Zero divisors and topological divisors of zero in certain Banach algebras
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913229505036288 |
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| author | Patel, Anurag Kumar Chandra, Harish |
| author_facet | Patel, Anurag Kumar Chandra, Harish |
| contents | In this paper we prove that an element $f\in \mathcal{A}(\mathbb{D})$ is a topological divisor of zero(TDZ) if and only if there exists $z_0 \in \mathbb{T}$ such that $f(z_0)=0.$ We also give a characterization of TDZ in the Banach algebra $L^\infty(μ).$ Further, we prove that the multiplication operator $M_h$ is a TDZ in $\mathcal{B}(L^p(μ))~(1\leq p\leq\infty)$ if and only if $h$ is a TDZ in $L^\infty(μ).$ Subsequently, we show that a composition operator $C_ϕ$ is a TDZ in $\mathcal{B}(L^2(μ))$ if and only if $\frac{dμϕ^{-1}}{dμ}$ is a TDZ in $L^{\infty}(μ).$ Lastly, we determine composition operators on the Hardy spaces $\mathbb{H}^p(\mathbb{D})$ and $\ell^p$ spaces which are zero-divisors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06303 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zero divisors and topological divisors of zero in certain Banach algebras Patel, Anurag Kumar Chandra, Harish Functional Analysis 43A99, 47B33, 30H10, 28A20 In this paper we prove that an element $f\in \mathcal{A}(\mathbb{D})$ is a topological divisor of zero(TDZ) if and only if there exists $z_0 \in \mathbb{T}$ such that $f(z_0)=0.$ We also give a characterization of TDZ in the Banach algebra $L^\infty(μ).$ Further, we prove that the multiplication operator $M_h$ is a TDZ in $\mathcal{B}(L^p(μ))~(1\leq p\leq\infty)$ if and only if $h$ is a TDZ in $L^\infty(μ).$ Subsequently, we show that a composition operator $C_ϕ$ is a TDZ in $\mathcal{B}(L^2(μ))$ if and only if $\frac{dμϕ^{-1}}{dμ}$ is a TDZ in $L^{\infty}(μ).$ Lastly, we determine composition operators on the Hardy spaces $\mathbb{H}^p(\mathbb{D})$ and $\ell^p$ spaces which are zero-divisors. |
| title | Zero divisors and topological divisors of zero in certain Banach algebras |
| topic | Functional Analysis 43A99, 47B33, 30H10, 28A20 |
| url | https://arxiv.org/abs/2402.06303 |