Recurrence and mixing recurrence of multiplication operators

Mohamed Amouch; Hamza Lakrimi

Mathematica Bohemica (2024)

  • Issue: 1, page 1-11
  • ISSN: 0862-7959

Abstract

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Let X be a Banach space, ℬ ( X ) the algebra of bounded linear operators on X and ( J , ∥ · ∥ J ) an admissible Banach ideal of ℬ ( X ) . For T ∈ ℬ ( X ) , let L J , T and R J , T ∈ ℬ ( J ) denote the left and right multiplication defined by L J , T ( A ) = T A and R J , T ( A ) = A T , respectively. In this paper, we study the transmission of some concepts related to recurrent operators between T ∈ 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 T ⊕ T is recurrent on X ⊕ 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.

How to cite

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Amouch, Mohamed, and Lakrimi, Hamza. "Recurrence and mixing recurrence of multiplication operators." Mathematica Bohemica (2024): 1-11. <http://eudml.org/doc/299245>.

@article{Amouch2024,
abstract = {Let $X$ be a Banach space, $\mathcal \{B\}(X)$ the algebra of bounded linear operators on $X$ and $(J, \Vert \{\cdot \}\Vert _\{J\})$ 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 $T\oplus T$ is recurrent on $X\oplus X$. 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.},
author = {Amouch, Mohamed, Lakrimi, Hamza},
journal = {Mathematica Bohemica},
keywords = {hypercyclicity; recurrent operator; left multiplication operator; right multiplication operator; tensor product; Banach ideal of operators},
language = {eng},
number = {1},
pages = {1-11},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Recurrence and mixing recurrence of multiplication operators},
url = {http://eudml.org/doc/299245},
year = {2024},
}

TY - JOUR
AU - Amouch, Mohamed
AU - Lakrimi, Hamza
TI - Recurrence and mixing recurrence of multiplication operators
JO - Mathematica Bohemica
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 1
EP - 11
AB - Let $X$ be a Banach space, $\mathcal {B}(X)$ the algebra of bounded linear operators on $X$ and $(J, \Vert {\cdot }\Vert _{J})$ 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 $T\oplus T$ is recurrent on $X\oplus X$. 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.
LA - eng
KW - hypercyclicity; recurrent operator; left multiplication operator; right multiplication operator; tensor product; Banach ideal of operators
UR - http://eudml.org/doc/299245
ER -

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