$J$-class weighted translations on locally compact groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918046347558912 |
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| author | Azimi, M. R. Akbarbaglu, I. Fard, A. R. Imanzadeh |
| author_facet | Azimi, M. R. Akbarbaglu, I. Fard, A. R. Imanzadeh |
| contents | A bounded linear operator $T$ on a Banach space $X$ (not necessarily separable) is said to be $J$-class operator whenever the extended limit set, say $J_T(x)$ equals $X$ for some vector $x\in X$. Practically, the extended limit sets localize the dynamical behavior of operators. In this paper, using the extended limit sets we will examine the necessary and sufficient conditions for the weighted translation $T_{a,ω}$ to be $J$-class on a locally compact group $G$, within the setting of $ L^p$-spaces for $ 1 \leq p < \infty $. Precisely, we delineate the boundary between $J$-class and hypercyclic behavior for weighted translations. Then, we will show that for torsion elements in locally compact groups, unlike the case of non-dense orbits of weighted translations, we have $J_{T_{a,ω}}(0)=L^p(G)$. Finally, we will provide some examples on which the weighted translation $ T_{a,ω}$ is $J$-class but it fails to be hypercyclic. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_04730 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $J$-class weighted translations on locally compact groups Azimi, M. R. Akbarbaglu, I. Fard, A. R. Imanzadeh Functional Analysis Dynamical Systems A bounded linear operator $T$ on a Banach space $X$ (not necessarily separable) is said to be $J$-class operator whenever the extended limit set, say $J_T(x)$ equals $X$ for some vector $x\in X$. Practically, the extended limit sets localize the dynamical behavior of operators. In this paper, using the extended limit sets we will examine the necessary and sufficient conditions for the weighted translation $T_{a,ω}$ to be $J$-class on a locally compact group $G$, within the setting of $ L^p$-spaces for $ 1 \leq p < \infty $. Precisely, we delineate the boundary between $J$-class and hypercyclic behavior for weighted translations. Then, we will show that for torsion elements in locally compact groups, unlike the case of non-dense orbits of weighted translations, we have $J_{T_{a,ω}}(0)=L^p(G)$. Finally, we will provide some examples on which the weighted translation $ T_{a,ω}$ is $J$-class but it fails to be hypercyclic. |
| title | $J$-class weighted translations on locally compact groups |
| topic | Functional Analysis Dynamical Systems |
| url | https://arxiv.org/abs/2506.04730 |