A Katznelson-Tzafriri theorem for analytic Besov functions of operators
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866929490854150144 |
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| author | Batty, Charles Seifert, David |
| author_facet | Batty, Charles Seifert, David |
| contents | Let $T$ be a power-bounded operator on a Banach space $X$, $\mathcal{A}$ be a Banach algebra of bounded holomorphic functions on the unit disc $\mathbb{D}$, and assume that there is a bounded functional calculus for the operator $T$, so there is a bounded algebra homomorphism mapping functions $f \in \mathcal{A}$ to bounded operators $f(T)$ on $X$. Theorems of Katznelson-Tzafriri type establish that $\lim_{n\to\infty} \|T^n f(T)\| = 0$ for functions $f \in \mathcal{A}$ whose boundary functions vanish on the unitary spectrum $σ(T)\cap \mathbb{T}$ of $T$, or sometimes satisfy a stronger assumption of spectral synthesis. We consider the case when $\mathcal{A}$ is the Banach algebra $\mathcal{B}(\mathbb{D})$ of analytic Besov functions on $\mathbb{D}$. We prove a Katznelson-Tzafriri theorem for the $\mathcal{B}(\mathbb{D})$-calculus which extends several previous results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_12076 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A Katznelson-Tzafriri theorem for analytic Besov functions of operators Batty, Charles Seifert, David Functional Analysis Let $T$ be a power-bounded operator on a Banach space $X$, $\mathcal{A}$ be a Banach algebra of bounded holomorphic functions on the unit disc $\mathbb{D}$, and assume that there is a bounded functional calculus for the operator $T$, so there is a bounded algebra homomorphism mapping functions $f \in \mathcal{A}$ to bounded operators $f(T)$ on $X$. Theorems of Katznelson-Tzafriri type establish that $\lim_{n\to\infty} \|T^n f(T)\| = 0$ for functions $f \in \mathcal{A}$ whose boundary functions vanish on the unitary spectrum $σ(T)\cap \mathbb{T}$ of $T$, or sometimes satisfy a stronger assumption of spectral synthesis. We consider the case when $\mathcal{A}$ is the Banach algebra $\mathcal{B}(\mathbb{D})$ of analytic Besov functions on $\mathbb{D}$. We prove a Katznelson-Tzafriri theorem for the $\mathcal{B}(\mathbb{D})$-calculus which extends several previous results. |
| title | A Katznelson-Tzafriri theorem for analytic Besov functions of operators |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2201.12076 |