In this paper, we introduce and study the diskcyclicity and disk transitivity of a set of operators. We establish a diskcyclicity criterion and give the relationship between this criterion and the ...diskcyclicity. As applications, we study the diskcyclicty of
C
0
-semigroups and
C
-regularized groups. We show that a diskcyclic
C
0
-semigroup exists on a complex topological vector space
X
if and only if dim(
X
) = 1 or dim(
X
) = ∞ and we prove that diskcyclicity and disk transitivity of
C
0
-semigroups (resp
C
-regularized groups) are equivalent.
An operator
in a Banach space
is said to be recurrent if the set
is dense in
. The operator
is said to be weakly sequentially recurrent if the set
is weakly dense in
. Costakis et al. Complex Anal. ...Oper. Theory
8
(8), 1601–1643 ask if
should be recurrent whenever so is
. This question has been answered negatively by Grivaux et al. arXiv: 2212.03652. In this paper, we prove the existence of an operator
weakly sequentially recurrent such that
is not.
On recurrent sets of operators Amouch, Mohamed; Benchiheb, Otmane
Boletim da Sociedade Paranaense de Matemática,
05/2024, Letnik:
42
Journal Article
Recenzirano
Odprti dostop
An operator $T$ acting on a Banach space $X$ is said to be recurrent if for each $U$; a nonempty open subset of $X$, there exists $n\in\mathbb{N}$ such that $T^n(U)\cap U\neq\emptyset.$ In the ...present work, we generalize this notion from a single operator to a set $\Gamma$ of operators. As application, we study the recurrence of $C$-regularized group of operators.
Let $X$ be a Banach space, $\mathcal{B}(X)$ the algebra of bounded linear operators on $X$ and $(J, \|{\cdot}\|_J)$ an admissible Banach ideal of $\mathcal{B}(X)$. For $T\in\mathcal{B}(X)$, let ...$L_{J, T}$ and $R_{J, T}\in\mathcal{B}(J)$ denote the left and right multiplication defined by $L_{J, T}(A)=TA$ and $R_{J, T}(A)=AT$, respectively. In this paper, we study the transmission of some concepts related to recurrent operators between $T\in\mathcal{B}(X)$, and their elementary operators $L_{J, T}$ and $R_{J, T}$. In particular, we give necessary and sufficient conditions for $L_{J, T}$ and $R_{J, T}$ to be sequentially recurrent. Furthermore, we prove that $L_{J, T}$ is recurrent if and only if $Tøplus T$ is recurrent on $Xøplus X$. Moreover, we introduce the notion of a mixing recurrent operator and we show that $L_{J, T}$ is mixing recurrent if and only if $T$ is mixing recurrent.
Let X be a complex Banach space with dim X > 1 such that its topological dual X∗ is separable and B(X) the algebra of all bounded linear operators on X. In this paper, we study the passage of ...property of being supercyclic from T ∈ B(X) to the left and the right multiplication induced by T on an admissible Banach ideal of B(X). Also, we give a sufficient conditions for the tensor product T ⊗bR of two operators on B(X) to be supercyclic.
In this paper, we introduce and study the notion of weak recurrence on a Banach space
X
. In one hand, we prove some properties of this class of operators. On the other hand, we study the weak ...recurrence of power bounded and diagonal operators. This leads us to deduce the relationship between this notion and other notion in linear dynamics.
A Furstenberg family
F
is a collection of infinite subsets of the set of positive integers such that if
A
⊂
B
and
A
∈
F
, then
B
∈
F
. For a Furstenberg family
F
, finitely many operators
T
1
,
.
.
.
...,
T
N
acting on a common topological vector space
X
are said to be disjoint
F
-transitive if for every non-empty open subsets
U
0
,
.
.
.
,
U
N
of
X
the set
{
n
∈
N
:
U
0
∩
T
1
-
n
(
U
1
)
∩
.
.
.
∩
T
N
-
n
(
U
N
)
≠
∅
}
belongs to
F
. In this paper, depending on the topological properties of
Ω
, we characterize the disjoint
F
-transitivity of
N
≥
2
composition operators
C
ϕ
1
,
…
,
C
ϕ
N
acting on the space
H
(
Ω
)
of holomorphic maps on a domain
Ω
⊂
C
by establishing a necessary and sufficient condition in terms of their symbols
ϕ
1
,
.
.
.
,
ϕ
N
.
Orbits of random dynamical systems Zaou, Ahmed; Benchiheb, Otmane; Amouch, Mohamed
Boletim da Sociedade Paranaense de Matemática,
01/2024, Letnik:
42
Journal Article
Recenzirano
Odprti dostop
In this paper we introduce and study the notions of hypercyclicity and transitivity for random dynamical systemsand we establish the relation between them in a topological space. We also introduce ...the notions of mixing andweakly mixing for random dynamical systems and give some of their properties.
An operator
T
acting on a Banach space
X
is said to be super-recurrent if for each open subset
U
of
X
, there exist
λ
∈
K
and
n
∈
N
such that
λ
T
n
(
U
)
∩
U
≠
∅
. In this paper, we introduce and ...study the notions of super-rigidity and uniform super-rigidity which are related to the notion of super-recurrence, we also investigate the basic properties of these two notions. In addition, we discuss the spectrum of these two operators. At the end, we study the super-recurrence, the super-rigidity and the uniform super-rigidity behaviors on finite-dimensional spaces.
On compositional dynamics on hardy space Benchiheb, Otmane; Karim, Noureddine; Amouch, Mohamed
Boletim da Sociedade Paranaense de Matemática,
01/2023, Letnik:
41
Journal Article
Recenzirano
Odprti dostop
In this work, we examine super-recurrence and super-rigidity of composition operators acting on $H(\mathbb{D})$ the space of holomorphic functions on the unit disk $\mathbb{D}$ and on ...$H^2(\mathbb{D})$ the Hardy-Hilbert space. We characterize the symbols that generate super-recurrent and super-rigid composition operators acting on $H(\mathbb{D})$ and $H^2(\mathbb{D})$.
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