Linear dynamics of the adjoint of a unilateral weighted shift operator
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| Format: | Preprint |
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2024
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| author | Das, Bibhash Kumar Mundayadan, Aneesh |
| author_facet | Das, Bibhash Kumar Mundayadan, Aneesh |
| contents | This paper is a sequel to our work in \cite{Das-Mundayadan}. Here, we primarily study the dynamics of the adjoint of a weighted forward shift operator $F_w$ on the analytic function space $\ell^p_{a,b}$ having a normalized Schauder basis of the form $\{(a_n+b_nz)z^n:~n \geq 0\}$. We obtain sufficient conditions for $F_w$ to be continuous, and show, under certain conditions, that the operator $F_w$ is similar to a compact perturbation of a weighted forward shift on $\ell^p(\mathbb{N}_0)$. This also allows us to obtain the essential spectrum of $F_w$. Further, we study when the adjoint $F_w^*$ is hypercyclic, mixing, and chaotic, and provide a class of chaotic operators that are compact perturbations of weighted shifts on $\ell^p(\mathbb{N}_0)$. Finally, it is proved that the adjoint of a shift on the dual of $\ell^p_{a,b}$ can have non-trivial periodic vectors, without being even hypercyclic. Also, the zero-one law of orbital limit points fails for $F_w^*$, which means that, under certain conditions, the adjoint $F_w^*$ is non-hypercyclic, but it has an orbit possessing non-zero norm limit points. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_05509 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear dynamics of the adjoint of a unilateral weighted shift operator Das, Bibhash Kumar Mundayadan, Aneesh Functional Analysis Dynamical Systems 47A16 This paper is a sequel to our work in \cite{Das-Mundayadan}. Here, we primarily study the dynamics of the adjoint of a weighted forward shift operator $F_w$ on the analytic function space $\ell^p_{a,b}$ having a normalized Schauder basis of the form $\{(a_n+b_nz)z^n:~n \geq 0\}$. We obtain sufficient conditions for $F_w$ to be continuous, and show, under certain conditions, that the operator $F_w$ is similar to a compact perturbation of a weighted forward shift on $\ell^p(\mathbb{N}_0)$. This also allows us to obtain the essential spectrum of $F_w$. Further, we study when the adjoint $F_w^*$ is hypercyclic, mixing, and chaotic, and provide a class of chaotic operators that are compact perturbations of weighted shifts on $\ell^p(\mathbb{N}_0)$. Finally, it is proved that the adjoint of a shift on the dual of $\ell^p_{a,b}$ can have non-trivial periodic vectors, without being even hypercyclic. Also, the zero-one law of orbital limit points fails for $F_w^*$, which means that, under certain conditions, the adjoint $F_w^*$ is non-hypercyclic, but it has an orbit possessing non-zero norm limit points. |
| title | Linear dynamics of the adjoint of a unilateral weighted shift operator |
| topic | Functional Analysis Dynamical Systems 47A16 |
| url | https://arxiv.org/abs/2412.05509 |