Recurrence and mixing recurrence of multiplication 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...
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| Format: | Article |
| Language: | English |
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Institute of Mathematics of the Czech Academy of Science
2024-04-01
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| Series: | Mathematica Bohemica |
| Subjects: | |
| Online Access: | https://mb.math.cas.cz/full/149/1/mb149_1_1.pdf |
| _version_ | 1797266225133256704 |
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| author | Mohamed Amouch Hamza Lakrimi |
| author_facet | Mohamed Amouch Hamza Lakrimi |
| author_sort | Mohamed Amouch |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-25T00:57:18Z |
| format | Article |
| id | doaj.art-8ca12431b0f7483bb651f5cc246d10d9 |
| institution | Directory Open Access Journal |
| issn | 0862-7959 2464-7136 |
| language | English |
| last_indexed | 2024-04-25T00:57:18Z |
| publishDate | 2024-04-01 |
| publisher | Institute of Mathematics of the Czech Academy of Science |
| record_format | Article |
| series | Mathematica Bohemica |
| spelling | doaj.art-8ca12431b0f7483bb651f5cc246d10d92024-03-11T09:20:43ZengInstitute of Mathematics of the Czech Academy of ScienceMathematica Bohemica0862-79592464-71362024-04-01149111110.21136/MB.2023.0047-22MB.2023.0047-22Recurrence and mixing recurrence of multiplication operatorsMohamed AmouchHamza LakrimiLet $X$ 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.https://mb.math.cas.cz/full/149/1/mb149_1_1.pdf hypercyclicity recurrent operator left multiplication operator right multiplication operator tensor product banach ideal of operators |
| spellingShingle | Mohamed Amouch Hamza Lakrimi Recurrence and mixing recurrence of multiplication operators Mathematica Bohemica hypercyclicity recurrent operator left multiplication operator right multiplication operator tensor product banach ideal of operators |
| title | Recurrence and mixing recurrence of multiplication operators |
| title_full | Recurrence and mixing recurrence of multiplication operators |
| title_fullStr | Recurrence and mixing recurrence of multiplication operators |
| title_full_unstemmed | Recurrence and mixing recurrence of multiplication operators |
| title_short | Recurrence and mixing recurrence of multiplication operators |
| title_sort | recurrence and mixing recurrence of multiplication operators |
| topic | hypercyclicity recurrent operator left multiplication operator right multiplication operator tensor product banach ideal of operators |
| url | https://mb.math.cas.cz/full/149/1/mb149_1_1.pdf |
| work_keys_str_mv | AT mohamedamouch recurrenceandmixingrecurrenceofmultiplicationoperators AT hamzalakrimi recurrenceandmixingrecurrenceofmultiplicationoperators |