rules
obtained from $(\\{Sys7,Dw6\\},Sys4)$ for row $000$ are regarded as irrational
and should be deleted. For row $003$, rules obtained from
$(\\{Sys7,Dw6\\},Sys4)$ don’t satisfy any of the constraints. Therefore, row
$003$ of Table 2 is regarded as part of the rational set.

####  3.2.2 Filtering algorithm for rules containing only support relations

###### Definition 14

Let $P:2^{\mathrm{A}}\rightarrow[0,1]$, argument $a$ be an influence target
and $Sup(a)$ be a set of influencers that support $a$.

  * (a)

$P$ is non-conclusive if any argument in $Sup(a)$ is believed but $a$ is
disbelieved.

  * (b)

$P$ is non-grounded if all arguments in $Sup(a)$ are disbelieved but $a$ is
believed.

Figure 3: The epistemic graph corresponding to Table 3 with the influence
tuple $((Im1,Im2),Qu1)$. Here, $Qu1$ denotes "Articles in Wikipedia are
reliable", $Im1$ denotes "The use of Wikipedia is well considered among
colleagues", $Im2$ denotes "My colleagues use Wikipedia". The $+$ label
denotes support.

| Qu1 | Im1 | Im2
---|---|---|---
001 | 0.7 | 0.1 | 0.2
002 | 0.3 | 0.3 | 0.7

Table 3: Containing columns and rows of data obtaining from the Spanish study
after changing the Likert values to probability values. Note that the
epistemic graph for the data is stated in Figure 3.

The two constraints defined in Definition 14 are generated according to the
deductive support and necessary support. On the one hand, if the relations
between the influence target and influencers are supports, then the acceptance
of an influencer leads to the acceptance of the influence target, which means
that if an influencer is believed then the influence target is also believed.
On the other hand, according to the necessary support, if one disbelieves all
influencers, then the influence target is also disbelieved when the relations
between the influencers and the influence target are supports. Any rule that
satisfies either the non-conclusive or the non-grounded constraint will be
regarded as irrational.

###### Example 4

Consider Table 3. We regard $Qu1$ as the influence target and $Im1$, $Im2$ as
supporters for $Qu1$, the influence tuple can be presented as
$(\\{Im1,Im2\\}$, $Qu1)$. The epistemic graph for $(\\{Im1,Im2\\},Qu1)$ can be
referred to Figure 3. For row 002, we can see that $P(Qu1)\leq 0.5$ , i.e.,
$Qu1$ is disbelieved. Meanwhile, $Im2$ is believed. This result satisfies the
non-conclusive constraint, and rules corresponding to $(\\{Im1,Im2\\},Qu1)$
are regarded as irrational and should be deleted. For row 001, the rule is
irrational. The result satisfies the non-grounded constraint since $Qu1$ is
believed while $Im1$ and $Im2$ are disbelieved.

####  3.2.3 Filtering algorithm for rules containing both support and attack
relations

###### Definition 15

Let $P:2^{\mathrm{A}}\rightarrow[0,1]$, argument $a$ be an influence target
and $Att(a)$ and $Sup(a)$ be sets of influencers that attack and support $a$
respectively, where $Att(a)\cap Sup(a)=\emptyset$.

  1. (a)

$P$ is gen-inconclusive if all attackers in $Att(a)$ are disbelieved but any
supporter in $Sup(a)$ is believed, and $a$ is disbelieved at the same time.

  2. (b)

$P$ gen-incoherent if all supporters in $Sup(a)$ are disbelieved but any
attacker in $Att(a)$ is believed, and $a$ is believed at the same time.

The gen-incoherent in Definition 15 is derived from the coherent constraint
and is also a generalization of the incoherent constraint defined in
Definition 13. The gen-inconclusive generalization is a more general
constraint derived based on the non-conclusive constraint. Any rule doesn’t
satisfy the constraints defined in Definition 15 is regarded as irrational.

###### Example 5

Consider Table 1. We regard Dw6 as the influence target. $Dw2$, $Dw3$ are
supporters for $Dw6$ while $Dw5$ is an attacker for $Dw6$, the influence tuple
can be denoted as $(\\{Dw2,Dw5,Dw3\\}$, $Dw6)$. The epistemic graph for
$(\\{Dw2,Dw5,Dw3\\},Dw6)$ can be refer to Figure 1. For row 026, we can see
that $P(Dw6)\leq 0.5$ , i.e., Dw6 is disbelieved. Meanwhile, the attacker
$Dw5$ is disbelieved, and both of the supporters $Dw2$ and $Dw3$ are believed
which satisfies that one of the supporters should be believed. This result
satisfies the gen-inconclusive constraint and the rule is regarded as
irrational. Therefore, rules with tuple $((Dw2,Dw5,Dw3),Dw6)$ in row 026
should be deleted. For row 111, the rule of $(\\{Dw2,Dw5,Dw3\\},Dw6)$ is
irrational. The result satisfies the gen-incoherent constraint since Dw6 is
believed while all supporters of Dw6 are disbelieved and the attacker Dw5 is
believed.

####  3.2.4 Filtering Algorithm

The following is the filtering algorithm which generates a set of rational
rules. We do this process of filtering rules before we generate rules based on
the restricted value set.

###### Definition 16

Let $I$ be an influence tuple and $Rel(I)$ be a set of relation items among
the influence tuple $I$. We set a function
Filtering${\mathrm{(}I,Rel(I))}$ to decide whether a rule derived from a data
set corresponding to an influence tuple $I$ satisfies the incoherent, non-
reinstated constraints if there only exist attack relations between arguments,
the non-conclusive and non-grounded constraints if there only exist support
relations between arguments, and the gen-inconclusive and gen-incoherent
constraints if there exist both attack and support relations between
arguments. If the rule satisfies one of the constraints, then the rule is
regarded as irrational and should be deleted. The outcome of the function is a
rational data set corresponding to various influence tuples.

###  3.3 An improved generating algorithm

This section provides a generating algorithm that generates a set of rational
best simplest rules from a dataset.

####  3.3.1 Multiple-way generalization step

This generalization step is based on the two-way generalization step defined
in [11]. The multiple-way generalization step contains a precondition and a
postcondition. The precondition is above the line, which is the input data.
The postcondition is below the line and the epistemic atoms of the rule are
compared to the restricted values in the restricted value set, and the
epistemic atoms can be either larger than, less than or equal to the
restricted values.

###### Definition 17

Let $d$ be a data item, $v_{i}$ be the input value for a data item,
$\Pi=\\{0,\pi_{1},...,\pi_{t},...,\pi_{n},1\\}$ be the restricted value set
where $\pi_{k},\pi_{r},\pi_{l}\in\Pi\backslash\\{0,1\\},$
$I=(\\{\alpha_{1},...,\alpha_{i},...,\alpha_{n}\\},\beta)$ be an influence
tuple, and $Rel(I)$ be a set containing all the relation items among the
influence tuple $I$. The multiple-way generalization step is defined as
follows where for each i, if $v_{i}>\pi_{t}$, then $\\#_{i}$ is "$>$", else if
$v_{i}\leq\pi_{t}$, then $\\#_{i}$ is "$\leq$".

$\displaystyle\frac{d(\alpha_{1})=v_{1},...,d(\alpha_{n})=v_{n},d(\beta)=v_{n+1}}{p(\alpha_{1})\\#_{1}\pi_{k}\wedge...\wedge
p(\alpha_{n})\\#_{n}\pi_{r}\rightarrow P(\beta)\\#_{n+1}\pi_{l}}$ (1)

We use the algorithm Filtering${\mathrm{(}I,Rel(I))}$ to decide whether rules
obtained from the precondition, i.e., rules obtained from the influence tuple
$I$ are rational.

We define $\mathrm{MultiWayGen}(d,I,\Pi)$ as follows. If a data item $d$
satisfies the precondition, then the function $\mathrm{MultiWayGen}(d,I,\Pi)$
first return a rational data set corresponding to $I$ using the algorithm
Filtering${\mathrm{(}I,Rel(I))}$. Afterwards, it returns a set of rational
rules out of a data item based on the restricted value set. If the data item
$d$ doesn’t satisfy the precondition or rules are all irrational, then the
postcondition returns nothing.

###### Example 6

Consider row 004 in Table 1. Here we set the influence tuple $I$ as
$(\\{Dw2,Dw5,Dw3\\},Dw6)$, the restricted value set $\Pi$ as
$\\{0,0.25,0.5,0.75,1\\}$ and the relation item set $Rel(I)$ as $\\{1,0,1\\}$.
After applying the multi-way generalization step to the data item, an example
of two possible rules is presented as follows: $p(Dw2)\leq 0.5\wedge
p(Dw5)\leq 0.5\wedge p(Dw3)\leq 0.5\rightarrow p(Dw6)\leq 0.5\\\
p(Dw2)>0.25\wedge p(Dw5)>0.25\wedge p(Dw3)>0.25\rightarrow p(Dw6)\leq 0.25$

Note that the influence target $Dw6$ and all three influencers are
disbelieved. This doesn’t satisfy any of the irrational constraints.
Therefore, the above two rules which are derived from tuple I are regarded as
rational and they can be two possible outputs of
$\mathrm{MultiWayGen}(d,I,\Pi)$.

###### Definition 18

Let D be a dataset, $\Pi$ be a restricted value set containing a set of
values, $I$ be an influence tuple and $Rel(I)$ be a set of relation items. The
generalization function $\mathrm{Generalize}(D,I,\Pi,Rel(I))$ returns the set
$\\{\mathrm{MultiWayGen}(d,I$, $\Pi,Rel(I))\mid d\in D\\}$.

####  3.3.2 Generating algorithm

The methods for generating the best and simplest rules are the same as [11].

The improved generating algorithm is presented in Figure 4.

Figure 4: The generating algorithm where $D$ is a dataset, $I$ is a set of
influence tuples, $\Pi$ is a constraint set and $\tau_{support}$,
$\tau_{confidence}$ are thresholds for support and confidence respectively.

##  4 Evaluation

In this paper, we apply the filtering-based general approach to two databases
of two published studies respectively, the dataset about political attitudes
in Italy [19] and the dataset about the appropriateness of using Wikipedia in
teaching in Higher Education [16]. Our algorithm is being implemented using
the Pycharm of python 3.8 version and the Lenovo R9000X laptop of Windows 10
with AMD Ryzen7 5800 and 16GB RAM. Each of the evaluations is being
implemented with ten repetitions, and we set $\tau_{support}$ as 0.4 and
$\tau_{confidence}$ as 0.8 for our evaluation. 111The link of the GitHub
repository for the code: https://github.com/cx3506/SAC.git.

There are two reasons for choosing these two datasets to evaluate our
framework. The first one is that our framework requires the beliefs and
disbelieves of participants towards arguments. Note that both the political
and Wikipedia studies used Likert scales to record the responses of
participants towards certain statements. The Likert scale is a technique
raised by Likert [15] to measure the attitude of participants. The respondents
of participants specify the level of agreement or disagreement towards
statements [15], which can be regarded as the beliefs and disbelieves of
participants towards arguments in our framework. Another reason is that
arguments in each dataset are influenced by other arguments, which allows us
to form influence tuples.

To implement our improved framework on each of the databases, we first
construct the remaining inputs for the framework by hand: 1) a set of
influence tuples, 2) a set of relation items based on the influence tuples, 3)
a restricted value set. Then, we split the dataset into a training dataset and
a testing dataset by randomly selecting 80 percent and 20 percent of the
dataset respectively, and generate a set of rules out of the training dataset.
Note that we set the maximum conditions per rule as four to avoid over-
fitting. After generating a set of rules, we evaluate rules using parameters
Lift, Support, Confidence and Irrational on the testing value set. Irrational
measures the number of irrational rules in the resulting rule set while the
definitions for Lift, Support, and Confidence are given in Definition 10.

In this section, we make comparisons between frameworks with and without the
filtering algorithm in the aspect of multiple restricted values. We set the
restricted value set as $\\{0,0.25,0.5,0.75,1\\}$

Figure 5: Results for the Spanish and Italian data sets implemented using the
framework without the filtering algorithm. Figure 6: Results for the Spanish
and Italian data sets implemented using the framework with the filtering
algorithm.

We can see from Figures 5 and 6 that both approaches perform well on the two
databases since the $Confidence$, $Support$, $Lift$ and $Irrational$ are all
in suitable constraints, i.e. $\mathrm{Confidence}>\tau_{confidence}=0.8$,
$\mathrm{Support}>\tau_{support}=0.4$, $\mathrm{Lift}>1$ and $Irrational$
$=0$. We then calculate the overall averages of Confidence for the two
approaches, and obtain 0.921 and 0.92 for Figure 5 and Figure 6 respectively.
The accuracy for both approaches are high and reasonable. We then calculate
the overall average Support for both approaches and obtain 0.775 and 0.778 for
Figure 5 and Figure 6 respectively, which means that targets in the Spain
database and the Italy database are influenced by influencers in a number of
ways. We also notice that for both approaches, the parameter Irrational equals
zero for all situations. This implies our filtering-based approach filters
irrational rules sufficiently.

To compare the time consumption for both approaches on the Spain database, we
draw a histogram to have a better visualization of the time consumed by
different approaches.

Figure 7: The consuming time of two frameworks on the Italian and Spanish data
set.

We can observe from the histogram in Figure 7 that the consuming time reduces
remarkably after improving the algorithm by adding a rational rule filtering
algorithm for both the Italian and Spanish data sets. Meanwhile, when the
complexity of the algorithm increased, i.e., when the number of influencers
increased, the gap of time performances between the filtering-based approach
and the approach without the rational rule filtering algorithm continues to
widen.

We notice that the number of resulting rules generated by approaches with and
without the rational rule filtering algorithm are approximately the same,
which means that most of the simplest and best rules generated using the
framework proposed by Hunter [11] are rational when $\tau_{confidence}$ equals
0.8. We find out that for the original approach proposed by Hunter [11], it
tends to generate rational rules if the value for the Confidence is large.
Now, let us first analyze this observation by looking at the definition of
Confidence. We obtain Confidence by calculating the number of rules that
agrees with $d$ out of the number of rules that are fired by $d$, where $d$ is
a row of data in a dataset. Therefore, a rule $R$ with a large Confidence
tends to be correct with more rows of data in a dataset. This means for a
large Confidence, the number of rows in a dataset that has the same nature of
beliefs in both the influencers and the influence target as $R$ is large.
Since most of the beliefs of agents towards different arguments in a dataset
obey relations contained in epistemic graphs, most of the rules in a dataset
are rational. Then, if the Confidence of a rule is large, the rule tends to be
rational. This is also shown in Example 7. We can also use the above
observation to evaluate the rationality of a dataset. A dataset is rational if
for a small value of $\tau_{confidence}$, the number of resulting best rules
which are rational is approximately zero. This is because there don’t exist
rules with a small Confidence value when their Support values are large.
Therefore, in the dataset, data that have the same nature of beliefs in
influencers as the resulting rules also have the same nature of beliefs in the
influence target. Since almost all resulting best rules are rational, almost
all corresponding data in the dataset are rational.

Figure 8: The number of irrational rules under different values of
$\tau_{confidence}$ for the Spanish and Italian data sets implemented using
the framework proposed by [11] on the restricted value set ${0,0.5,1}$.

###### Example 7

Consider the rule $P(Dw2)<0.5\rightarrow P(Dw6)>0.5$ for tuple $(\\{Dw2\\}$,
$Dw6)$ with data from Table 1. This rule is irrational. The rule is correct
with $row111$, and fired with $row111$ and $row004$. We assume the dataset
only has these three rows, then Confidence here equals 0.5. If
$\tau_{Confidence}$ equals 0.2, then the rule satisfies the standard for best
rules in respect of Confidence. If $\tau_{Confidence}$ equals 0.8, then the
rule cannot be a best rule.

The above observations are shown in Table 8. We observe that the value of
Confidence is positively related to a rule being rational, and the $Spain$
dataset is more rational than the "Italy" dataset.

We also notice that our framework with the filtering algorithm performs better
on the Spanish data set than the Italian data set overall since the framework
filters more irrational rules for the Spanish data set than the Italian data
set for every number of influencers except for two influencers. In other
words, the consuming time is positively related to the complexity of
generating rules and negatively related to the number of irrational rules
generated from the data set. We prove this observation in the following
analysis.

Note that the consuming time for the framework without the filtering algorithm
is the time without deleting any rules. We denote the consuming time for the
framework with the filtering algorithm as $t_{1}$, and the time for the
framework without the filtering algorithm as $t_{0}$. We calculate the time
difference $(t_{0}-t_{1})$ of the two databases, and denote the time
difference as $t_{I}$ for the Italian database while $t_{S}$ for the Spanish
database. We are interested in the difference $(t_{S}-t_{I})$ for each row
with the same number of influencers. We are also interested in comparing the
number of rules being deleted for each row of the two datasets. We denote the
number of rules being deleted for each row in the Italy dataset by $D_{I}$ and
the number of rules being deleted for each row in the Spain dataset by
$D_{S}$. We calculate $(D_{S}-D_{I})$ for each row with the same number of
influencers. By comparing the $(t_{S}-t_{I})$ and the $(D_{S}-D_{I})$, we
obtain the relationship between the consuming time and the number of deleted
rules. Figure 9 shows the relationship between $(D_{S}-D_{I})$ and
$(t_{S}-t_{I})$ as the number of influencers increased. We observed that the
number of irrational rules is positively related to the time performance of
the rule-generating framework.

Figure 9: The relationship between $(D_{S}-D_{I})$ and $(t_{S}-t_{I})$ as the
number of influencers increases.

##  5 Conclusions and Future Work

In this paper, we have addressed the three research questions mentioned in the
introduction. The main contributions include the following. We set forward a
filtering-based approach using the multiple-way generalization step method to
generate a set of rational rules whose probability constraints are able to
represent various degrees of beliefs in better efficiency. We proposed a
filtering algorithm to generate a set of rational rules based on both the
tabular data and the natural language data. This filtering algorithm also
improves the efficiency of generating rules. We also set forward a multiple-
way generalization step method to deal with the case that the 2-way
generalization step cannot work for more than one restricted value situation.

However, there still remain two main limitations of the current approach. The
first one is that we decide the influence tuples and the relations between two
arguments by hand. Nonetheless, in many cases, different agents can have
different views on arguments and the relations between them. To deal with this
limitation, one can refer to the stance detection method to automatically
apply positive or negative labels between two arguments. One can also consider
the causal inference to decide the relations between two arguments.

Another limitation of the current approach is that if a data set doesn’t
contain many irrational rules, the efficiency of applying the improved
framework will not be as good as one expected since the efficiency of the new
method depends on the number of irrational rules in a rule set. Also, even
though the filtering algorithm gets rid of a large number of rules, the
consuming time is still large. It is essential to raise a framework that
improves the efficiency fundamentally apart from filtering and deleting
unsatisfied rules.

## References

  * [1] Leila Amgoud, Claudette Cayrol, Marie-Christine Lagasquie-Schiex, and P.Livet. 2008. On bipolarity in argumentation frameworks. Int. J. Intell. Syst. 23, 10 (2008), 1062–1093. https://doi.org/10.1002/int.20307
  * [2] Pietro Baroni, Martin Caminada, and Massimiliano Giacomin. 2011. An introduction to argumentation semantics. The Knowledge Engineering Review 26, 4 (2011), 365–410. https://doi.org/10.1017/S0269888911000166
  * [3] Guido Boella, Dov M.Gabbay, Leendert W.N.vanderTorre, and Serena Villata. 2010. Support in Abstract Argumentation. In Computational Models of Argument: Proceedings of COMMA 2010, Desenzano del Garda, Italy, September 8-10, 2010 (Frontiers in Artificial Intelligence and Applications), Pietro Baroni, Federico Cerutti, Massimiliano Giacomin, and Guillermo Ricardo Simari (Eds.), Vol. 216. IOS Press, 111–122.
  * [4] Claudette Cayrol and Marie-Christine Lagasquie-Schiex. 2005. On the Acceptability of Arguments in Bipolar Argumentation Frameworks. In Symbolic and Quantitative Approaches to Reasoning with Uncertainty, 8th European Conference, ECSQARU 2005, Barcelona, Spain, July 6-8, 2005, Proceedings (Lecture Notes in Computer Science), Lluís Godo (Ed.), Vol. 3571. Springer, 378–389.
  * [5] Claudette Cayrol and Marie-Christine Lagasquie-Schiex. 2010. Coalitions of arguments: A tool for handling bipolar argumentation frameworks. Int. J. Intell. Syst.25,1(2010),83–109. https://doi.org/10.1002/int.20389
  * [6] Claudette Cayrol and Marie-Christine Lagasquie-Schiex. 2013. Bipolarity in argumentation graphs: Towards a better understanding. Int. J. Approx. Reason. 54,7(2013),876–899. https://doi.org/10.1016/j.ijar.2013.03.001
  * [7] Phan Minh Dung. 1995. On the Acceptability of Arguments and its Fundamental Role in Nonmonotonic Reasoning, Logic Programming and n-Person Games. Artif. Intell. 77, 2 (1995), 321–358. https://doi.org/10.1016/0004- 3702(94)00041- X
  * [8] Phan Minh Dung and Phan Minh Thang. 2010. Towards(Probabilistic)Argumentation for Jury-based Dispute Resolution. In Computational Models of Argument: Proceedings of COMMA 2010, Desenzano del Garda, Italy, September 8-10, 2010 (Frontiers in Artificial Intelligence and Applications), Pietro Baroni, Federico Cerutti, Massimiliano Giacomin, and Guillermo Ricardo Simari (Eds.), Vol. 216. IOS Press, 171–182.
  * [9] Anthony Hunter. 2012. Some Foundations for Probabilistic Abstract Argumentation. In Computational Models of Argument - Proceedings of COMMA 2012, Vienna, Austria, September 10-12, 2012 (Frontiers in Artificial Intelligence and Applica- tions), Bart Verheij, Stefan Szeider, and Stefan Woltran (Eds.), Vol. 245. IOS Press, 117–128. https://doi.org/10.3233/978-1-61499-111-3-117
  * [10] Anthony Hunter. 2013. A probabilistic approach to modelling uncertain logical arguments.Int.J.Approx.Reason.54,1(2013),47–81. https://doi.org/10.1016/j. ijar.2012.08.003
  * [11] Anthony Hunter. 2020. Learning Constraints for the Epistemic Graphs Approach to Argumentation. In Computational Models of Argument - Proceedings of COMMA 2020, Perugia, Italy, September 4-11, 2020 (Frontiers in Artificial Intelligence and Applications), Henry Prakken, Stefano Bistarelli, Francesco Santini, and Carlo Taticchi(Eds.),Vol.326.IOSPress,239–250. https://doi.org/10.3233/FAIA200508
  * [12] Anthony Hunter, Sylwia Polberg, and Matthias Thimm. 2020. Epistemic graphs for representing and reasoning with positive and negative influences of arguments. Artif.Intell.281(2020),103236. https://doi.org/10.1016/j.artint.2020.103236
  * [13] Anthony Hunter and Matthias Thimm. 2014. Probabilistic Argumentation with Epistemic Extensions and Incomplete Information. CoRR abs/1405.3376 (2014). arXiv:1405.3376 http://arxiv.org/abs/1405.3376
  * [14] Hengfei Li, Nir Oren, and Timothy J. Norman. 2011. Probabilistic Argumentation Frameworks. In Theorie and Applications of Formal Argumentation - First International Workshop, TAFA 2011. Barcelona, Spain, July 16-17, 2011, Revised Selected Papers (Lecture Notes in Computer Science), Sanjay Modgil, Nir Oren, and Francesca Toni (Eds.), Vol. 7132. Springer, 1–16.
  * [15] Rensis Likert. 1932. A technique for the measurement of attitudes. Archives of psychology (1932).
  * [16] AntoniMeseguer-Artola, EduardAibar, JosepLladós, JuliàMinguillón, and Maura Lerga. 2016. Factors that influence the teaching use of Wikipedia in higher education. Journal of the Association for Information Science and Technology 67, 5 (2016),1224–1232. https://doi.org/10.1002/asi.23488
  * [17] Farid Nouioua and Vincent Risch. 2010. Bipolar Argumentation Frameworks with Specialized Supports. In 22nd IEEE International Conference on Tools with Artificial Intelligence, ICTAI 2010, Arras, France, 27-29 October 2010 - Volume 1. IEEEComputerSociety,215–218. https://doi.org/10.1109/ICTAI.2010.37
  * [18] Farid Nouioua and Vincent Risch. 2011. Argumentation Frameworks with Necessities. In Scalable Uncertainty Management - 5th International Conference, SUM 2011, Dayton, OH, USA, October 10-13, 2011. Proceedings (Lecture Notes in Computer Science), Salem Benferhat and John Grant (Eds.), Vol. 6929. Springer, 163–176.
  * [19] Valerio Pellegrini, Luigi Leone, and Mauro Giacomantonio. 2019. Dataset about populist attitudes, social world views, socio-political dispositions, conspiracy beliefs, and anti-immigration attitudes in an Italian sample. Data in Brief 25 (2019),104144. https://doi.org/10.1016/j.dib.2019.104144
  * [20] Matthias Thimm. 2012. A Probabilistic Semantics for abstract Argumentation. In ECAI 2012 - 20th European Conference on Artificial Intelligence. Including Pres- tigious Applications of Artificial Intelligence (PAIS-2012) System Demonstrations Track, Montpellier, France, August 27-31 , 2012 (Frontiers in Artificial Intelligence and Applications), Luc De Raedt, Christian Bessiere, Didier Dubois, Patrick Do- herty, Paolo Frasconi, Fredrik Heintz, and Peter J. F. Lucas (Eds.), Vol. 242. IOS Press, 750–755. https://doi.org/10.3233/978-1-61499-098-7-750


# Codiskcyclic sets of operators on complex topological vector spaces

Mohamed Amouch and Otmane Benchiheb  Mohamed Amouch and Otmane Benchiheb,
University Chouaib Doukkali. Department of Mathematics, Faculty of science
Eljadida, Morocco<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $X$ be a complex topological vector space and $L(X)$ the set of all
continuous linear operators on $X.$ In this paper, we extend the notion of the
codiskcyclicity of a single operator $T\in L(X)$ to a set of operators
$\Gamma\subset L(X).$ We prove some results for codiskcyclic sets of operators
and we establish a codiskcyclicity criterion. As an application, we study the
codiskcyclicity of $C_{0}$-semigroups of operators.

###### Key words and phrases:

Hypercyclity, supercyclicity, diskcyclicity, codiskcyclicity,
$C_{0}$-semigroup.

###### 2010 Mathematics Subject Classification:

47A16

##  1\. Introduction and Preliminary

Let $X$ be a complex topological vector space and $L(X)$ the set of all
continuous linear operators on $X.$ By an operator, we always mean a
continuous linear operator.

The most studied notion in the linear dynamics is that of hypercyclicity$:$ an
operator $T$ acting on $X$ is called hypercyclic if there exits a vector $x\in
X$ such that the orbit of $x$ under $T$;

$Orb(T,x)=\\{T^{n}x\mbox{ : }n\in\mathbb{N}\\},$

is dense in $X$, such a vector $x$ is called hypercyclic for $T$. The set of
all hypercyclic vectors for $T$ is denoted by $HC(T)$. Another important
notion in the linear dynamics is that of supercyclicity which was introduced
in [12]. We say that $T$ is supercyclic if there exists $x\in X$ such that

$\mathbb{C}Orb(T,x)=\\{\alpha T^{n}x\mbox{ : }\alpha\in\mathbb{C}\mbox{,
}n\in\mathbb{N}\\},$

is dense in $X$. The vector $x$ is called a supercyclic vector for $T$. We
denote by $SC(T)$ the set of all supercyclic vectors. For more information
about hypercyclic and supercyclic operators, see [7, 11, 10].

Another notion in the linear dynamics which was been studied by many authors
is that of codiscyclicity$:$ an operator $T$ is called codiskcyclic if there
is $x\in X$ such that the codisk orbit of $x$ under $T;$

$\mathbb{U}Orb(T,x)=\\{\alpha T^{n}x\mbox{ : }\alpha\in\mathbb{U}\mbox{,
}n\geq 0\\},$

is dense in $X$, when $\mathbb{U}:=\\{\alpha\in\mathbb{C}\mbox{ :
}|\alpha|\geq 1\\}$. In this case, the vector $x$ is called a codiskcyclic
vector for $T$. The set of all codiskcyclic vectors for $T$ is denoted by
$\mathbb{U}C(T)$. In the case of a separable complex Banach space, an operator
$T$ is codiskcyclic if and only if it is codisk transitive; that is for each
pair $(U,V)$ of nonempty open sets there exist some $\alpha\in\mathbb{U}$ and
some $n\geq 0$ such that $\alpha T^{n}(U)\cap V\neq\emptyset.$ For a general
overview of the codiskcyclicity, see [13, 14, 17, 19].

Recently, some notions of the linear dynamical system were introduced for a
set $\Gamma$ of operators instead of a single operator $T,$ see [1, 2, 3, 4,
5, 6]$:$ A set $\Gamma$ of operators is called hypercyclic if there exists a
vector $x$ in $X$ such that its orbit under $\Gamma$;
$Orb(\Gamma,x)=\\{Tx\mbox{ : }T\in\Gamma\\},$ is a dense subset of $X$. If
there exits a vector $x\in X$ such that $\mathbb{C}Orb(\Gamma,x)=\\{\alpha
Tx\mbox{ : }T\in\Gamma\mbox{, }\alpha\in\mathbb{C}\\},$ is a dense subset of
$X$, then $\Gamma$ is supercyclic. If $x\in X$ is a vector such that
$\mbox{span}\\{Orb(\Gamma,x)\\}=\mbox{span}\\{Tx\mbox{ : }T\in\Gamma\\}$ is
dense in $X$, then $\Gamma$ is cyclic. If there exists a vector $x\in X$ such
that its disk orbit under $T$; $\mathbb{D}Orb(\Gamma,x)=\\{\alpha Tx\mbox{ :
}T\in\Gamma\mbox{, }\alpha\in\mathbb{D}\\},$ is a dense subset $X$, then
$\Gamma$ is called a diskcyclic, when $\mathbb{D}$ is the unit closed disk. In
each case, the vector $x$ is called a hypercyclic, a supercyclic, a cyclic and
a diskcyclic vector for $\Gamma$, respectively.

In this paper, we continue the study of the dynamics of a set of operator by
introducing the concept of codiskcyclicity for a set of operators.

In Section $2$, we introduce and study the codiskcyclicity for a set of
operators. In particular, we show that the set of codiscyclic vectors of a set
$\Gamma$ is a $G_{\delta}$ type and we prove that codiskcyclicity is preserved
under quasi-similarity.

In Section $3$, we extend the notion of codisk transitivity of a single
operator to a set of operators. We give the relation between this notion and
the concept of codiskcyclic and we establish a codiskcyclic criterion.

In Section $4$, we study the codiskcyclicity of a $C_{0}$-semigroup of
operators. We show that the codiskcyclicity and the codisk transitivity are
equivalent and we prove that a codiskcyclic $C_{0}$-semigroup of operators
exists on $X$ if and only if dim$(X)=1$ or dim$(X)=\infty$.

##  2\.  Codiskcyclic Sets of Operators

In the following definition, we introduce the notion of the codiskcyclicity of
a set of operators instead of a single operator.

######  Definition 2.1.

We say that $\Gamma$ is codiskcyclic if there exists $x\in X$ for which the
codisk orbit of $x$ under $\Gamma;$

$\mathbb{U}Orb(\Gamma,x):=\\{\alpha Tx\mbox{ : }\alpha\in\mathbb{U}\mbox{,
}T\in\Gamma\\},$

is dense in $X$. The vector $x$ is called a codiskcyclic vector for $\Gamma$.
The set of all codiskcyclic vectors for $\Gamma$ is denoted by
$\mathbb{U}C(\Gamma)$.

######  Remark 2.2.

An operator $T$ is codiskcyclic if and only if the set $\Gamma=\\{T^{n}\mbox{
: }n\geq 0\\}$ is codiskcyclic.

######  Example 2.3.

Let $f$ be a nonzero linear form on a locally convex space $X$ and $D$ be a
subset of $X$ such that the set $\mathbb{U}D:=\\{\alpha x\mbox{ :
}\alpha\in\mathbb{U}\mbox{, }x\in D\\}$ is a dense subset of $X$. For all
$x\in X$, let $T_{x}$ defined by $T_{x}y=f(y)x,$ for all $y\in X.$ Put
$\Gamma_{f}=\\{T_{x}\mbox{ : }x\in D\\}$ and let $y$ be a vector of $X$ such
that $f(y)\neq 0.$ Then $\mathbb{U}Orb(\Gamma_{f},y)=\\{\alpha T_{x}y\mbox{ :
}x\in D\mbox{, }\alpha\in\mathbb{U}\\}=\\{\alpha f(y)x\mbox{ : }x\in D\mbox{,
}\alpha\in\mathbb{U}\\}=\mathbb{U}D.$ Hence, $\Gamma_{f}$ is codiskcyclic.

A necessary condition for the codiskcyclicity is given by the following
proposition.

######  Proposition 2.4.

Let $X$ be a complex normed space and $\Gamma$ a subset of $L(X)$. If $x$ is a
codiskcyclic vector for $\Gamma)$, then $\sup\\{\|\alpha Tx\|\mbox{ :
}\alpha\in\mathbb{U}\mbox{, }T\in\Gamma\\}=+\infty.$

###### Proof.

Let $x\in\mathbb{U}C(\Gamma)$. Assume that $\sup\\{\|\alpha Tx\|\mbox{ :
}\alpha\in\mathbb{U}\mbox{, }T\in\Gamma\\}=m<+\infty,$ and let $y\in X$ such
that $\|y\|>m$. Since $x\in\mathbb{U}C(\Gamma)$, there exist
$\\{\alpha_{k}\\}\subset\mathbb{U}$ and $\\{T_{k}\\}\subset\Gamma$ such that
$\alpha_{k}T_{k}x\longrightarrow y.$ Hence, $\|y\|\leq m$, which is a
contradiction. ∎

We denote by $\\{\Gamma\\}^{{}^{\prime}}$ the set of all elements of $L(X)$
which commutes with every element of $\Gamma.$

######  Proposition 2.5.

Let $X$ be a complex topological vector space and $\Gamma$ a subset of $L(X).$
Assume that $\Gamma$ is codiskcyclic and let $T\in L(X)$ be with dense range.
If $T\in\\{\Gamma\\}^{{}^{\prime}}$, then $Tx\in\mathbb{U}C(\Gamma)$, for all
$x\in\mathbb{U}C(\Gamma)$.

###### Proof.

Let $O$ be a nonempty open subset of $X$. Since $T$ is continuous and of dense
range, $T^{-1}(O)$ is a nonempty open subset of $X$. Let
$x\in\mathbb{U}C(\Gamma)$, then there exist $\alpha\in\mathbb{U}$ and
$S\in\Gamma$ such that $\alpha Sx\in T^{-1}(O)$, that is $\alpha T(Sx)\in O$.
Since $T\in\\{\Gamma\\}^{{}^{\prime}}$, it follows that $\alpha S(Tx)=\alpha
T(Sx)\in O.$ Hence, $\mathbb{U}Orb(\Gamma,Tx)$ meets every nonempty open
subset of $X$. From this, $\mathbb{U}Orb(\Gamma,Tx)$ is dense in $X$. That is,
$Tx\in\mathbb{U}C(\Gamma)$. ∎

Recall from [2], that $\Gamma\subset L(X)$ and $\Gamma_{1}\subset L(Y)$ are
called quasi-similar if there exists a continuous map $\phi$ :
$X\longrightarrow Y$ with dense range such that for all $T\in\Gamma,$ there
exists $S\in\Gamma_{1}$ satisfying $S\circ\phi=\phi\circ T$. If $\phi$ can be
chosen to be a homeomorphism, then $\Gamma$ and $\Gamma_{1}$ are called
similar.

In the following, we prove that the codiskcyclicity is preserved under quasi-
similarity.

######  Proposition 2.6.

If $\Gamma$ and $\Gamma_{1}$ are quasi-similar, then $\Gamma$ is codiskcyclic
in $X$ implies that $\Gamma_{1}$ is codiskcyclic in $Y$. Moreover,
$\phi(\mathbb{U}C(\Gamma)\subset\mathbb{U}C(\Gamma_{1}).$

###### Proof.

Assume that $\Gamma$ is codiskcyclic in $X$. Let $O$ be a nonempty open subset
of $Y$, then $\phi^{-1}(O)$ is a nonempty open subset of $X$. If
$x\in\mathbb{U}C(\Gamma)$, then there exist $\alpha\in\mathbb{U}$ and
$T\in\Gamma$ such that $\alpha Tx\in\phi^{-1}(O)$, that is $\alpha\phi(Tx)\in
O$. Let $S\in\Gamma_{1}$ such that $S\circ\phi=\phi\circ T$. Hence, $\alpha
S(\phi x)=\alpha\phi(Tx)\in O.$ Hence, $\Gamma_{1}$ is codiskcyclic and $\phi
x\in\mathbb{U}C(\Gamma_{1})$. ∎

######  Proposition 2.7.

Let $(c_{T})_{T\in\Gamma}\subset\mathbb{R}_{+}^{*}$. If $\\{c_{T}T\mbox{ :
}T\in\Gamma\\}$ is codiskcyclic and $(k_{T})_{T\in\Gamma}$ is such that
$c_{T}\geq k_{T}>0$ for all $T\in\Gamma,$ then the set $\\{k_{T}T\mbox{ :
}T\in\Gamma\\}$ is codiskcyclic.

###### Proof.

Let $x$ be a codiskcyclic vector for $\\{c_{T}T\mbox{ : }T\in\Gamma\\}$. Since
$c_{T}\geq k_{T}$ for all $T\in\Gamma$, we have $\mathbb{U}Orb(\\{c_{T}T\mbox{
: }T\in\Gamma\\},x)\subset\mathbb{U}Orb(\\{k_{T}T\mbox{ : }T\in\Gamma\\},x).$
Since $\mathbb{U}Orb(\\{c_{T}T\mbox{ : }T\in\Gamma\\},x)$ is dense in $X$, it
follows that $\mathbb{U}Orb(\\{k_{T}T\mbox{ : }T\in\Gamma\\},x)$ is dense in
$X$, this means that $\\{k_{T}T\mbox{ : }T\in\Gamma\\}$ is codiskcyclic in
$X$. ∎

######  Proposition 2.8.

Let $\\{X_{i}\\}_{i=1}^{n}$ be a family of complex topological vector spaces
and $\Gamma_{i}$ a subset of $L(X_{i}),$ for $1\leq i\leq n$. If
$\bigoplus_{i=1}^{n}\Gamma_{i}$ is a codiskcyclic set in
$\bigoplus_{i=1}^{n}X_{i}$, then $\Gamma_{i}$ is a codiskcyclic set in
$X_{i}$, for all $1\leq i\leq n$.

###### Proof.

If $1\leq j\leq n$, then $\bigoplus_{i=1}^{n}\Gamma_{i}$ is quasi-similar to
$\Gamma_{j}$, and the result follow by Proposition 2.6. ∎

Let $X$ be a complex topological vector space. The following proposition gives
a characterization of the set of codiskcyclic vector of set of operators using
a countable basis of the topology of $X$. Note that the set $\mathbb{D}$ is
the unit closed disk defined by $\mathbb{D}=\\{\alpha\in\mathbb{C}\mbox{ :
}|\alpha|\leq 1\\}.$

######  Proposition 2.9.

Let $X$ be a second countable complex topological vector space and $\Gamma$ a
subset of $L(X).$ If $\Gamma$is codiskcyclic, then

$\mathbb{U}C(\Gamma)=\bigcap_{n\geq
1}\left(\bigcup_{\beta\in\mathbb{D}}\bigcup_{T\in\Gamma}T^{-1}(\beta
U_{n})\right),$

where $(U_{n})_{n\geq 1}$ is a countable basis of the topology of $X$. As a
consequence, $\mathbb{U}C(\Gamma)$ is a $G_{\delta}$ type set.

###### Proof.

Let $x\in X$. Then, $x\in\mathbb{U}C(\Gamma)$ if and only if
$\overline{\mathbb{U}Orb(\Gamma,x)}=X$. Equivalently, for all $n\geq 1,$
$U_{n}\cap\mathbb{U}Orb(\Gamma,x)\neq\emptyset,$ that is for all $n\geq 1$,
there exist $\lambda\in\mathbb{U}$ and $T\in\Gamma$ such that $\lambda Tx\in
U_{n}$. This is equivalent to the fact that for all $n\geq 1,$ there exist
$\beta\in\mathbb{D}$ and $T\in\Gamma$ such that $x\in T^{-1}(\beta U_{n})$.
Hence, $\displaystyle x\in\bigcap_{n\geq
1}\bigcup_{\beta\in\mathbb{D}}\bigcup_{T\in\Gamma}T^{-1}(\beta U_{n}).$ ∎

##  3\. Density and Codisk Transitivity of Sets of Operators

In the following definition, we introduce the notion of a codisk transitive
set of operators which generalize the codisk transitivity of a single
operator.

######  Definition 3.1.

We say that $\Gamma$ is a codisk transitive set of operators, if for any pair
$(U,V)$ of nonempty open subsets of $X$, there exist $\alpha\in\mathbb{U}$ and
$T\in\Gamma$ such that $T(\alpha U)\cap V\neq\emptyset.$

######  Remark 3.2.

An operator $T\in L(X)$ is codisk transitive if and only if
$\Gamma=\\{T^{n}\mbox{ : }n\geq 0\\}$ is codisk transitive.

######  Example 3.3.

Assume that $X$ is a locally convex space. Let $x$, $y\in X$ and let $f_{y}$
be a linear form on $X$ such that $f_{y}(y)\neq 0$. Let $T_{f_{y},x}$ be an
operator defined by $T_{f_{y},x}z=f_{y}(z)x.$ Define
$\Gamma=\\{T_{f_{y},x}\mbox{ : }x\mbox{, }y\in X\mbox{ such that }f_{y}(y)\neq
0\\}.$ Let $U$ and $V$ be two nonempty open subsets of $X$. There exist $x,$
$y\in X$ such that $x\in U$ and $y\in V$. We have $T_{f_{y},x}(y)=f_{y}(y)x.$
Since $0<f_{y}(y)<1$, it follows that $x=\frac{1}{f_{y}(y)}T_{f_{y},x}(y)$.
Hence $x\in U$ and $x\in\frac{1}{f_{y}(y)}T_{f_{y},x}(V)$, which implies that
$U\cap\frac{1}{f_{y}(y)}T_{f_{y},x}(V)\neq\emptyset.$ Thus $\Gamma$ is a
codisk transitive.

In the following proposition, we prove that the codisk transitivity of sets of
operators is preserved under quasi-similarity.

######  Proposition 3.4.

Assume that $\Gamma\subset L(X)$ and $\Gamma_{1}\subset L(Y)$ are quasi-
similar. If $\Gamma$ is codisk transitive in $X$, then $\Gamma_{1}$ is codisk
transitive in $Y$.

###### Proof.

Let $U$ and $V$ be nonempty open subsets of $X$. Since $\phi$ is continuous
and of dense range, $\phi^{-1}(U)$ and $\phi^{-1}(V)$ are nonempty and open
sets. Since $\Gamma$ is codisk transitive in $X$, there exist
$y\in\phi^{-1}(U)$ and $\alpha\in\mathbb{U},$ $T\in\Gamma$ with $\alpha
Ty\in\phi^{-1}(V)$, which implies that $\phi(y)\in U$ and $\alpha\phi(Ty)\in
V$. Let $S\in\Gamma$ such that $S\circ\phi=\phi\circ T$. Then, $\phi(y)\in U$
and $\alpha S\phi(y)\in V$. Thus, $\alpha S(U)\cap V\neq\emptyset.$ Hence,
$\Gamma_{1}$ is codisk transitive in $Y.$ ∎

In the following result, we give necessary and sufficient conditions for a set
of operators to be codisk transitive.

######  Theorem 3.5.

Let $X$ be a complex normed space and $\Gamma$ a subset of $L(X)$. The
following assertions are equivalent$:$

  * $(i)$

$\Gamma$ is codisk transitive;

  * $(ii)$

For each $x$, $y\in X,$ there exists sequences $\\{x_{k}\\}$ in $X$,
$\\{\alpha_{k}\\}$ in $\mathbb{U}$ and $\\{T_{k}\\}$ in $\Gamma$ such that
$x_{k}\longrightarrow x$ and $\alpha_{k}T_{k}(x_{k})\longrightarrow y;$

  * $(iii)$

For each $x$, $y\in X$ and for $W$ a neighborhood of $0$, there exist $z\in
X$, $\alpha\in\mathbb{U}$ and $T\in\Gamma$ such that $x-z\in W$ and $\alpha
T(z)-y\in W.$

###### Proof.

$(i)\Rightarrow(ii)$ Let $x$, $y\in X$. For all $k\geq 1$, let
$U_{k}=B(x,\frac{1}{k})$ and $V_{k}=B(y,\frac{1}{k})$. Then $U_{k}$ and
$V_{k}$ are nonempty open subsets of $X$. Since $\Gamma$ is codisk transitive,
there exist $\alpha_{k}\in\mathbb{U}$ and $T_{k}\in\Gamma$ such that
$\alpha_{k}T_{k}(U_{k})\cap V_{k}\neq\emptyset$. For all $k\geq 1$, let
$x_{k}\in U_{k}$ such that $\alpha_{k}T_{k}(x_{k})\in V_{k}$, then
$\|x_{k}-x\|<\frac{1}{k}$ and $\|\alpha_{k}T_{k}(x_{k})-y\|<\frac{1}{k},$ this
implies that $x_{k}\longrightarrow x\hskip 8.5359pt\mbox{ and }\hskip
8.5359pt\alpha_{k}T_{k}(x_{k})\longrightarrow y.$

$(ii)\Rightarrow(iii)$ Clear.

$(iii)\Rightarrow(i)$ Let $U$ and $V$ be two nonempty open subsets of $X$.
Then there exists $x$, $y\in X$ such that $x\in U$ and $y\in V$. Since for all
$k\geq 1$, $W_{k}=B(0,\frac{1}{k})$ is a neighborhood of $0$, there exist
$z_{k}\in X,$ $\alpha_{k}\in\mathbb{U}$ and $T_{k}\in\Gamma$ such that
$\|x-z_{k}\|<\frac{1}{k}$ and $\alpha_{k}T_{k}(z_{k})-y\|<\frac{1}{k}.$ This
implies that $z_{k}\longrightarrow x$ and
$\alpha_{k}T_{k}(z_{k})\longrightarrow y.$ Since $U$ and $V$ are nonempty open
subsets of $X$, $x\in U$ and $y\in V$, there exists $N\in\mathbb{N}$ such that
$z_{k}\in U$ and $\alpha_{k}T_{k}(z_{k})\in V$, for all $k\geq N.$ ∎

Let $\Gamma$ be a subset of $L(X)$. In whats follows, we prove that $\Gamma$
is codisk transitive if and only if it admits a dense subset of codiskcyclic
vectors.

######  Theorem 3.6.

Let $X$ be a second countable Baire complex topological vector space and
$\Gamma$ a subset of $L(X)$. The following assertions are equivalent$:$

  * $(i)$

$\mathbb{U}C(\Gamma)$ is dense in $X$;

  * $(ii)$

$\Gamma$ is codisk transitive.

As a consequence, a codisk transitive set is codiskcyclic.

###### Proof.

$(i)\Rightarrow(ii):$ Assume that $\mathbb{U}C(\Gamma)$ is dense in $X$ and
let $U$ and $V$ be two nonempty open subsets of $X$. By Proposition 2.9, we
have $\mathbb{U}C(\Gamma)=\bigcap_{n\geq
1}\left(\bigcup_{\beta\in\mathbb{D}}\bigcup_{T\in\Gamma}T^{-1}(\beta
U_{n})\right).$ Hence, for all $n\geq 1,$ $\displaystyle
A_{n}:=\bigcup_{\beta\in\mathbb{D}}\bigcup_{T\in\Gamma}T^{-1}(\beta U_{n})$ is
dense in $X$. Thus, for all $n,$ $m\geq 1,$ we have $A_{n}\cap
U_{m}\neq\emptyset$ which implies that for all $n,$ $m\geq 1,$ there exist
$\beta\in\mathbb{U}$ and $T\in\Gamma$ such that $T(\beta U_{m})\cap
U_{n}\neq\emptyset$. Hence, $\Gamma$ is a codisk transitive set.
$(ii)\Rightarrow(i):$ Assume that $\Gamma$ is codisk transitive. Let $n$,
$m\geq 1$, then there exist $\beta\in\mathbb{U}$ and $T\in\Gamma$ such that
$T(\beta U_{m})\cap U_{n}\neq\emptyset$, which implies that
$T^{-1}(\frac{1}{\beta}U_{n})\cap U_{m}\neq\emptyset$. Hence, for all $n\geq
1,$ we have
$\displaystyle\bigcup_{\beta\in\mathbb{D}}\bigcup_{T\in\Gamma}T^{-1}(\beta
U_{n})$ is dense in $X$. Since $X$ is a Baire space, it follows that
$\mathbb{U}C(\Gamma)=\bigcap_{n\geq
1}\left(\bigcup_{\beta\in\mathbb{D}}\bigcup_{T\in\Gamma}T^{-1}(\beta
U_{n})\right)$ is a dense subset of $X$. ∎

The converse of Theorem 3.6 holds with some additional assumption.

######  Theorem 3.7.

Let $X$ be a complex topological vector space and $\Gamma$ a subset of $L(X)$.
Assume that for all $T$, $S\in\Gamma$ with $T\neq S$, there exists
$A\in\Gamma$ such that $T=AS$. The following assertions are equivalent$:$

  * $(i)$

$\Gamma$ is codiskcyclic;

  * $(ii)$

$\Gamma$ is codisk transitive.

###### Proof.

$(i)\Rightarrow(ii)$ This implication is due to Theorem 3.6.
$(ii)\Rightarrow(i)$ Since $\Gamma$ is codiskcyclic, there exists $x\in X$
such that $\mathbb{U}Orb(\Gamma,x)$ is a dense subset of $X$. Let $U$ and $V$
be two nonempty open subsets of $X$, then there exist $\alpha$,
$\beta\in\mathbb{U}$ with $|\alpha|\geq|\beta|$, and $T$, $S\in\Gamma$ such
that $\alpha Tx\in U$ and $\beta Sx\in V.$ There exists $A\in\Gamma$ such that
$T=AS$. Hence, $\alpha A(Sx)\in U$ and $\beta A(Sx)\in A(V)$, which implies
that $U\cap A(\frac{\alpha}{\beta}V)\neq\emptyset$. Hence, $\Gamma$ is codisk
transitive. ∎

In the following definition we introduce the notion of strictly codisk
transitivity of a set of operators. The case of hypercyclicity (resp,
supercyclicity, diskcyclicity) were introduced in [1, 4, 5].

######  Definition 3.8.

We say that $\Gamma$ is strictly codisk transitive if for each pair of nonzero
elements $x,$ $y$ in $X$, there exist some $\alpha\in\mathbb{U}$ and
$T\in\Gamma$ such that $\alpha Tx=y.$

######  Remark 3.9.

An operator $T\in L(X)$ is strictly codisk transitive if and only if the set
$\Gamma=\\{T^{n}\mbox{ : }n\geq 0\\}$ is a strictly codisk transitive.

######  Proposition 3.10.

If $\Gamma$ is strictly codisk transitive set, then it is codisk transitive.
As a consequence, if $\Gamma$ is strictly codisk transitive set, then it is
codiskcyclic.

###### Proof.

Assume that $\Gamma$ is a strictly codisk transitive set. If $U$ and $V$ are
two nonempty open subsets of $X$, then there exist $x,$ $y\in X$ such that
$x\in U$ and $y\in V$. Since $\Gamma$ is strictly codisk transitive, it
follows that there exist $\alpha\in\mathbb{U}$ and $T\in\Gamma$ such that
$\alpha Tx=y.$ Hence, $\alpha Tx\in\alpha T(U)$ and $\alpha Tx\in V.$ Thus,
$\alpha T(U)\cap V\neq\emptyset,$ which implies that $\Gamma$ is codisk
transitive. By Theorem 3.6, we deduce that $\Gamma$ is codiskcyclic. ∎

In the following proposition, we prove that the strictly codisk transitivity
of sets of operators is preserved under similarity.

######  Proposition 3.11.

If $\Gamma\subset L(X)$ and $\Gamma_{1}L(Y)$ are similar, then $\Gamma$ is
strictly codisk transitive in $X$ if and only if $\Gamma_{1}$ is strictly
codisk transitive in $Y.$

###### Proof.

Let $x$, $y\in Y$. There exist $a,$ $b\in X$ such that $\phi(a)=x$ and
$\phi(b)=y$. Since $\Gamma$ is strictly codisk transitive in $X$, there exist
$\alpha\in\mathbb{U}$ and $T\in\Gamma$ such that $\alpha Ta=b,$ this implies
that $\alpha\phi\circ T(a)=\phi(b)$. Let $S\in\Gamma_{1}$ such that
$S\circ\phi=\phi\circ T$. Hence, $\alpha Sx=y$. Hence $\Gamma_{1}$ is strictly
codisk transitive in $Y$. ∎

Recall that the strong operator topology (SOT for short) on $L(X)$ is the
topology with respect to which any $T\in L(X)$ has a neighborhood basis
consisting of sets of the form

$\Omega=\\{S\in L(X)\mbox{ : }Se_{i}-Te_{i}\in U\mbox{, }i=1,2,\dots,k\\},$

where $k\in\mathbb{N}$, $e_{1},e_{2},\dots e_{k}\in X$ are linearly
independent and $U$ is a neighborhood of zero in $X$, see [8].

Let $x$ be an element of a complex topological vector space $X$. Note that
$\mathbb{U}_{x}$ is the subset of $X$ defined by
$\mathbb{U}\\{x\\}:=\mathbb{U}_{x}=\\{\alpha x\mbox{ :
}\alpha\in\mathbb{U}\\}.$

In the following theorem, the proof is also true for norm-density if $X$ is
assumed to be a normed linear space.

######  Theorem 3.12.

For each pair of nonzero vectors $x$, $y\in X$ with $y\notin\mathbb{U}_{x}$,
there exists a SOT-dense set $\Gamma_{xy}\subset L(X)$ which is not strictly
codisk transitive. Furthermore, $\Gamma\subset L(X)$ is a dense nonstrictly
codisk transitive set if and only if $\Gamma$ is a dense subset of
$\Gamma_{xy}$ for some $x$, $y\in X.$

###### Proof.

Fix nonzero vectors $x,$ $y\in X$ such that $y\notin\mathbb{U}_{x}$ and let
$\Gamma_{xy}$ the set defined by

$\Gamma_{xy}=\\{T\in L(X)\mbox{ : }y\notin\mathbb{U}_{Tx}\\}.$

Then $\Gamma_{xy}$ is not strictly codisk transitive. Let $\Omega$ be a
nonempty open set in $L(X)$ and $S\in\Omega$. If $Sx$ and $y$ are such that
$y\notin\mathbb{U}_{Sx}$, then $S\in\Omega\cap\Gamma_{xy}$. Otherwise, putting
$S_{n}=S+\frac{1}{n}I$, we see that $S_{k}\in\Omega$ for some $k$, but
$S_{k}x$ and $y$ are such that $y\notin\mathbb{U}_{S_{k}x}$. Hence,
$\Omega\cap\Gamma_{xy}\neq\emptyset$ and the proof is completed.

We prove the second assertion of the theorem. Suppose that $\Gamma$ is a dense
subset of $L(X)$ that is not strictly codisk transitive. Then there are
nonzero vectors $x,$ $y\in X$ such that $y\notin\mathbb{U}_{Tx}$ for all
$T\in\Gamma$ and hence $\Gamma\subset\Gamma_{xy}$. To show that $\Gamma$ is
dense in $\Gamma_{xy}$, assume that $\Omega_{0}$ is an open subset of
$\Gamma_{xy}$. Thus, $\Omega_{0}=\Gamma_{xy}\cap\Omega$ for some open set
$\Omega$ in $L(X)$. Then $\Gamma\cap\Omega_{0}=\Gamma\cap\Omega\neq\emptyset$.

For the converse, let $\Gamma$ be a dense subset of $\Gamma_{xy}$ for some
$x,$ $y\in X$. Then $\Gamma$ is not strictly codisk transitive. Also, since
$\Gamma_{xy}$ is a dense open subset of $L(X)$, we conclude that $\Gamma$ is
also dense in $L(X)$. Indeed, if $\Omega$ is any open set in $L(X)$ then
$\Omega\cap\Gamma_{xy}\neq\emptyset$ since $\Gamma_{xy}$ is dense in $L(X)$.
On the other hand, $\Omega\cap\Gamma_{xy}$ is open in $\Gamma_{xy}$ and so it
must intersect $\Gamma$ since $\Gamma$ is dense in $\Gamma_{xy}$. Thus,
$\Omega\cap\Gamma\neq\emptyset$ and so $\Gamma$ is dense in $L(X)$. ∎

######  Corollary 3.13.

There is a subset $\Gamma_{1}$ of $\Gamma$ such that
$\overline{\Gamma_{1}}=L(X)$ and $\Gamma_{1}$ is not strictly codisk
transitive.

###### Proof.

For nonzero $x,$ $y$ such that $y\notin\mathbb{U}_{x},$ put
$\Gamma_{1}=\Gamma\cap\Gamma_{xy}$. ∎

In the following definition, we introduce that notion of codiskcyclic
transitivity of set of operators. The case of hypercyclicity (resp,
supercyclicity, diskcyclicity) were introduced in [1, 4, 5].

######  Definition 3.14.

We say that $\Gamma$ is a codiskcyclic transitive set or codiskcyclic
transitive if $\mathbb{U}C(\Gamma)=X\setminus\\{0\\}.$

######  Remark 3.15.

An operator $T\in L(X)$ is codiskcyclic transitive if and only if the set
$\Gamma=\\{T^{n}\mbox{ : }n\geq 0\\}$ is codiskcyclic transitive.

It is clear that a codiskcyclic transitive set is codiskcyclic. Moreover, the
next proposition shows that codiskcyclic transitivity of sets of operators
implies codisk transitivity.

######  Proposition 3.16.

If $\Gamma$ is codiskcyclic transitive, then $\Gamma$ is codisk transitive.

###### Proof.

Let $U$ and $V$ be two nonempty open subsets of $X$. There exists $x\in
X\setminus\\{0\\}$ such that $x\in U$. Since $\Gamma$ is codiskcyclic
transitive, there exists $\alpha\in\mathbb{U}$ and $T\in\Gamma$ such that
$\alpha Tx\in V$. This implies that $\alpha T(U)\cap V\neq\emptyset.$ Hence,
$\Gamma$ is codisk transitive. ∎

In the following proposition, we prove that the codiskcyclic transitivity is
preserved under similarity.

######  Proposition 3.17.

Assume that $\Gamma$ and $\Gamma_{1}$ are similar, then $\Gamma$ is
codiskcyclic transitive on $X$ if and only if $\Gamma_{1}$ is codiskcyclic
transitive on $Y$.

###### Proof.

If $\Gamma$ is a codiskcyclic transitive on $X$, then by Proposition 2.6,
$\phi(\mathbb{U}C(\Gamma))\subset\mathbb{U}C(\Gamma_{1})$. Since $\phi$ is
homeomorphism, the result holds. ∎

Assume that $X$ is a topological vector space and $\Gamma$ a subset of $L(X)$.
The following result shows that the SOT-closure of $\Gamma$ is not large
enough to have more codiskcyclic vectors than $\Gamma$.

######  Proposition 3.18.

If $\overline{\Gamma}$ stands for the SOT-closure of $\Gamma$ then
$\mathbb{U}C(\Gamma)=\mathbb{U}C(\overline{\Gamma}).$

###### Proof.

We only need to prove that
$\mathbb{U}C(\overline{\Gamma})\subset\mathbb{U}C(\Gamma)$. Fix
$x\in\mathbb{U}C(\overline{\Gamma})$ and let $U$ be an arbitrary open subset
of $X$. Then there is some $\alpha\in\mathbb{U}$ and $T\in\overline{\Gamma}$
such that $\alpha Tx\in U$. The set $\Omega=\\{S\in L(X)\mbox{ : }\alpha Sx\in
U\\}$ is a SOT-neighborhood of $T$ and so it must intersect $\Gamma$.
Therefore, there is some $S\in\Gamma$ such that $\alpha Sx\in U$ and this
shows that $x\in\mathbb{U}C(\Gamma)$. ∎

######  Corollary 3.19.

Let X be a topological vector space and $\Gamma$ a subset of $L(X)$. Then
$\Gamma$ is codiskcyclic transitive if and only if $\overline{\Gamma}$ is
codiskcyclic transitive.

###### Proof.

Assume that $\Gamma$ is codiskcyclic transitive, then
$\mathbb{U}C(\overline{\Gamma})=X\setminus\\{0\\}$. Since by Proposition 3.18,
we have $\mathbb{U}C(\overline{\Gamma})=\mathbb{U}C(\Gamma)$, it follows that
$\mathbb{U}C(\Gamma)=X\setminus\\{0\\}$. Hence, $\Gamma$ is codiskcyclic
transitive. ∎

In the next definition, we introduce the notion of codiskcyclic criterion of a
set of operators which generalizes the definition of codiskcyclic criterion of
a single operator.

######  Definition 3.20.

We say that $\Gamma$ satisfies the criterion of codiskcyclicity if there exist
two dense subsets $X_{0}$ and $Y_{0}$ in $X$ and sequences $\\{\alpha_{k}\\}$
of $\mathbb{U}$, $\\{T_{k}\\}$ of $\Gamma$ and a sequence of maps $S_{k}$ :
$Y_{0}\longrightarrow X$ such that$:$

  * $(i)$

$\alpha_{k}T_{k}x\longrightarrow 0$ for all $x\in X_{0}$;

  * $(ii)$

$\alpha_{k}^{-1}S_{k}x\longrightarrow 0$ for all $y\in Y_{0}$;

  * $(iii)$

$T_{k}S_{k}y\longrightarrow y$ for all $y\in Y_{0}$.

######  Remark 3.21.

An operator $T\in L(X)$ satisfies the criterion of codiskcyclicity for
operators if and only if the set $\Gamma=\\{T^{n}\mbox{ : }n\geq 0\\}$
satisfies the criterion of codiskcyclicity for sets of operators, see [19].

######  Theorem 3.22.

Let $X$ be a second countable Baire complex topological vector space and
$\Gamma$ a subset of $L(X)$. If $\Gamma$ satisfies the criterion of
codiskcyclicity, then $\mathbb{U}C(\Gamma)$ is a dense subset of $X$. As
consequence; $\Gamma$ is codiskcyclic.

###### Proof.

Let $U$ and $V$ be two nonempty open subsets of $X$. Since $X_{0}$ and $Y_{0}$
are dense in $X$, there exist $x_{0}$ and $y_{0}$ in $X$ such that $x_{0}\in
X_{0}\cap U$ and $y_{0}\in Y_{0}\cap V.$ For all $k\geq 1,$ let
$z_{k}=x_{0}+\alpha_{k}^{-1}S_{k}y$. We have
$\alpha_{k}^{-1}S_{k}y\longrightarrow 0$, which implies that
$z_{k}\longrightarrow x_{0}$. Since $x_{0}\in U$ and $U$ is open, there exists
$N_{1}\in\mathbb{N}$ such that $z_{k}\in U$, for all $k\geq N_{1}$. On the
other hand, we have
$\alpha_{k}T_{k}z_{k}=\alpha_{k}T_{k}x_{0}+T_{k}(S_{k}y_{0})\longrightarrow
y_{0}$. Since $y_{0}\in V$ and $V$ is open, there exists $N_{2}\in\mathbb{N}$
such that $\alpha_{k}T_{k}z_{k}\in V$, for all $k\geq N_{2}$. Let
$N=$max$\\{N_{1},N_{2}\\}$, then $z_{k}\in U$ and $\alpha_{k}T_{k}z_{k}\in V$,
for all $k\geq N$, that is $\alpha_{k}T_{k}(U)\cap V\neq\emptyset,$ for all
$k\geq N$. Hence, $\Gamma$ is codisk transitive. By Theorem 3.6 we deduce that
$\mathbb{U}C(\Gamma)$ is a dense subset of $X$. We use again Theorem 3.6 to
conclude that $\Gamma$ is codiskcyclic and this complete the proof. ∎

##  4\. Codiskcyclic $C_{0}$-Semigroups of Operators

In this section we will study the particular case when $\Gamma$ is a
$C_{0}$-semigroup of operators.

Recall that a family $(T_{t})_{t\geq 0}$ of operators is called a
$C_{0}$-semigroup of operators if the following three conditions are
satisfied$:$

  * $(i)$

$T_{0}=I$ the identity operator on $X$;

  * $(ii)$

$T_{t+s}=T_{t}T_{s}$ for all $t,$ $s\geq 0$;

  * $(iii)$

$\lim_{t\rightarrow s}T_{t}x=T_{s}x$ for all $x\in X$ and $t\geq 0$.

For more informations about the theory of $C_{0}$-semigroups the reader may
refer to [16].

######  Example 4.1.

Let $X=\mathbb{C}$. For all $t\geq 0$, let $T_{t}x=\exp(t)x$, for all
$x\in\mathbb{C}$. Then $(T_{t})_{t\geq 0}$ is a $C_{0}$-semigroup and we have
$\mathbb{U}Orb((T_{t})_{t\geq 0},1)=\\{\alpha T_{t}(1)\mbox{ : }t\geq 0\mbox{,
}\alpha\in\mathbb{U}\\}=\\{\alpha y\mbox{ : }y\in\mathbb{R}^{+}\mbox{,
}\alpha\in\mathbb{U}\\}.$ Let $x\in\mathbb{C}\setminus\\{0\\}$. Then
$\displaystyle x=|x|\frac{x}{|x|}\in\mathbb{U}Orb((T_{t})_{t\geq 0},1)$.
Hence, $\overline{\mathbb{U}Orb((T_{t})_{t\geq 0},1)}=\mathbb{C}.$ Thus,
$(T_{t})_{t\geq 0}$ is a codiskcyclic $C_{0}$-semigroup of operators and $1$
is a codiskcyclic vector for $(T_{t})_{t\geq 0}$.

Recall from [18, Lemma 5.1], that if $X$ is a complex topological vector space
such that $2\leq$ dim$(X)<\infty$, then $X$ supports no supercyclic
$C_{0}$-semigroups of operators.

In the following theorem we will prove that the same result holds in the case
of codiskcyclicity on a complex topological vector space.

######  Theorem 4.2.

Assume that $2\leq$ dim$(X)<\infty$. Then $X$ supports no codiskcyclic
$C_{0}$-semigroups.

###### Proof.

By using [18, Lemma 5.1] and the fact that
$\mathbb{U}Orb(\Gamma,x)\subset\mathbb{C}Orb(\Gamma,x).$ ∎

A necessary and sufficient condition for a $C_{0}$-semigroup of operators to
be codiskcyclic is given in the next lemma and theorem.

######  Lemma 4.3.

Let $(T_{t})_{t\geq 0}$ be a codiskcyclic $C_{0}$-semigroup of operators on a
Banach infinite dimensional space $X$. If $x\in X$ is a codiskcyclic vector of
$(T_{t})_{t\geq 0}$, then the following assertions hold:

  * $(1)$

$T_{t}x\neq 0$, for all $t\geq 0$;

  * $(2)$

The set $\\{\alpha T_{t}x\mbox{ : }t\geq s\mbox{, }\alpha\in\mathbb{U}\\}$ is
dense in $X$, for all $s\geq 0$.

###### Proof.

$(1)$ Suppose that $t_{0}>0$ is minimal with the property that $T_{t_{0}}x=0$.
We show first that each $y\in X$ is of the form $y=\alpha T_{t}x$ for some
$t\in[0,t_{0}]$ and $\alpha\in\mathbb{U}$. Since $x\in\mathbb{U}C(\Gamma)$,
there exist a sequence $(t_{n})_{n\in\mathbb{N}}\subset[0,t_{0}]$ and a
sequence $(\alpha_{n})_{n\in\mathbb{N}}\subset\mathbb{U}$ such that
$\alpha_{n}T_{t_{n}}x\longrightarrow y$. Without loss of generality we may
assume that $(t_{n})_{n\in\mathbb{N}}$ converges to some $t$. By compactness
we may assume that $(\alpha_{n})_{n\in\mathbb{N}}$ converges to some $\alpha$
and we infer that $y=\alpha T_{t}x$.

Now take three vectors $y_{i}=\alpha_{i}T_{t_{i}}x\in X$, spanning a two-
dimensional subspace, such that each pair $y_{i}$, $y_{j}$, $i\neq j$, is
linearly independent. Assume that $t_{1}>t_{2}>t_{3}$. We have then
$y_{3}=c_{1}y_{1}+c_{2}y_{2}$. Now we arrive at the contradiction

$\displaystyle 0$
$\displaystyle\neq\alpha_{3}T_{(t_{0}+t_{3}-t_{2})}x=T_{(t_{0}-t_{2})}y_{3}=c_{1}T_{(t_{0}-t_{2})}y_{1}+c_{2}T_{(t_{0}-t_{2})}y_{2}$
$\displaystyle=c_{1}\alpha_{1}T_{(t_{0}+t_{1}-t_{2})}x+c_{2}\alpha_{2}T_{t_{0}}x=0.$

$(2)$ Suppose that there exists $s_{0}>0$ such that $\\{\alpha T_{t}x\mbox{ :
}t\geq s_{0}\mbox{, }\alpha\in\mathbb{U}\\}$ is not dense in X. Hence there
exists a bounded open set $U$ such that $U\cap\overline{A}=\emptyset$.
Therefore we have $U\subset\overline{\\{\alpha T_{t}x\mbox{ : }0\leq t\leq
s_{0}\mbox{, }\alpha\in\mathbb{U}\\}}$ by using the relation

$X=\overline{\\{\alpha T_{t}x\mbox{ : }t\geq 0\mbox{,
}\alpha\in\mathbb{U}\\}}=\overline{\\{\alpha T_{t}x\mbox{ : }t\geq
s_{0}\mbox{, }\alpha\in\mathbb{U}\\}}\cup\overline{\\{\alpha T_{t}x\mbox{ :
}0\leq t\leq s_{0}\mbox{, }\alpha\in\mathbb{U}\\}}.$

Thus, $\overline{U}$ is compact. Hence $X$ is finite dimensional, which
contradicts that $X$ is infinite dimensional. ∎

######  Theorem 4.4.

Let $(T_{t})_{t\geq 0}$ be a $C_{0}$-semigroup of operators on a separable
Banach infinite dimensional space X. Then the following assertions are
equivalent:

  * $(1)$

$(T_{t})_{t\geq 0}$ is codiskcyclic;

  * $(2)$

for all $y$, $z\in X$ and all $\varepsilon>0$, there exist $v\in X$, $t>0$ and
$\alpha\in\mathbb{U}$ such that $\|y-v\|<\varepsilon$ and $\|z-\alpha
T_{t}v\|<\varepsilon;$

  * $(3)$

for all $y$, $z\in X$, all $\varepsilon>0$ and for all $l\geq 0$, there exist
$v\in X$, $t>l$ and $\alpha\in\mathbb{U}$ such that $\|y-v\|<\varepsilon$ and
$\|z-\alpha T_{t}v\|<\varepsilon.$

###### Proof.

$(1)\Rightarrow(3)$: Let $x\in X$ such that $\\{\alpha T_{t}x\mbox{ : }t\geq
0\mbox{, }\alpha\in\mathbb{U}\\}$ is dense in $X$ and let $\varepsilon>0$. For
any $y\in X$, there exist $s_{1}>0$ and $\alpha_{1}\in\mathbb{U}$ such that
$\|y-\alpha_{1}T_{s_{1}}x\|<\varepsilon$. If $l\geq 0$, then by Lemma 4.3, the
set $\alpha_{1}\\{\alpha T_{t}x\mbox{ : }t\geq s+l\mbox{,
}\alpha\in\mathbb{U}\\}:=\\{\alpha_{1}\alpha T_{t}x\mbox{ : }t\geq s+l\mbox{,
}\alpha\in\mathbb{U}\\}$ is dense in $X$. For any $z\in X$, there exist
$s_{2}>l+s_{1}$ and $\alpha_{2}\in\mathbb{U}$ such that
$\|z-\alpha_{1}\alpha_{2}T_{s_{2}}x\|<\varepsilon$. Put
$v=\alpha_{1}T_{s_{1}}x$, $t=s_{2}-s_{1}>l$ and $\alpha=\alpha_{2}$. Then we
have $\|y-v\|<\varepsilon$ and $\|z-\alpha T_{t}v\|<\varepsilon.$
$(3)\Rightarrow(2)$: It is obvious.
$(2)\Rightarrow(1)$: Let $\\{z_{1},z_{2},z_{3},...\\}$ be a dense sequence in
$X$. we construct sequences $\\{y_{1},y_{2},y_{3},...\\}\subset X$,
$\\{t_{1},t_{2},t_{3},...\\}\subset[0,+\infty)$ and
$\\{\alpha_{1},\alpha_{2},\alpha_{3},...\\}\subset\mathbb{U}$ inductively:

  * •

Put $y_{1}=z_{1}$, $t_{1}=0$;

  * •

For $n>1$, find $y_{n}$, $t_{n}$ and $\alpha_{n}$ such that

$\|y_{n}-y_{n-1}\|\leq\frac{2^{-n}}{\sup\\{\|T_{t_{j}}\|\mbox{ : }j<n\\}},$
(4.1)

and

$\|z_{n}-\alpha_{n}T_{t_{n}}y_{n}\|\leq 2^{-n}.$ (4.2)

In particular, (4.1) implies that $\|y_{n}-y_{n-1}\|\leq 2^{-n}$, so that the
sequence $(y_{n})_{n\geq 1}$ has a limit $x$. Applying (4.2) and once again
(4.1) we infer that

$\displaystyle\|z_{n}-\alpha_{n}T_{t_{n}}x\|$
$\displaystyle=\|z_{n}-\alpha_{n}T_{t_{n}}y_{n}+\alpha_{n}T_{t_{n}}y_{n}-\alpha_{n}T_{t_{n}}x\|$
$\displaystyle\leq\|z_{n}-\alpha_{n}T_{t_{n}}y_{n}\|+\|\alpha_{n}T_{t_{n}}(y_{n}-x)\|$
$\displaystyle\leq\|z_{n}-\alpha_{n}T_{t_{n}}y_{n}\|+\|\alpha_{n}T_{t_{n}}\|\sum_{i=n+1}^{+\infty}\|y_{i}-y_{i-1}\|$
$\displaystyle\leq 2^{-n}+\sum_{i=n+1}^{+\infty}2^{-i}=2^{-n+1}.$

Given $z\in X$ and $\varepsilon>0$ there are arbitrarily large $n$ such that
$\|z_{n}-z\|<\frac{\varepsilon}{2}$. Choosing $n$ large enough such that
$2^{-n+1}<\frac{\varepsilon}{2}$, we obtain
$\|\alpha_{n}T_{t_{n}}x-z\|\leq\|z-z_{n}\|+\|z_{n}-\alpha_{n}T_{t_{n}}x\|<\varepsilon.$
Therefore, $\\{\alpha T_{t}x\mbox{ : }t\geq 0\mbox{, }\alpha\in\mathbb{U}\\}$
is dense in $X$. ∎

As a corollary we obtain a sufficient condition of codiskcyclicity of a
$C_{0}$-semigroup of operators.

Let $X$ be a separable Banach infinite dimensional space. Denote $X_{0}$ the
set of all $x\in X$ such that $\lim_{t\longrightarrow\infty}T_{t}x=0,$ and
$X_{\infty}$ the set of all $x\in X$ such that for each $\varepsilon>0$ there
exist some $w\in X$ , $\alpha\in\mathbb{U}$ and some $t>0$ with
$\|w\|<\varepsilon$ and $\|\alpha T_{t}w-x\|<\varepsilon.$

######  Theorem 4.5.

Let $(T_{t})_{t\geq 0}$ be a $C_{0}$-semigroup of operators on a separable
Banach infinite dimensional space $X$. If both $X_{\infty}$ and $X_{0}$ are
dense subsets, then $(T_{t})_{t\geq 0}$ is codiskcyclic.

###### Proof.

Let $z\in X_{\infty}$ and $y\in X_{0}$. Then for each $\varepsilon>0$ there
are arbitrarily large $t>0$, $\alpha\in\mathbb{U}$ and $w\in X$ such that
$\|w\|<\varepsilon$ and $\|\alpha T_{t}w-x\|<\frac{\varepsilon}{2}.$ Since
$y\in X_{0}$, for sufficiently large $t$ we have $\|\alpha
T_{t}y\|<\frac{\varepsilon}{2}$. We put $v=y+w$ and infer
$\|z-T_{t}v\|\leq\|z-T_{t}w\|+\|\alpha T_{t}y\|<\varepsilon,$ and
$\|y-v\|=\|w\|<\varepsilon.$ By Theorem 4.4, the result holds. ∎

We use Theorem 3.7 to prove that the codiskcyclicity and codisk transitivity
of a $C_{0}$-semigroup of operators on a complex topological vector space are
equivalent.

######  Theorem 4.6.

Let $(T_{t})_{t\geq 0}$ be a $C_{0}$-semigroup of operators on a complex
topological vector space $X$. Then, the following assertions are equivalent$:$

  * $(i)$

$(T_{t})_{t\geq 0}$ is codiskcyclic;

  * $(ii)$

$(T_{t})_{t\geq 0}$ is codisk transitive.

###### Proof.

By remarking that if $t_{1}>t_{2}\geq 0$, then there exists $t=t_{1}-t_{2}$
such that $T_{t_{1}}=T_{t}T_{t_{2}}$, and using Theorem 3.7. ∎

## References

  * [1] M. Amouch, O. Benchiheb, Diskcyclicity of sets of operators and applications. Acta Mathematica Sinica, English Series., (2020). 36, 1203-1220.
  * [2] M. Amouch, O. Benchiheb, On cyclic sets of operators. Rend. Circ. Mat. Palermo, 2. Ser., (2019), 68, 521-529.
  * [3] M. Amouch, O. Benchiheb, On linear dynamics of sets of operators. Turk. J. Math., (2019), 43(1), 402-411.
  * [4] M. Amouch, O. Benchiheb, Some versions of supercyclicity of a s set of operators. Accepted for publication in Filomat Journal.
  * [5] M. Ansari, K. Hedayatian, B. Khani-robati, On the density and transitivity of sets of operators, Turk. J. Math., 42 (2018), 181-189.
  * [6] M. Ansari, K. Hedayatian, B. Khani-robati, A. Moradi, A note on topological and strict transitivity. Iran. J. Sci. Technol. Trans. Sci. 42(1), 59-64 (2018)
  * [7] F. Bayart, E. Matheron, Dynamics of linear operators. New York, NY, USA: Cambridge University Press, (2009).
  * [8] J.B. Conway, A course in Functional Analysis. Springer Graduate Texts in Math Series, 1985.
  * [9] W. Desch, W. Schappacher, G. F. Webb, Hypercyclic and chaotic semigroups of linear operators, Ergod. Th. Dynam. Sys.,1997, 17, 793-819.
  * [10] K.-G. Grosse-Erdmann, A. Peris Manguillot, Linear chaos, Universitext, Springer, London, 2011.
  * [11] K. G. Grosse-Erdmann, (1999). Universal families and hypercyclic operators, Bulletin of the American Mathematical Society, 36(3), 345-381.
  * [12] H. M. Hilden, L. J. Wallen, Some cyclic and non-cyclic vectors of certain operators, Indiana Univ. Math. J. 23 (1973/74), 557-565.
  * [13] Y. X. Liang, Z. H. Zhou Disk-cyclic and Codisk-cyclic tuples of the adjoint weighted composition operators on Hilbert spaces, Bull. Belg. Math. Soc. Simon Stevin., 23(2), 203-215 (2016).
  * [14] Y. X. Liang, Z. H. Zhou, Disk-cyclic and Codisk-cyclic of certain shift operators, operators and matrices., 9(4), 831-846 (2015).
  * [15] M. Matsui, M. Yamada. and F. Takeo, Erratum to Supercyclic and chaotic translation semigroups, Proc. Amer. Math. Soc., 2004, 132, 3751-3752.
  * [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
  * [17] Y. Wang, H. G. Zeng, Disk-cyclic and codisk-cyclic weighted pseudo-shifts, Bull. Belg. Math. Soc. Simon Stevin., 25(2), 209-224 (2018).
  * [18] J. Wengenroth, Hypercyclic operators on non-locally convex spaces, Proc. Amer. Math. Soc. 131 (2003) 1759-1761.
  * [19] Z. J. Zeana, Cyclic phenomena of operators on Hilbert space, Thesis, University of Baghdad (2002).